Statistical Analysis of Wind Turbine's Output Power

Self-consumption enhancement on a low-voltage grid-connected photovoltaic system

Ciprian Nemes, Mihaela Adochitei, Florin Munteanu, Alexandra Ciobanu, Octav Neagu Faculty of Electrical Engineering, “Gheorghe Asachi” Technical University of Iasi, Romania Bd. Profesor Dimitrie Mangeron no. 21-23, 700050, Iasi, Romania [email protected]

Abstract— The electricity production from photovoltaic reducing the outage time of prosumers. Actually, the reliability systems is seen as a real alternative to the electric power supply. of power supply depends on the characteristics of PV system. The current global trend is to financially support the self- The PV systems with energy storage could supply the consumption through a remuneration of the PV energy prosumer without the contribution of low voltage distribution production that is locally consumed by the owner of PV system. network. This paper conducts an analysis of the self-consumption indices related with the energy production of a low-voltage grid- Having in view these benefits, in the last time the self- connected PV system, installed on Faculty of Electrical consumption is highly encouraged. The current global trend is Engineering, Technical University of Iasi, Romania. to use the electricity from renewable sources in order to improve the energy performance of buildings [6]. EU Keywords—photovoltaic system; self-consumption; self- legislation on renewable energy (in particular from sufficency; prosumer. photovoltaic sources) raises the issue of financial support for owners of residential photovoltaic systems in order to I. INTRODUCTION encourage the self-consumption from their own sources. This financial support is based on a remuneration of self-consumed In the last decades, the electricity production from electricity [3-7]. In order to encourage the PV owners to photovoltaic systems has permanently increased being increase their self-consumption, the remuneration is encouraged by EU energy policies and supported through substantially increased if the rate of self-consumption is higher various incentives mechanisms. In European Union, in 2016, than 30%. Consequently, the interest in self-consumption of the photovoltaic capacity has increased with 6122,8 MW [1], p residential grid-connected PV systems is increasing among PV achieving at the end of 2016 a total cumulated PV capacity system owners and also in the scientific community. around to 100,935 GWp, which includes around 100,65 GWp installed on grid-connected PV systems and 0,285 GWp Usually, in the residential PV systems without dedicated installed on stand-alone PV systems. consumption control system, the self-consumption can reach a level of 30%, depending on behavior of load profile. A According to Solar Power Europe [2], the photovoltaic literature survey indicates that among different technologies market is currently in transition, changing from a market whose used to increase the PV self-consumption, only two of them are growth was based on the incentives, to a new market structure, suitable [8-14]. The first one includes technologies based on where photovoltaic owners are encouraged to use solar energy storage, mainly using batteries, and the second one is electricity for self-consumption in residential, commercial or based on the concept of demand side management, namely an industrial sectors. The self-consumption is defined as part of active load shifting. The conclusions of these works show that PV energy production that is locally consumed by the owner of it is possible to increase the relative self-consumption by 13– PV system, as a consequence of overlap between PV power 24% with a battery storage capacity of 0.5–1 kWh per installed generation and load demand. Thus, a new concept is present in kW PV power and between 2% and 15% with a demand side the literature, that of prosumer. A prosumer is a producing management, both compared to the original rate of self- consumer of PV energy that will use the solar electricity for consumption [10]. self-consumption. Considering these issues, a quantitative and qualitative The main benefits of local energy production could be analysis of the self-consumption and self-sufficiency values classified as technical benefits, referring to reducing of related with the energy production of a low-voltage grid- requirement stress in distribution network, as well as the connected photovoltaic system is conducted in the paper. The economic benefits which refers to that the local generate paper has the following structure. In order to have an overview energy is usually much cheaper than that bought from the of main self-consumption indices, a brief literature survey is suppliers. Usually, the cost of electricity in residential sector presented in Section 2. After the grid-connected PV system has a higher value than the cost of electricity generated on PV involved in this study is presented in Section 3, in the next systems, thus the advantages of self-consumption are more section a numerical analysis based on the measurements of the substantial. Thus, the self-consumption is an alternative to PV output power and load demand is conducted in order to incentives that could bring an economic benefit to PV systems calculate self-consumption and self-sufficiency indices. The owners [3-5]. Moreover, the PV systems used for self- main conclusions are given at the end of paper. consumption will improve the reliability in power supply by II. THE SELF-CONSUMPTION CONCEPTS More accurately assessment of these indices requires to The self-consumption concept is related to the generated indicate the instantaneous load demand L(t) and also the PV energy directly used by the PV system owner. The aim of instantaneous PV power generation P(t). Based on this section is to give an overview of main indices of self- instantaneous values of load and generated power, the self- consumption for a prosumer with a grid-connected PV system. consumption and self-sufficiency indices can be defined as follows: Concerning the metrics of self-consumption concept, a t 2 t2 literature survey indicates that the most works refers to self- minP(t), L(t) dt minP(t), L(t) dt t  consumption and self-sufficiency indices [8-14]. The self- 1 , t1 (3) SCt ,t  SSt ,t  1 2 t 2 1 2 t 2 consumption index is defined as the percentage of the total PV P(t)dt L(t)dt  t energy production directly consumed by the owner of PV t1 1 system, whereas the self-sufficiency index is defined as the self-produced energy relative to the total load demand. The prosumers intend to install large PV systems in order to maximize the percent of consumption that is locally generated Based on the timely overlapping between the generated PV by the owner PV system, which means an optimization of both power and load demand of residential prosumer, a part of the indices, the self-consumption and also the self-sufficiency energy generated by the PV system is directly locally index. Unfortunately these indices are complementary. As can consumed. Thus, the generated PV power is primarily used to be seen, the self-consumption is normalized by total generated directly cover the load demand. If the generated power is energy, whereas the self-sufficiency is normalized by total load greater than the load, the power is injected into grid, otherwise, demand. It is clearly that increasing the PV system capacity, if the power is less than the power, the deficit is drawn from which means an increasing of the PV power generation relative the grid. to demanded load, will lead to a decrease of the self- consumption and an increase of the self-sufficiency. Generated PV A solution to improve the self-consumption and the power P(t) Power Power PV economic feasibility of the PV systems is to store the excess of energy into battery storage systems. In this case, the excess of self-consumption Load demand L(t) energy from the PV system could be stored in the battery and A used later when the PV power is insufficient to cover the load. Another solution is to accurate manage the residential PV C system in order to reduce the imbalance between required load C B and generated power to a minimum value, considering a demand side management system designed to control the 12AM 9AM 12PM 9PM 12AM electric appliances of individual prosumer. Fig. 1. Daily generated power and load profiles. In order to find the optimal value of self-consumption, from previous relations, we will remove area B, obtaining thus the In accordance with Fig. 1, the self-consumption (SC) and energy exchanged with the grid. self-sufficiency (SS) indices can be defined as ratio of areas depicted in the previous figure. The self-consumption indicates how much energy from PV production (A+B) is locally Power injected consumed by the prosumer (B). Instead, the self-sufficiency into grid indicates how much from load demand (B+C) is locally Power Power drawn generated by the own PV system (B). from the grid B B SC  and SS  (1) A  B B  C Besides the self-consumption and self-sufficiency indices, another index is usually used to measure the overlapping 12AM 9AM 12PM 9PM 12AM between load and PV power, thus the size of the PV system can be adjusted by taking into account the ratio of the PV energy Fig. 2. Daily exchanged power wirh the grid. production to the energy consumption [10]: A perfect annual balance between generated power and A  B load demand is obtained when the self-consumption is equal to EnergyRatio  (2) B  C the self-sufficiency, which means that total PV generated power is equal with load demand. If the energy exchanged with Actually, the ratio between self-consumption and self- the grid is taken into account, the perfect balance between sufficiency indices is the ratio of energy consumed and PV generated power and load demand leads to an equal amount of energy produced in residential PV system for same reference energy injected into the electrical grid with the amount of period. Therefore, these indices are able to be defined for energy drawn from the grid. various reference periods, such as hourly, daily, monthly or yearly reference. III. LACARP PHOTOVOLTAIC SYSTEM The inverter synchronizes the output power with the system The PV system under analysis is connected to the local frequency, converting direct current to alternating current with utility grid of Faculty of Electrical Engineering, Technical a maximum efficiency of 97%. The inverter (see Fig. 5) is University of Iasi, the generated electricity being used to equipped with an OptiTrack Global Peak, a maximum power supply the own consumers. The faculty has three photovoltaic point tracking controller which ensure that the operating point of inverter is all times accurately chosen, identifying the systems with a total installed capacity of 15 kWp, two systems being operated as stand-alone PV systems and one of them as a presence of several maximum power points in an available grid-connected PV system. operating range, as may be the case of partially shaded photovoltaic modules. The aim of this section is to present the grid-connected photovoltaic system [16], as part of “Laboratory for Applied Research and Prototype Design” (LACARP), a research laboratory of the Power Engineering Department. The PV system (called LACARP PV system) is located in back yard of Electrical Engineering Faculty (see Fig. 3), the geographical coordinates of location being 47°9'26" (47°.1572) North, 27°35'29" (27°.5914).

The rated capacity of LACARP PV system is 3 kWp, the system being composed on 12 photovoltaic modules Fig. 5. The tie-grid inverter of LACARP PV system. interconnected in series. The PV modules are based on polycrystalline silicon technology with a maximum power of The PV system has been installed in 10th December 2013, 250Wp and an average efficiency around 15%. The PV followed by a period of time in that the PV system was the modules are mounted on a dual axis solar tracker (DEGER subject of a various number of interruptions due technical tracker 3000NT/HD/CT), which adjust the position of PV adjustments of tracking system and BOS interface. The PV panels in order to optimize the received solar radiation. The output energy database has been obtained through continuously control unit which manages the dual axis solar tracker detects, monitored of the grid-connected PV system, covering over four through two irradiation sensors, the position with maximum years, from January 2014 till now. Complete operation data is solar radiation and automatically adjusts the azimuth and averaged over every 15 min, the hourly, daily, monthly and elevation angles of the tracking system. The PV panels position annual energy productions being stored on a SD card and can also be manually adjusted using a remote control. uploaded on the internet (on the SMA’ Sunny Portal). For instance, the monthly energy injected into grid is reported, for years 2014-2017, in the Fig. 6.

Fig. 3. LACARP photovoltaic system.

The PV modules are interconnected in one sting to a Sunny Boy 3000TL tie-grid inverter, which provides the energy in the electrical utility grid to 0.4 kV. The three-phase electrical diagram of grid-connected PV system is shown in Fig. 4.

L1 L2 L3 GND

PCC

L1 L2 L3 GND

Fig. 6. LACARP PV energy production.

The average annual energy production of PV system is around 3250 kWh/year, whilst the total energy production from Fig. 4. Three-line diagram of LACARP PV system. December 2013 till now is around 13145 kWh. IV. SELF-CONSUMPTION ANALYSIS, RESULTS AND In order to evaluate the self-consumption and self- DISCUSSIONS sufficiency indices, the previous energies database has been involved in an analysis, thus, the hourly values of exchanged Actually, the analyses of self-consumption and self- energy with the grid have been calculated. Figures 9 and 10 sufficiency indices require the measurements resources related show the hourly values of the load profile and of PV energy to load demand profile and output power of PV system. production, as well as the hourly values of energy exchanged Therefore, in first plot of Fig. 7 are shown the hourly values of with the grid for a summer week and for a winter week, load demand of all consumers from faculty whereas, in the respectively. As can be seen in following figures and taking second plot are shown the hourly values of output energy from into account the analyses of energy exchanged with the grid for the grid-connected PV system. For a comparison of these two whole year, it can be concluded that the entire amount of PV sizes, the monthly values of load demand and PV output energy is locally consumed by the own consumers, without an energy are shown in the same coordinates in Fig.8. excess of energy that have to be injected into grid. The lower values of PV output energy leads to a deficit of energy, which have to be drawn from the electrical grid.

Fig. 7. Hourly values of load demand and PV output energy.

Fig. 9. Hourly values of load, PV output energy and excanged energy with the grid for 32th week. Energy (kWh) Energy

Fig. 8. Monthly values of load demand and PV output energy.

As can be seen, the hourly and monthly PV output energies Fig. 10. Hourly values of load, PV output energy and excanged energy with have lower values relative to consumption profiles, especially the grid for 10th week. during the winter season. The PV output energy is considerably lower from the end of September to the beginning of March. The self-consumption indicates how much from PV The most important causes of lower values of PV output production is directly used in the faculty utility grid to supply energy are the lower amount of solar radiation in the winter the own customers. Having in view the lower amount of PV season, as well as the permanently shading effect of energy production, all energy is locally consumed, which surrounding buildings and the temporary shading of the snow. means that the self-consumption index of PV energy is 100% In the winter season, when the sun path has lower heights, the for whole year. The self-sufficiency indicates how much from solar radiation is obstructed by the surrounding buildings, faculty consumption is locally generated by the PV system. For especially in the morning and in the evening. Furthermore, the assessment of self-sufficiency index, the daily, monthly and PV system during its continuous operation is sometimes annual reference periods have been considered. Thus, the covered with snow, decreasing the amount of solar radiation hourly and monthly values of self-sufficiency index have been reaching on the PV panels. calculated over one year period, these values being depicted in Fig. 11. three PV system of the faculty, with a total PV capacity of 15kWp, the self-sufficiency will increase from 2,914% to 13,707%, which means an annual energy locally generated and used to supply the own customers around 15,965 MWh/year, decreasing the amount of energy drawn from the grid, from 152,947 MWh to 140,296 MWh.

Fig. 11. Hourly and monthly values of self-sufficency index.

As can be seen, the self-sufficiency index has lower values, with the higher values around 15% during the summer and lower values in winter, the annual self-sufficiency index calculated as an average of daily values, has been found around to 2,914%. Fig. 12. Self-consumption and self-sufficency indices with respect to the PV Concerning the enhancement of the self-consumption system capacity. indices, a quantitative analysis has been carried out related to the effect of increasing the capacity of PV system on the self- With regards to the quantitative dependence of self- consumption indices. Increasing the PV system capacity, much consumption indices on PV capacity, in Fig. 12 are shown in more load demand is covered by the PV production, which same coordinates the evolution of the self-consumption and leads to an increase of the self-sufficiency index. However, if self-sufficiency indices as a function of the PV system the load cannot be covered by own PV energy production, the capacity. The lower values of PV capacity lead to a higher deficit of energy will be drawn from the electrical grid. Thus, value of self-consumption and also to a lower value of self- various scenarios with different rated capacities of grid- sufficiency, which means that there is a lower amount of excess connected PV system have been considered and for each of energy that have to be injected into grid as well as a large scenario the self-consumption indices and energy exchanged amount of deficit energy that have to be drawn from the grid in with the grid have been calculated. This assessment has been order to cover the load demand. Instead, increasing the PV conducted as an average of daily reference periods. Table I capacity, the self-sufficiency is able to be increased and self- shows the self-consumption and self-sufficiency indices, as consumption will decrease. well as the energy drawn from the grid, energy locally In terms of effects of the PV system capacity on the self- generated and energy injected into grid for some scenarios with consumption indices, a qualitative analysis has been developed. different PV system capacities. The analysis indicates the weight of the increasing of PV system capacity on the self-consumption enhancement. TABLE I. THE ANNUAL SELF-CONSUMPTION INDICES AND ENERGY Considering the qualitative analysis, it can be seen how much EXCHANGED WITH THE GRID the variation of the PV capacity changes the final values of PV system Self- Self- Energy Energy Energy self-consumption indices. Thus, for lower values of the PV capacity consumption sufficiency drawn from locally injected system capacity (up to 20 kWp), the self-sufficiency is (kWp) index index the grid generated into grid increased with around 1% for each 1 kWp of PV power added (%) (%) (MWh) (MWh) (MWh) to the PV capacity and with around 0.1%/kWp for the PV 3 100,00 2,914 152,947 3,313 0,000 system capacities larger than 50 kWp. 5 99,987 4,855 150,740 5,521 0,001 7 99,977 6,796 148,533 7,728 0,003 9 99,910 8,716 146,338 9,923 0,017 V. CONCLUSIONS 11 99,597 10,540 144,205 12,056 0,093 The owners of PV systems design their systems in order to 13 98,924 12,204 142,189 14,072 0,286 15 98,011 13,707 140,296 15,965 0,603 cover as much as possible of their consumption. A common 17 96,960 15,071 138,521 17,740 1,036 way to increase the self-consumption from PV system is to 19 95,826 16,308 136,858 19,403 1,582 increase the grid-connected PV system capacity. The quantitative and qualitative effect of increasing the self- Nevertheless, over a PV capacity threshold, the generated consumption is dependent by the PV capacity, therefore when PV energy will be higher than the load consumption, this the self-consumption of PV system is considered, it is excess of energy being injected into grid, leading to a decrease important to know how much PV capacity is actually needed in of the self-consumption index. In case of grid-connecting of all order to achieve the optimal self-consumption indices. This could maximize the self-consumption indices for the lowest PV markets through investment profitability”, Renewable Energy, vol. 87, system capacity. A proper size of PV capacity is an important 2016, pp. 42-53. [6] Directive 2010/31 / EU of the European Parliament and of the Council issue to ensure the optimal investment on PV system. of 19 May 2010 on the Energy Performance of Buildings, Official Quantifying the variability of self-consumption and self- Journal of the European Union L 153/13. sufficiency indices of a PV prosumer is important for assessing [7] EU publications, „First workshop on identification of future emerging technologies for low carbon energy supply”, JRC, Ispra, Italy, 1st the PV system capacity. In this work, the self-consumption December 2016, ISBN 978-92-79-69764-7, on site: indices for a low-voltage grid-connected PV system have been https://publications.europa.eu/en/publication-detail/-/publication/d22ece40-6a9a-11e7-b2f2- calculated. The values of self-sufficiency have been found very 01aa75ed71a1/language-en/format-PDF [8] T. Beck, H. Kondziella, G. Huard, T. Bruckner, „Assessing the influence low, which means that the PV system capacity is insufficient to of the temporal resolution of electrical load and PV generation profiles cover the load, much more capacity has to be installed in order on self-consumption and sizing of PV-battery”, Systems Applied Energy to achieve the optimal value of self-consumption indices. 173, 2016, pp. 331–342. [9] Salom J, Marszal AJ, Widen J, Candanedo J, Lindberg KB. „Analysis of load match and grid interaction indicators in net zero energy buildings ACKNOWLEDGMENT with simulated and monitored data”, Applied Energy 136, 2014, pp. This work was supported by a grant of the Romanian Ministry 119–131. [10] Rasmus Luthander, Joakim Widén, Daniel Nilsson, Jenny Palm, of Research and Innovation, CCCDI - UEFISCDI, project „Photovoltaic self-consumption in buildings: A review”, Applied number PN-III-P2-2.1-CI-2017-0823, within PNCDI III. Energy, vol. 142, 2015, pp. 80–94. [11] M. Senol, S. Abbasoglu, O. Kukrer, A.A. Babatunde, „A guide in installing large-scale PV power plant for self consumption mechanism”, REFERENCES Solar Energy 132, 2016, pp. 518–537. [1] European Commission. EU Energy in figures, statistical pocketbook, EU [12] A Ayala-Gilardón, L. Mora-Lopez, M. Sidrach-de-Cardona, „Analysis Brussels, Belgium, 2016. ISSN 2363-247X, on site: of a photovoltaic self-consumption facility with different net metering http://ec.europa.eu/energy/en/data-analysis/energy-statistical-pocketbook schemes”, International Journal of Smart Grid and Clean Energy, vol. 6, [2] SolarPower Europe, Global Market Outlook 2016-2020, available on no. 1, 2017, pp. 47-53. http://www.solarpowereurope.org/home/ [13] G. Merei, J. Moshövel, D. Magnor, „Optimization of self-consumption [3] International Energy Agency (IEA)–PVPS, Gaetan Masson, Jose Ignacio and techno-economic analysis of PV-battery systems in commercial Briano, Maria Jesus Baez, Review and Analysis of PV self-consumption applications”, Applied Energy 168, 2016, pp. 171–178. policies, IEAPVPS T1-28, 2016, ISBN 978-3-906042-33-6. [14] Antimo Barbato and Antonio Capone, „Optimization Models and [4] Tillmann Lang, David Ammann, Bastien Girod, „Profitability in absence Methods for Demand-Side Management of Residential Users: A of subsidies: A techno-economic analysis of rooftop photovoltaic self- Survey”, Energies 7, 2014, pp. 5787-5824. consumption in residential and commercial buildings”, Renewable [15] CEI EN 61724, Photovoltaic system performance monitoring. Energy, vol. 87, 2016, pp. 77-87. Guidelines for measurement, data exchange and analysis, 1999. [5] L. De Boeck, S. Van Asch, P. De Bruecker, A. Audenaert, „Comparison [16] LACARP laboratory: of support policies for residential photovoltaic systems in the major EU www.sunnyportal.com/Templates/PublicPage.aspx?page=e2999c2d- f99d-490a-bcf6-9c8f3e217c1f

Smart Cities Symposium Prague 2018

Probabilistic Analysis of Sky Clearness Index for Solar Energy Systems Planning

Ciprian Nemes, Romeo Ciobanu, Calin Rugina

probability density function. Actually, the estimation of this Abstract—Proper planning of solar energy systems require index requires the solar data measurements. The solar accurate knowledge concerning the effective values of irradiation measurement database behind of this study has been recorded incident on solar system. One of the important parameters that using a weather station belonging to the Faculty of Electrical provide information concerning the effective value of irradiation Engineering, Technical University of Iasi, Romania. at the ground level is the clearness index, this index being directly related to the attenuation effect of the earth’s atmosphere and the The study presented in this work is conducted according to clouds. In this paper, a solar radiation database recorder in Iasi, the following structure. Section 2 is focused on notions and Romania has been involved in a statistical analysis in order to assessment techniques used to evaluate the clearness index. In evaluate the theoretical probability density function associated to Section 3, the main probability density functions used to the daily clearness index values. model the statistical behaviour of the clearness index are Keywords-clearness index, solar radiation, probability density reported. In Section 4 a numerical analysis is developed in function, cumulative distribution function. order to evaluate which distribution function gives the best I. INTRODUCTION estimation of clearness index frequency. Finally, the main conclusions are given in Section 5. CCURATE knowledge of the availability of solar radiation has fundamental importance for studying, A II. SKY CLEARNESS INDEX planning and designing of the solar energy systems. The solar energy systems include that technology that convert the solar Clearness index is an important parameter that provides energy in a useful form of energy. The energy converted by a information concerning the real solar radiation that is solar system is directly related to the amount of solar radiation transmitted through the atmosphere, compared with the falling on the system’s surface. available extraterrestrial solar radiation. Based on the An important factor that affects the solar radiation value at literature in solar engineering [1-5, 11, 13], the clearness index the ground level is the attenuation effect of the earth’s is defined as the ratio of the global solar radiation at ground atmosphere and cloudiness level. When the solar radiation level on a horizontal surface and the extraterrestrial solar passes through the earth’s atmosphere, it is reduced due to irradiation. The values of clearness index range between 0 and reflection, scattering and absorption, thus, the solar radiation 1, being a dimensionless size. In the literature, there are that reaches the earth's surface is much reduced in intensity. A defined three indices that involves the clearness index, parameter that provides information concerning the effective namely: the monthly average clearness index, the daily value of irradiation at the ground level is the clearness index, clearness index and the hourly clearness index, respectively. this indicator being directly related to the attenuation effect of The irradiance value reached on a given location of the earth’s the earth’s atmosphere and the clouds. For solar system surface depends on the irradiance values outside of studies is important to assess and understand the distribution atmosphere as well as on the attenuation effects of the earth's of the frequencies of different values of clearness index for atmosphere and clouds. various time frame intervals. In this order, the present paper A. Extraterrestrial Solar Radiation focuses on an assessment and a statistical analysis of daily In order to assess the clearness index values, an clearness index values in order to evaluate their theoretical extraterrestrial solar radiation model based on the mathematical equations published in the European Solar Received: 1st February 2018 Radiation Atlas (ESRA) [7] has been used and adapted for This work was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CCCDI – UEFISCDI, project number location under analysis. 83/2016 - Smart Urban Isle, within PNCDI III. Concerning the solar radiation, two terms are commonly Ciprian Nemes is with the Department of Electric Power Engineering, used in literature, namely the irradiance and the irradiation. “Gheorghe Asachi” Technical University of Iasi, Bd. Prof. Dimitrie Mangeron There is important to understand the difference between these no. 21-23, 700050, Iasi, Romania (e-mail: [email protected]) 2 Romeo Ciobanu is with the Department of Electrical Measurements and two terms. The irradiance, G (W/m ), is defined as the power Materials, “Gheorghe Asachi” Technical University of Iasi, Bd. Prof. Dimitrie received by the earth from the sun rays that fall on a unit Mangeron no. 21-23, 700050, Iasi, Romania (e-mail: [email protected]) surface on the ground. At the edge of the earth’s atmosphere, Calin Rugina is the PhD student at Faculty of Electrical Engineering, “Gheorghe Asachi” Technical University of Iasi, Bd. Prof. Dimitrie Mangeron the irradiance received from the sun is nearly constant. no. 21-23, 700050, Iasi, Romania (e-mail: [email protected])

978-1-5386-5017-2/18/$31.00 ©2018 European Union The average of the irradiance at the highest level of the and provides this information as a measurements database. 2 atmosphere is defined as the solar constant, GS0 (W/m ). Its The measurements include information concerning wind speed value has been adopted by the World Radiometric Center as a and wind direction, the global solar radiation, ultraviolet constant equal with 1367 W/m2 [6]. Instead, as a consequence radiation dose and index, temperature, humidity, atmospheric of the elliptical orbit of the earth, the irradiance at the edge of pressure, precipitation amounts and other special indices the earth’s atmosphere fluctuates around of the solar constant which combine the parameters of these measurements, like value, during a year. According with ESRA model [7], since dew point, evapotranspiration heat index, rainfall, rain rate, the astronomical position of the earth is known, the THSW index, wind chill, sunrise and sunset, moon phase and extraterrestrial irradiance value for a given day of the year can so on. More details and technical informations for each type of be evaluated using following equation: instrument and sensor that equip the station are reported in [8]. ª ⋅ º = + ⋅ § 360 Nd · GN GS 0 «1 0.033 cos¨ ¸» (1) d ¬ © 365 ¹¼ where Nd is the number of the day in year, starting from the first of January. Thus, during a year, the extraterrestrial irradiance varies between ±3.34% from the solar constant. The second term, widely used to account the amount of solar radiation, is the irradiation. The irradiation, H (Wh/m2), is a measure of energy received from the sun over a given time interval, being calculated as the irradiance integrated over a Fig. 1. The Integrated Sensor Suite and the Console modules of the Vantage certain period of time. On the top of the earth’s atmosphere, Pro2™ weather station. the daily irradiation received on a horizontal surface depends The Console of weather station has a software preinstalled on the latitude of the location and the number of the day in which ensures the interface with a computer, in order to log year. The average daily extraterrestrial irradiation on a the weather data and to upload weather information on the horizontal surface, (kWh/m2), can be computed by H 0 database. The weather station is in operation since March integrating the eq. (1) from sunrise to sunset, getting following 2013, the previous weather parameters being continuously formula for the representative day of each month: monitored till now, complete operation database being ª πω º 24 S available for one minute acquisition interval and averaged in H 0 = G «cos(φ)cos(δ)sin(ω ) + sin(φ)sin(δ)» (2) π Nd S order to calculate the hourly, daily and monthly average ¬« 180 ¼» values. Concerning the solar radiation database, this is φ where the latitude of the location on the earth, , has to be commonly available in two forms. The first form is the δ measured in degree. The solar declination angle, , represents average values of global irradiation on a horizontal surface, the angle between sun rays and the equatorial plane, gradually whereas the second one is the average value of global changing from +23.45° (or north) on June 21, to -23.45° (or irradiance on same surface. south) on December 21. The declination angle can be found using the following formula: III. PROBABILITY DENSITY FUNCTIONS FOR CLEARNESS INDEX δ = ⋅ § 360 + · 23.45 sin ¨ (284 Nd )¸ (3) A number of studies have been published in scientific © 365 ¹ literature related to clearness index distribution, which The sunset hour angle, Ȧs, for the representative day of each propose to use a variety of probability density functions [9- month is defined as: 11]. A literature survey indicates various probability density ω = −1()− φ δ functions used to model the statistical behaviour of clearness S cos tan( ) tan( ) (4) The fundamental equations above mentioned are available index. Firstly, Liu and Jordan [12] suggested the existence of in ESRA model, as well as in the standard textbooks on solar a generalized family of probability density functions for daily engineering [1-5]. clearness index, dependent only on monthly average clearness index, regardless of location and month. B. Solar Measurements Database Then, Bendt [13] proposed a probability density function of The solar measurement used in this analysis has been daily clearness index based on monthly average of daily recorded using a Vantage Pro2 wireless weather station. The clearness index. They proposed the following expression for weather station belongs of LACARP Laboratory, Faculty of the probability density function: Electrical Engineering, Technical University of Iasi, Romania, γ ⋅exp(γ k ) f (k , k ) = d (5) the geographical coordinates of weather station being 47°9'26" d d γ − γ exp( kd ) exp( kd ) North, 27°35'29" East, and 44 m above the sea level. max min The corresponding cumulative distribution function is: The weather station, shown in Fig. 1, consists of two exp(γ k ) − exp(γ k ) modules, namely the “Integrated Sensor Suite” and “Console” = dmin d F(kd , k ) (6) modules. The first module is equipped with instruments and d exp(γk ) − exp(γ k ) dmin dmax sensors for measuring weather parameters with high accuracy daily clearness index values obtained by dividing the observed where kd is the daily clearness index, kd is the monthly global irradiation by the extraterrestrial global irradiation. average clearness index, kdmin and kdmax being lower and upper limits of daily clearness index calculated for each month. The Week:12 / 2016 1000

parameter Ȗ is related on clearness index through the equation: )

2 800 § − 1 · γ − § − 1 · γ 600 ¨kd ¸exp( kd ) ¨kd ¸exp( kd ) ¨ min γ ¸ min ¨ max γ ¸ max 400 = © ¹ © ¹ k d (7) 200 γ − γ Irradiation (W/m exp( kd ) exp( kd ) min max 0 Another widely used probability density function is that Monday Tuesday Wednesday Thursday Friday Saturday Sunday proposed by Hollands and Huget [14], which have proposed a 0.8 modified gamma distribution to generate the random 0.6 sequences of daily clearness index: 0.4 § k · λ = ⋅¨ − d ¸⋅ kd 0.2 f (kd , k d ) C 1 e (8) ¨ ¸ Daily Clearness Index kd max 0 © ¹ Monday Tuesday Wednesday Thursday Friday Saturday Sunday The associate cumulative distribution function is: Fig. 2. Extraterrestrial and global irradiation measurements (a) and daily C λ k 2 F(k , k ) == []1− e d ()1− λ ⋅k + γ ⋅k (9) clearness index (b) for 12th week of 2016. d d ⋅ ⋅λ ⋅γ d d kd max k d The parameter C is analytically obtained from conditions The daily clearness index has been calculated for entire period and it has been involved in the following analysis as a required to f (k ,k ) to be a probability density function: d d numerical database. Thus, the Fig. 3 and 4 show the box-plots λ2 ⋅ k of the daily clearness index through a monthly representation. C = d max λ k (10) As can be seen, the lowest values are close to 0 while the e d max −1− λ ⋅ k d max highest values are around to 0.6-0.7. Furthermore, the lowest, The γ and Ȝ parameters are interconnected through the mean and highest values are calculated (in %) and tabulated in upper limit of daily clearness index: Table 1. λ2 γ = (11) + λ ⋅ 2 kd max whereas Ȝ depends on kdmax and kd , being established through the following expressions:

λ k λ k index clearness Daily ()2 / λ + k (1 −e d max )+ 2k e d max k = d max d max (12) d λ k d max − − λ e 1 kd max

IV. CLEARNESS INDEX NUMERICAL FREQUENCY ANALYSIS This section focuses to evaluate the random behaviour of Daily clearness index clearness Daily irradiation at ground level taking into account the stochastic features [15] of daily clearness index. In this order, the daily clearness index values have been calculated considering the Fig. 3. The yearly box-plots of the daily clearness index for 2013-2014. ration between daily values of global irradiation on horizontal surface available on database and daily values of extraterrestrial irradiation evaluated based on analytical method. In this work, the solar radiation database covers the period between 1st March 2013 and 31th December 2016. In this order, the values of extraterrestrial irradiation have been Daily crearness index Daily calculated taking into account the diurnal and seasonal changes in the position of the sun relative to the earth. In order to assess the irradiance values, as well as the amount of the irradiation over a given period of time, a software application has been developed to perform all these calculations. For instance, in first plot of Fig. 2 are drawn, in the same Daily crearness index Daily system coordinates, the calculated extraterrestrial irradiation, as well as the measurements of global irradiation for few consecutive days, whereas in the second plot are shown the Fig. 4. The yearly box-plots of the daily clearness index for 2015-2016. TABLE I TABLE III MONTHLY STATISTICAL REPORT OF DAILY CLEARNESS INDEX (%) MONTHLY VALUES OF HOLLANDS AND HUGET PDF COEFFICIENTS 2013 2014 2015 2016 2013 2014 2015 2016

Month kdmin kd kdmax kdmin kd kdmax kdmin kd kdmax kdmin kd kdmax Month C Ȝ C Ȝ C Ȝ C Ȝ Jan 4.28 28.75 61.83 8.23 29.13 61.41 5.15 28.26 61.69 Jan 1.3788 3.46882 1.297 3.72853 1.4531 3.30049 Feb 10.66 30.89 54.37 5.73 33.44 67.05 5.33 31.27 57.80 Feb 0.6711 6.96163 0.9884 3.97867 0.8084 5.77827 Mar 4.34 36.31 80.37 4.30 35.34 61.32 8.38 35.43 62.64 5.65 36.74 60.31 Mar 1.1659 2.41216 0.5496 6.39994 0.5965 5.97616 0.3952 7.47644 Apr 5.41 41.76 67.23 5.69 33.82 58.24 5.25 39.81 62.55 7.26 41.27 59.30 Apr 0.3093 7.03661 0.5539 6.86777 0.2758 8.03819 0.121 10.6941 May 8.18 46.54 67.26 6.01 41.14 69.55 15.39 46.15 58.55 13.87 39.39 55.72 May 0.1145 9.27735 0.4153 6.02243 0.0124 16.0646 0.1056 11.8897 Jun 16.92 43.55 60.62 22.07 45.44 68.89 18.01 49.04 62.62 6.91 41.67 56.31 Jun 0.0778 11.4397 0.1839 7.98948 0.0139 14.665 0.0524 13.4575 Jul 11.55 45.50 67.24 19.39 39.55 56.41 8.71 45.60 65.39 39.92 45.23 48.92 Jul 0.1465 8.7462 0.1161 11.4747 0.1072 9.74957 4E-09 54.2303 Aug 14.13 45.22 64.00 10.24 44.20 65.63 10.28 44.15 67.66 7.14 45.16 59.75 Aug 0.0928 10.3312 0.1575 8.85078 0.2066 7.91343 0.0332 13.5727 Sep 13.23 39.90 57.67 16.72 48.21 56.81 4.36 38.97 63.09 9.31 42.84 55.53 Sep 0.1334 10.8229 0.0006 23.2364 0.3428 7.39747 0.0229 15.6537 Oct 8.12 38.17 65.52 4.83 37.62 58.59 4.61 37.87 55.74 2.81 27.38 57.40 Oct 0.4826 6.15793 0.2739 8.77493 0.1695 10.6602 1.3606 4.06088 Nov 4.50 30.44 57.27 3.79 22.52 56.56 0.00 29.82 57.35 3.04 29.39 56.25 Nov 0.8855 5.56404 2.3691 1.93916 0.974 5.2367 0.9737 5.40654 Dec 1.71 31.62 57.80 3.57 25.71 65.78 0.00 35.16 57.60 5.02 30.20 54.68 Dec 0.7671 5.94427 2.1343 1.48791 0.4089 7.86133 0.7722 6.44701

This paper attempts to find an answer to the question: which Using previous parameters, in Fig. 5 are drawn in same probability density function (pdf) gives the best estimation of diagram the empirical cumulative distribution function (cdf), clearness index frequency? Among of the probability density drawn based on the real numerical values, and theoretical functions of clearness index, the most frequently studied cumulative distribution functions based on Bendt and functions take into account the daily clearness index, followed Hollands and Huget expressions. The cumulative frequency by the functions that consider the hourly clearness index. The distributions of the daily clearness index have been calculated most important studies on the daily clearness index, cited in from the previous available data, separately for all months. most works, refer to Bendt as well as Hollands and Huget functions. Based on these functions, considered as generalized Empirical vs. Theoretical CDFs (December 2016) probability density functions associated to clearness index, 1 Empirical several works have been carried out using various daily 0.9 Theoretical Bendt Theoretical Holland clearness index databases and taking into account the previous 0.8 functions as reference probability density functions [11,15]. In 0.7 this order, among previous probability density functions, these two functions have been taken into account in the study in 0.6 ) d 0.5

order to describe the clearness index frequency distributions. F(k Based on the clearness index database, the parameters of the 0.4 theoretical probability density functions that approximate the 0.3 empirical distribution of clearness index frequency have been 0.2 estimated. Table II shows the parameters of the Bendt pdf for 0.1 analyzed database, the gamma parameter being determined 0 using previous equation (7) related to lowest, average and 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 k highest values of clearness index for each month. d Fig. 5. Empirical and theoretical cumulative distribution plots for daily clearness index values (December 2016). TABLE II MONTHLY VALUES OF BENDT PDF COEFFICIENTS To show how well the cumulative distribution functions fit 2013 2014 2015 2016 with empirical data, two kinds of tests are employed in this Month Ȗ Ȗ Ȗ Ȗ Jan -1.5810 -2.4823 -1.9800 study. In this order, to evaluate the performance of these two Feb -1.0274 -0.9481 -0.1263 cumulative distribution functions, the mean bias error (MBE) Mar -1.2752 0.9363 -0.0329 1.5293 and root mean square error (RMSE) have been used to Apr 1.7412 0.8062 2.2183 3.7601 evaluate the accuracy of estimated cumulative distribution May 3.2061 1.0049 6.6872 3.2457 Jun 3.0935 -0.0210 5.8280 5.5313 function to empirical distribution [16]. The MBE and RMSE Jul 2.4326 1.4525 3.3824 12.2909 are statistical tests widely used to evaluate the difference Aug 3.0835 2.5295 1.9270 5.8264 between values provided by a theoretical cumulative Sep 2.7721 10.9873 1.8629 6.7368 distribution function and the real values of the database Oct 0.4949 2.5301 3.7396 -1.1040 Nov -0.1902 -3.4793 0.4197 -0.1102 distribution. The MBE provides information about the model’s Dec 0.7105 -2.9293 2.3690 0.1737 performance, a lower MBE value being desirable. Positive values indicate overestimated values, while negative values Table III shows the parameters of the Hollands and Huget indicate underestimated values. The RMSE is always positive, probability density function, the C and lambda parameters a lower value being desirable, too. RMSE test provides also being determined using previous equations (10, 12) related to the information on the performance of the fitting considering average and highest values of clearness index for each month. the deviation between the theoretical values and the empirical values. The relative mean bias and root mean square errors have V. CONCLUSIONS been calculated for the entire period, from March 2013 to The most important requirement for the solar systems December 2016, based on the numerical values obtained from planning and operation is the accurate estimation of solar daily values of clearness index calculated based on the resources. Investigation of solar resources should be carefully database, the relative errors being reported in Table IV for performed in accordance with their clearness index both Bendt as well as the Hollands and Huget probability probabilistic features. Daily clearness index is required to density functions. estimate the incident solar irradiation in solar collectors, thus TABLE IV establishing the value of energy production (electrical or MONTHLY VALUES OF MBE STATISTICAL TEST MBE 2013 2014 2015 2016 thermal) in the analyzed area. This study aims on assessing of Hollands Hollands Hollands Hollands Bendt Bendt Bendt Bendt the degree of clearness index in the Iasi, Romania, using the Month and Huget and Huget and Huget and Huget Jan 0.013 -0.151 -0.006 -0.154 -0.006 -0.154 solar radiation database recorded for period 2013-2016. The Feb -0.028 -0.114 -0.051 -0.189 -0.051 -0.189 recorded numerical database has been involved in a statistical Mar -0.012 -0.178 0.015 -0.246 -0.003 -0.204 -0.003 -0.204 analysis, thus, the monthly average as well as the daily values Apr -0.012 -0.077 0.145 0.031 0.039 -0.024 0.039 -0.024 of clearness index have been calculated. May -0.026 -0.076 0.061 -0.028 0.101 -0.136 0.101 -0.136 Jun -0.015 -0.071 -0.059 -0.174 -0.150 -0.186 -0.150 -0.186 The probability density distributions have been derived Jul -0.040 -0.136 0.149 0.010 -0.048 -0.125 -0.048 -0.125 from this database and the distributional parameters have been Aug -0.027 -0.086 -0.013 -0.087 -0.013 -0.105 -0.013 -0.105 evaluated. The degree of accuracy of the probability density Sep -0.016 -0.083 -0.198 -0.227 -0.016 -0.117 -0.016 -0.117 function fitting is evaluated by two statistical tests, mean bias Oct -0.017 -0.129 0.005 -0.086 0.024 -0.116 0.024 -0.116 Nov -0.017 -0.121 0.117 -0.036 -0.003 -0.116 -0.003 -0.116 error (MBE) and root mean square error (RMSE). It has been Dec -0.019 -0.134 0.088 -0.095 -0.083 -0.174 -0.083 -0.174 shown, from computational results, that the probability density function which gives the lowest values of statistical tests is the As can be seen, the MBE test has the best estimation when Bendt function. However, from accuracy viewpoint, the the Bendt distribution is used. The percentage of MBE for all Hollands and Huget probability density function is also a good under evaluated cases varies between 0.3% and 19.8% relative probabilistic approach because its probability density function to Bendt distribution function and between 1% and 24.6% gives close values to Bendt function. relative to Hollands and Huget distribution function. In Table V are tabulated the root mean square errors REFERENCES between theoretical cumulative distribution functions, using [1] Duffie J.A, Beckman W.A., “Solar engineering of thermal processes”, Bendt as well as Hollands and Huget probability functions, Wiley and Sons Press, New York, 1991, pp. 90-96. [2] Zekai S., “Solar Energy Fundamentals and Modeling Techniques”, and empirical cumulative distribution function drawn based on Publisher Springer, 2008, pp. 65-95. numerical database values. As can be seen, the minimal [3] Gilbert M., “Renewable and Efficient Electric Power Systems”, John RMSE values which indicate the best estimation of cumulative Wiley and Sons Publishing, New Jersey, 2004, pp. 395-415. [4] M. Iqbal, “An introduction to solar radiation”, Academic Press, Toronto, distribution function are obtained in case of Bendt distribution. 1983, pp. 85-90. [5] Marius Paulescu, “Algoritmi de estimare a energiei solare”, Editura TABLE V Matrix Rom, Bucuresti 2005, pp. 20-31. MONTHLY VALUES OF RMSE STATISTICAL TEST [6] Huashan Li et al., “Solar constant values for estimating solar radiation”, RMSE 2013 2014 2015 2016 Energy 36, pp. 1785-1789, 2011. Hollands Hollands Hollands Hollands [7] Scharmer K., Greif J., “The European Solar Radiation Atlas”, vol. 1. Les Month Bendt Bendt Bendt Bendt and Huget and Huget and Huget and Huget Presses de l’École des Mines, Paris, 2000, pp. 28-38. Jan 0.034 0.185 0.037 0.189 0.033 0.178 [8] *** https://www.davisnet.com/solution/vantage-pro2 Feb 0.112 0.185 0.056 0.213 0.048 0.181 [9] Tian Pau Chang, “Investigation on frequency distribution of global Mar 0.044 0.208 0.130 0.754 0.108 0.537 0.170 0.568 radiation using different probability density functions”, International Apr 0.051 0.111 0.179 0.218 0.083 0.122 0.149 0.266 Journal of Applied Science and Engineering, no. 2, pp. 99-107, 2010. May 0.055 0.095 0.103 0.049 0.228 0.498 0.283 0.491 [10] G. Tina, S. Gagliano, “Probability analysis of weather data for energy th Jun 0.039 0.106 0.095 0.189 0.163 0.213 0.105 0.115 assessment of hybrid Solar/Wind power system”, 4 International Jul 0.128 0.166 0.198 0.553 0.127 0.149 0.242 0.472 Conference on Energy, Environment, Ecosystems and Sustainable Aug 0.073 0.103 0.081 0.104 0.092 0.122 0.083 0.107 Development, Algarve, Portugal, June 11-13, 2008, pp. 217-223. [11] Ibanez, M., Rosell, J.I., Beckman, W.A., 2003. A bi-variable probability Sep 0.039 0.113 0.247 0.284 0.075 0.129 0.123 0.158 density function for the daily clearness index. Solar Energy, vol. 75, pp. Oct 0.038 0.143 0.053 0.152 0.116 0.365 0.183 0.107 73–80, 2003. Nov 0.073 0.186 0.135 0.135 0.067 0.181 0.070 0.180 [12] Liu B. Y. H. and R. C. Jordan, “The Interrelationship and Characteristic Dec 0.046 0.168 0.124 0.111 0.102 0.208 0.051 0.152 Distribution of Direct, Diffuse and Total Solar Radiation”, Solar Energy, vol. 4, no. 3, issue 1, 1960. [13] Bendt P, Collares-Pereira M, Rabl A, “The frequency distribution of Based on both statistical tests, it has found that Bendt daily insolation value”, Solar Energy, vol. 27, no. 1, pp. 1–5, 1981. probability density function is superior in accuracy and has a [14] Hollands T., Huget G.,”A Probability Density Function for the clearness index, with applications”, Solar Energy, vol. 30, pp. 195-209, 1983. smaller error compared with the Holland and Huget function. [15] Achim Woyte, Ronnie Belmans, Johan Nijs, “Fluctuations in Thus, it can be summarized that the Bendt is the best instantaneous clearness index: Analysis and statistics”, Solar Energy, probability density function used to estimate the clearness vol. 81, pp. 195–206, 2007. [16] J.M Santos “Metodology for generating daily clearness index values index distributions taking into consideration the MBE and staring from monthly average daily value. Determining daily sequence RMSE as measurements of comparison for location under using stochastic models”, Renewable Energy vol. 29, pp. 1523-1544, analysis. 2003. A clear sky irradiation assessment using the European Solar Radiation Atlas model and Shuttle Radar Topography Mission database: A case study for Romanian territory Ciprian Nemes

Citation: Journal of Renewable and Sustainable Energy 5, 041807 (2013); doi: 10.1063/1.4813001 View online: https://doi.org/10.1063/1.4813001 View Table of Contents: http://aip.scitation.org/toc/rse/5/4 Published by the American Institute of Physics

Articles you may be interested in Benchmark of algorithms for solar clear-sky detection Journal of Renewable and Sustainable Energy 8, 033703 (2016); 10.1063/1.4950948

Analysis of solar direct irradiance models under clear-skies: Evaluation of the improvements for locally adapted models Journal of Renewable and Sustainable Energy 9, 023703 (2017); 10.1063/1.4981798

Applicability of ASHRAE clear-sky model based on solar-radiation measurements in Saudi Arabia AIP Conference Proceedings 1850, 140001 (2017); 10.1063/1.4984509

Monthly average daily global and diffuse solar radiation based on sunshine duration and clearness index for Brasov, Romania Journal of Renewable and Sustainable Energy 6, 053106 (2014); 10.1063/1.4896596

Calculation of direct normal irradiation from global horizontal irradiation AIP Conference Proceedings 1477, 391 (2012); 10.1063/1.4753912

Preface to Special Issue: Renewable Energy in South-Eastern Europe Journal of Renewable and Sustainable Energy 5, 041701 (2013); 10.1063/1.4818514 JOURNAL OF RENEWABLE AND SUSTAINABLE ENERGY 5, 041807 (2013)

A clear sky irradiation assessment using the European Solar Radiation Atlas model and Shuttle Radar Topography Mission database: A case study for Romanian territory Ciprian Nemesa) Department of Electrical Engineering, Technical University of Iasi, 700050 Iasi, Romania (Received 28 January 2013; accepted 24 May 2013; published online 3 July 2013)

The objectives of this paper are to develop a software application that allows to establish the clear-sky solar radiation on horizontal surface considering the shading effect of complex topography of terrain and to employ it in order to assess the amount of clear-sky irradiation over Romanian territory. In order to achieve the ﬁrst objective, a clear sky solar radiation model developed in accordance with the latest mathematical equations published in European Solar Radiation Atlas has been adopted and implemented together with a digital elevation model developed based on the Shuttle Radar Topography Mission database. In order to achieve the second objective, based on a digital elevation model of the Romanian territory, the monthly average daily clear-sky irradiation over whole territory has been established. The estimates have been validated using the solar radiation values from other two well known databases, the relative errors shown that the irradiance values have a good accuracy. The software application has been involved in order to build and analyze the maps of monthly average of daily irradiation, in the assumption of the clear-sky conditions and in horizontal surface, over whole Romanian territory. VC 2013 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4813001]

I. INTRODUCTION In the last time, some studies have been developed in order to evaluate the potential of solar energy over several countries or regions of the Earth. A literature survey indicates different techniques developed to predict the solar potential, based on recorded solar database1,2 or geo- physical calculations with the solar constant and attenuation effect of the Earth’s atmosphere.3,4 Thus, various parts of the Earth have been covered by these kinds of studies, such as for South- America,5 New Zealand,6 Africa,7 and so forth. Accurate knowledge of the amount of the solar radiation has a fundamental importance for studying, planning, and designing of these solar energy systems. The best information about the solar radiation is obtained from the real measurements on the system’s place, but, unfortunately, the solar radiation measurements are not yet available for many locations. Therefore, modeling is a proper solution for estimation of solar radiation at locations where measurements are not available, taking into account the amount of solar radiation received in clear-sky condition and applying a factor that parameterizes attenuation caused by cloudiness. In this paper, a software application based on a clear-sky model and on a digital elevation model is developed in order to evaluate the amount of clear-sky solar radiation over the Romanian territory. The clear-sky model is based on the latest mathematical equations published in the European Solar Radiation Atlas (ESRA),8,9 whereas the digital elevation model

a)Author to whom correspondence should be addressed. Electronic mail: [email protected]. Phone: þ40 232 278683. Fax: þ40 232 237627.

1941-7012/2013/5(4)/041807/12 5, 041807-1 VC Author(s) 2013 041807-2 Ciprian Nemes J. Renewable Sustainable Energy 5, 041807 (2013)

(DEM) is developed using the digital topographic database provided by Shuttle Radar 10 VR Topography Mission (SRTM). In order to evaluate the amount of solar radiation, a MATLAB software application is developed to perform all these calculations, the details of the mathematical model used in the software application are given in Sec. II. The main goal of this section is the simple parameterization of the incoming solar radiation in terms of site elevation, including the shading effect through those factors which take into account the horizon obstructions. The fundamental equations used below are based on ESRA model8,9 and some standard textbooks on solar engineering.11–14 The topographic information of Romanian territory is achieved from a DEM, which is developed and adapted in Sec. III in order to provide information about the elevation and horizon obstructions for each point of DEM. This DEM provides the opportunity to take into account the effect of topographic characteristics in spatial distribution of solar radiation over large areas based on the shading effect in zones with a complex topography. In order to create a solar radiation database over Romanian territory (in Sec. IV), the DEM database is joined with the software application, so as the irradiance and irradiation values are calculated for each point of the DEM. The accuracy of solar radiation database certainty depends on DEM resolution. Estimated solar radiation values are tested against values drawn out from others radiation database, namely, with those values drawn out from SoDa15,17 and Photovoltaic Geographic Information System (PVGIS)18 database, with the aim of evaluating the level of accuracy of this software application. Generally, for the whole area, the software application has a good performance in terms of standard errors. In Sec. V, the software application is used to build the maps of monthly average of daily irradiation, which in turn are involved in some spatial and temporal statistical analyses. Finally, the main conclusions of this paper are given in Sec. V.

II. CLEAR SKY MODEL DESCRIPTION The energy received by the Earth from the Sun rays can be defined as an instantaneous size (irradiance, G (W/m2)), as well as a cumulative size (irradiation, H (Wh/m2)) estimated over a given period of time. The irradiance value reached on a given location of the Earth surface depends on two main factors; namely, the irradiance values outside of atmosphere taking also into account the position of the Sun relative to the site location under evaluation, and also the attenuation effects of the Earth’s atmosphere. Outside the Earth’s atmosphere, at any given point in the space, the irradiance received from the Sun is nearly constant. The average of the irradiance at the highest level of the atmos- 2 2 phere is defined as the solar constant, GS0 (W/m ), and it is adopted to be equal with 1367 W/m . While the astronomical position of the Earth is known, the extraterrestrial irradiance value for a given day of the year can be evaluated based on the ESRA model.8,9 In order to take into account the position of the Sun relative to the site location, for any moment of day, the relative position of the Sun can be described in terms of its azimuth and altitude angles.8,9,11–14 The altitude angle is related to the angle of latitude, the Earth’s declination, and also the hour angle. The solar declination angle is gradually changed from þ23.45 (or North) on June 21, to 23.45 (or South) on December 21. The hour angle represents the number of degrees that the Earth must rotate before the Sun will be directly over the local meridian, being calculated as the difference between the noon and the local solar time, expressed in 2p rotation on the 24 h. The solar time is a correction applied to the local zone time, considering the rotation of the Earth about its axis and, also, the Earth’s revolution around the Sun. This correction of local zone time has two components. The first one is a correction for the difference between the local time meridian of the local zone time and the observer’s longitude, and, the second one takes into account the perturbations in the Earth revolution around the Sun, during which the Earth does not sweep equal areas. The second factor that affects the irradiance value at the ground level is the attenuation effect of the Earth’s atmosphere. In clear sky conditions, when the solar radiation passes through the Earth’s atmosphere, it is reduced only due to scattering and absorption, thus, the irradiance that reaches the Earth’s surface is reduced in intensity. Thus, under cloudless 041807-3 Ciprian Nemes J. Renewable Sustainable Energy 5, 041807 (2013) conditions, the radiation that is not scattered or absorbed will directly reaches on the Earth’s surface as a direct (beam) radiation, while the scatter radiation that reaches the ground is called diffuse radiation. In accordance with ESRA model,8,9 the clear sky direct irradiance on a hori- 2 zontal surface, GD (W/m ), is calculated as follows:

E0 GD ¼ G expð0:8662 TLK m dRðmÞÞ sinðaÞ; (1) where TLK is the Linke atmospheric turbidity factor for air mass 2, m is the optical air mass, dR(m) is the Rayleigh optical depth, and a is the solar altitude angle. The Linke turbidity factor is a convenient measure of the atmospheric absorption and scattering of the solar radiation under clear skies, ranging from TLK ¼ 2 for extremely clear cold air 19,20 in winter and 3 for clear warm air, to TLK greater than 6, for the polluted atmosphere. The air mass ratio, m, is the path length of the Sun’s rays when pass through atmosphere, divided by the minimum path length, which occurs when the Sun is to noon time. The optical air mass is affected by two main factors, the direction of the Sun’s rays and the site’s atmospheric pressure, which in turn is inﬂuenced by the site’s elevation above the sea level and can be expressed by the following formula:21

c c 1:6364 m ¼ðp=p0Þ=ðsina þ 0:50572 ða þ6:07995Þ Þ; (2) where ac (in degrees) is a correction of solar altitude angle

0:1594 þ 1:123 a þ 0:065656 a2 ac ¼ a þ 0:061359 ; (3) 1 þ 28:9344 a þ 277:3971 a2 and

ðp=p0Þ¼expðz=HRÞ (4) is the pressure correction with the site’s elevation of the observer location, z (in meters), relative to the sea level, and HR is the height of the Rayleigh atmosphere, equal to 8435.2 m. The Rayleigh optical depth, dR(m), depends on the optical air mass’s value, a widely used empirical equation is given by following expression20: 8 > 1 < ð6:6296 þ 1:7513 m 0:1202 m2 þ 0:0065 m3 0:00013 m4Þ ; for m 20 d ðmÞ¼ R :> ð10:4 þ 0:718 mÞ1; for m > 20: (5)

Returning to the ESRA model, the clear sky diffuse irradiance on a horizontal surface, Gd (W/m2), is the product between the normal extraterrestrial irradiance, the diffuse transmission function Tn, and the diffuse solar altitude function Fd, as follows:

E0 Gd ¼ G TnðTLKÞFdðaÞ: (6)

The diffuse transmission function is a function of Linke turbidity factor based on the following expression:8,9

2 TnðTLKÞ¼0:015843 þ 0:030543 TLK þ 0:0003797 TLK: (7)

The solar altitude function is a function of the solar altitude, evaluated using the following expression: 041807-4 Ciprian Nemes J. Renewable Sustainable Energy 5, 041807 (2013)

2 Fdða; TLKÞ¼A0 þ A1sin a þ A2sin a; (8)

12 where the values of the coefﬁcients A0, A1 and A2, depending on the Linke turbidity, are deﬁned by the following expressions: ( 0:0022=TnðTLKÞ; for A0 TnðTLKÞ < 0:0022 A0 ¼ 2 0:26463 0:061581 TLK þ 0:0031408 TLK; otherwise; (9) 2 A1 ¼ 2:0402 þ 0:018945 TLK 0:011161 TLK; 2 A2 ¼1:3025 þ 0:039231 TLK þ 0:0085079 TLK:

Consequently, all these sets of equations are used to evaluate the values of irradiance on horizontal surface, for a given location on the Earth. The clear sky global irradiation, for a given time interval, is calculated as a cumulative sum of the instantaneous irradiance in time. Therefore, in order to ﬁnd the daily irradiation values, the global irradiance falling on a horizontal surface have to be integrated between sunrise and sunset, the sunrise and sunset times11,23 being given by the following formula:

1 tr;s ¼ 12 : 006 cos1ðtanð/ ÞtanðdÞÞ: (10) 15 LA

Thus, the daily values of clear sky irradiation could be integrated in order to provide monthly or yearly values of the clear sky irradiation on horizontal surface.

III. DIGITAL ELEVATION MODEL OF ROMANIAN TERRITORY In order to create a solar radiation database for Romanian territory, a DEM based on digital topographic database has been developed. A DEM is a digital three-dimensional model of the Earth’s surface, containing the data concerning the terrain elevation for each geographical coordinates of ground position, sampled with a regularly spaced horizontal interval. For a DEM with a medium resolution, the SRTM database has been chosen to assure proper information about the terrain elevation. The SRTM is a joint international project developed by National Aeronautics and Space Administration and the National Imagery and Mapping Agency, whose main objective is to generate a near-global digital elevation model of the Earth using radar interferometry.10 The SRTM database is a non-commercial product and is freely available for download through the USGS Earth Resources Observation and Science (EROS) Data Centre.24 These data are intended for use with a geographic information system or other special application software, and are not directly viewable in a browser. The original data have a resolution of 3 arc sec (approximately 90 m), a higher resolution of SRTM database (1 arc sec, about 30 m) being available only for the United States and other few countries.10,24 Romania is located between 43370 and 48150 North latitude parallels and between 20150 and 29410 East longitude meridians, covering a geographical area of 238 391 km2. For these coordinates, there were downloaded 70 hgt.zip files from USGS website,24 covering different areas by files with the name that includes an extension of 1 latitude and 1 longitude situated between 43 and 48 North latitude parallels, respectively, 19 and 30 East longitude meridians, those files indicating the latitude and the longitude of the area. For instance, the N46E027.hgt.zip includes the area between 46 and 47 North latitudes, respectively, 27 and 28 East longitudes. VR A MATLAB routine has been used to draw out from the SRTM files (with 3 arc sec resolution) the data in the DEM format. The routine needs as input arguments the decimal degree coordinates, and it has as output argument the elevation assigned to longitude and latitude coordinates. Although the resolution of SRTM files is approximately 90 m, the resolution of developed DEM has been chosen to be in a step of 0.01 of longitudes and latitudes, between 20 and 29.99 meridians, respectively 43 and 48.99 parallels, meaning a number of 1000 600 cells. The 041807-5 Ciprian Nemes J. Renewable Sustainable Energy 5, 041807 (2013)

FIG. 1. DEM of Romanian territory. resolution of developed Romanian DEM is about 1 km 1km (shown in Fig. 1), being enough for purposes of this paper, even if a DEM with a higher resolution can be obtained but it is actually detrimental to the computing time. Giving the geographical coordinates and elevation for a given cell of the DEM, the direct and diffuse components of solar irradiance at a given time could be estimated. Nevertheless, the Sun obstructions and shading effects caused by the topographic features have to be taken into account for an accurate prediction of solar radiation. In zones with a complex topography, variability in elevation causes the Sun obstructions and shading effects, which leads to a local gradient of isolation. In this order, an angular distribution of Sun obstruction is computed at each moment for every cell of the DEM. It should be noted that the DEM’s cells present different relative positions to the Sun during the day. Based on an analysis of the surrounding topography of the DEM’s cell, the angular obstruction is evaluated by searching around the cell of interest the maximum angle of Sun obstruction for each moment, having in view the Sun direction (through solar azimuth angle). Fig. 2 depicts an intuitive view of the searching process.

FIG. 2. Sun obstruction caused by the topographic features. 041807-6 Ciprian Nemes J. Renewable Sustainable Energy 5, 041807 (2013)

FIG. 3. DEM without and with shading effect.

In order to evaluate the shading effect of a given cell from DEM at a moment h, the eleva- tions of the cells that surround the cell under evaluation, corresponding to solar azimuth wh, are compared with the elevation of given cell, in order to ﬁnd maximum angle of Sun obstruction. If the Sun obstruction angle is greater than the solar altitude angle ah, then the cell under evaluation is shaded at moment h. An example of shading effect in DEM is shown in Fig. 3. When the Sun is unobstructed, the global irradiance results from sum of both components, direct and diffuse components, whereas when the Sun is obstructed by terrain features, the global irradiance comprise only diffuse component. Daily solar irradiation is then computed by integrating the instantaneous global irradiance values between sunrise and sunset.

IV. SOFTWARE DEVELOPMENT AND RESULTS VALIDATION

VR The previous mathematical model has been implemented into a MATLAB software application and used together with the DEM, in order to calculate the irradiance as well as the daily, monthly and yearly irradiation. In order to obtain the direct and diffuse solar radiation, the model needs the geographical coordinates and elevation above the sea level for every cell of DEM and based on the shading information, the global irradiance take in account only diffuse or both components. In addition, the model requires information related to the sky condition, through the turbidity factor value.24 This parameter has been assumed through its monthly average values available on SoDa database. Therefore, using a cumulative sum of irradiance over the whole Romanian territory, the daily, monthly average, and yearly average of daily irradiation have been calculated, the results for yearly average daily global irradiation without shading effect being graphically represented in Fig. 4 and with shading effect in Fig. 5, respectively. In order to validate the results of the application, some comparisons with data from others solar radiation database have been conducted. Numerous solar database and estimation tools are available worldwide, a signiﬁcant progress being made in this direction in last time.25,26 Among of these should be mentioned PVGIS, SoDa, HelioClim, NASA-SSE (Surface Meteorology and Solar Energy programme), Meteonorm, Satel-light, etc, where, in order to account the spatial distributions of solar radiation, the solar radiation models have been integrated within geographical information systems, obtaining so very powerful solar radiation analysis tools and database. The SoDa project is based on the information acquired by processing the satellite images, most of the database resources being available in the graphical and tabular format, the database being available through SoDa services.15,16 Some information from this database have been drawn out and used in this paper to validate the results of the developed application. Another solar database used in this paper is from PVGIS, project developed by the Joint Research Centre of Europe Commission.17 This database includes, among other information, the monthly and yearly average values of the clear-sky global irradiation on horizontal surface. Its web interface is developed to provide interactive access to the maps and data solar radiation over European geographical regions. 041807-7 Ciprian Nemes J. Renewable Sustainable Energy 5, 041807 (2013)

FIG. 4. Map of yearly average daily global irradiation without shading effect.

Thus, for a numerical comparison of performance of the software application, the monthly average daily irradiation values have been computed for six cities of Romania, during the whole year, the cities being chosen in order to cover the all areas of Romanian territory, namely, Brasov, Bucuresti, Cluj, Constanta, Iasi, and Timisoara. The numerical values of VR monthly average daily irradiation obtained from MATLAB software application (MSA) and also from SoDa and PVGIS database are numerical reported in Table I. The degree of accuracy of the software application is evaluated by two statistical tests, mean bias error (MBE) and root mean square error (RMSE), these statistical tests being widely used tests in assessing the performance of the analytical models.22,27,28 The MBE provides

FIG. 5. Map of yearly average daily global irradiation with shading effect. 041807-8 Ciprian Nemes J. Renewable Sustainable Energy 5, 041807 (2013)

TABLE I. Monthly average daily irradiation (Wh/m2/day) for the cities under evaluation.

Monthly average daily irradiation (Wh/m2/day) City Database Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Brasov MSA 2286 3469 5288 7249 8697 9308 8984 7769 5943 3995 2541 1949 45.63N SoDa 2389 3685 5437 7217 8430 8947 8424 7342 5681 4112 2619 1999 25.58E PVGIS 2371 3703 5413 7277 8647 9378 8868 7656 5772 4198 2776 2031 Bucuresti MSA 2375 3533 5293 7370 8538 9110 8806 7660 5919 4041 2625 2050 44.43N SoDa 2447 3779 5652 7382 8432 8855 8409 7424 6005 4259 2674 2090 26.1N PVGIS 2439 3704 5658 7397 8367 8832 8354 7403 6018 4326 2659 2035 Cluj MSA 2115 3286 5110 7303 8591 9226 8892 7640 5776 3812 2367 1884 46.76N SoDa 2210 3570 5452 7396 8721 9341 8738 7506 5660 4025 2520 1856 23.6E PVGIS 2190 3503 5355 7360 8614 9348 8639 7549 5535 3997 2531 1990 Constanta MSA 2415 3573 5327 7491 8547 9111 8810 7676 5949 4079 2665 2080 44.18N SoDa 2473 3857 5681 7499 8600 9027 8604 7667 6177 4389 2741 2104 28.65E PVGIS 2453 3833 5646 7548 8551 9028 8553 7673 6190 4435 2721 2253 Iasi MSA 2027 3182 4992 6981 8473 9111 8776 7520 5657 3704 2275 1603 47.16N SoDa 2053 3212 5028 6811 8126 8602 8178 7090 5422 3746 2322 1698 27.6E PVGIS 2074 3105 4970 6709 8083 8500 8086 7103 5351 3688 2337 1687 Timisoara MSA 2201 3356 5138 7066 8492 9094 8775 7579 5782 3870 2450 1915 45.74N SoDa 2261 3463 5155 6793 7920 8450 8134 7023 5457 3903 2495 1970 21.22E PVGIS 2288 3408 5153 6775 7870 8442 8157 7084 5435 3909 2558 2040

information about the model’s performance, a lower MBE value being desirable. Positive values indicate overestimated values, while negative values indicate underestimated values. The RMSE is always positive, a lower value being desirable, too. RMSE test provides also the information on the performance of the model considering the deviation between the calculated values and the desired values. To obtain dimensionless statistical indicators, the MBE and RMSE have been normalized to average of database values, the relative errors being calculated with following expressions: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XN 1 XN ðx x^ Þ ðx x^ Þ2 N i i N i i MBEð%Þ¼ i¼1 100; RMSEð%Þ¼ i¼1 100; (11) 1 XN 1 XN ðx^ Þ ðx^ Þ N i N i i¼1 i¼1 where xi and xî are the ith values of software application and from database, respectively, all values used in previous equations referring to monthly average daily irradiation. The relative mean bias and root mean square errors have been computed for an entire year, from January to December, and for all cities under consideration, based on the numerical values obtained from software application and monthly average daily irradiation draw out from SoDa and PVGIS database, the relative errors being reported in Table II. Comparisons of the application results with those given by the SoDa and PVGIS database show that the software application gives fairly close results for a preliminary evaluation of the solar irradiation, the percentage of MBE for all under evaluated cases varies between 0.42% to 2.53% relative to SoDa database and between 0.46% to 2.36% relative to PVGIS database, while RMSE varies between 1.35% and 3.92% for SoDa database, and between 1.64% and 3.75% for PVGIS database, respectively. The underlined values indicate the maximal values of percentage of errors and, as can be seen, the relative errors are lower than 5% for all months of the year and for both SoDa and PVGIS database. 041807-9 Ciprian Nemes J. Renewable Sustainable Energy 5, 041807 (2013)

TABLE II. Monthly values of relative MBE and RMSE.

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

MBE (%) SoDa 1.50 2.71 1.94 0.42 1.10 1.63 2.53 2.03 0.91 1.91 1.46 1.01 PVGIS 1.43 2.02 1.63 0.46 1.20 1.34 2.36 1.55 1.06 2.14 2.11 2.31 RMSE (%) SoDa 2.26 4.26 3.36 1.35 2.56 3.07 3.92 3.41 2.77 3.21 2.31 1.96 PVGIS 2.10 3.71 3.00 1.64 2.61 3.07 3.75 2.77 3.07 3.77 3.55 3.83

Furthermore, the errors indicate that the application results are underestimated relative to database values during the winter period (from October to March), whereas in the in summer period (from April to September) the calculated values are slightly overestimated. The analysis of the errors shows that the results of developed software application are in agreement with those from SoDa and PVGIS database values, giving enough conﬁdence on the values provided by the developed software application. Concerning the fact that during the winter, the application results are underestimated relative to database values, whereas in summer the results are overestimated, it can be explained considering the relationships between direct and diffuse radi- ations and solar altitude. As is presented in Eq. (1), the direct radiation is related to solar altitude through the sinus function, whereas the diffuse radiation is related to solar altitude through a quadratic polynomial sinus function (Eq. (8)). It means that a higher value of solar radiation has a higher effect on the diffuse radiation in comparison with effect on direct radiation.

V. ANALYSIS, RESULTS, AND DISCUSSIONS The developed software application has been used to evaluate and analyze the amount of clear-sky solar radiation over Romanian territory. Therefore, the solar radiation has been estimated as the monthly maps of average values of daily clear-sky irradiation, evaluated in horizontal surface, without and with consideration of shading effect. The monthly maps have been plotted by calculating the spatial distribution of average values of daily irradiation obtained for each month. Figure 6 shows an example of such maps, which have been drawn in 1 1km2 spatial resolution, for four representative months (March, June, September, and December). These maps allow as the spatial and temporal analyses of monthly average daily irradiation to be conducted. As can be seen, the clear-sky solar radiation values show a north-south variation as well as the inﬂuence of the high-elevation zones. As can be seen, the amount of irradiation over Romanian territory increases from the low-elevation zones and northern sides, to high-elevation zones and to that located in southern sides of country. This variation of spatial irradiation is observed for all months of the year, the southern sides obviously receiving more irradiation than northern ones. If the shading effect is taken into account, the high-elevation zones, especially the mountains zones, are affected by the neighboring topographic features that surround the area, especially in the morning and evening, at lower solar altitude angles. These effects can be observed from the fact that the same areas are shaded for every month of the year. In order to evaluate the temporally variation of solar radiation, the distribution of monthly average daily irradiations has been evaluated over whole Romanian territory, without and with consideration of shading effect. In order to avoid overestimations, all the cells outside Romanian borders have been removed from the DEM. Simultaneous comparisons of distributions of monthly average daily irradiation are shown in Fig. 7; thus, for each month spatial distributions of monthly average daily irradiation have been statistical analyzed using the box-plot representation, for both analyzed cases, with and without shading effect. Monthly average daily irradiations vary during the year and are very low during the winter season. In the summer season values are the highest, while in the fall and spring seasons values are intermediate. Table III indicates the numerical values of monthly values of mean (M, kWh/ m2/day), standard deviation (SD, kWh/m2/day), relative standard deviation (RSD, %) (i.e., absolute value of coefﬁcient of variation) as well as the minimum (Min, kWh/m2/day) and maximum (Max, kWh/m2/day) values of daily irradiation. 041807-10 Ciprian Nemes J. Renewable Sustainable Energy 5, 041807 (2013)

FIG. 6. Examples of maps of monthly average daily clear-sky irradiation without and with shading effect (for March, June, September, and December).

Statistical analysis indicates that the amount of monthly average daily irradiation over whole Romanian territory, on horizontal surface and in the clear-sky conditions, varies between 2.04 kWh/m2/day and 9.25 kWh/m2/day if the shading effect is neglected and between 1.99 kWh/ m2/day and 9.23 kWh/m2/day with consideration of shading effect. On the other hand, the standard deviations of monthly average daily irradiation have the same tendency, with higher values in winter, around to 10.92% (without shading effect) and 13.15% (with shading effect), and 041807-11 Ciprian Nemes J. Renewable Sustainable Energy 5, 041807 (2013)

FIG. 7. Comparisons of monthly average daily irradiation, based on box plot representation. lower values in summer, around to 0.93% and 1.2%, respectively. It is observed that in winter months the standard deviation of clear-sky irradiation is larger that in the summer months, this being obviously the effect of the difference between solar altitudes angles during the year. In Sec. IV, Figs. 4 and 5 show the maps of yearly average daily clear-sky irradiation over Romanian territory evaluated using data without and with consideration of shading effect, respectively. The yearly average daily clear-sky irradiation for Romanian territory without the shading effect is around 5.6345 kWh/m2/day, with the minimum value to 5.3163 kWh/m2/day and the maximum value to 6.1788 kWh/m2/day. If the shading effect is considered in analysis, the average daily irradiation is around 5.6002 kWh/m2/day, with the minimum value to 0.968 kWh/m2/day and with the same maximum value. The amount of monthly losses caused by the shading effect, relative to whole Romanian territory, varies from maximum value of 45.66 Wh/m2/day in January to minimum value of 25.62 Wh/m2/day in July. Although absolute values appear comparable, the values relative to monthly average daily irradiation have a variation between 1.95% and 0.28%, respectively. The average value over the whole Romanian territory is around 34.22 Wh/m2/day, but unfortunately

TABLE III. Statistical analysis of monthly average daily irradiation.

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

2 Without shading effect (kWh/m /day) M 2.34 3.56 5.27 7.20 8.64 9.25 8.94 7.75 5.96 4.05 2.62 2.04 SD 0.23 0.24 0.22 0.16 0.10 0.09 0.09 0.13 0.19 0.23 0.23 0.22 RSD 9.78 6.70 4.10 2.17 1.19 0.93 1.01 1.70 3.26 5.75 8.88 10.92 Min. 1.93 3.12 4.87 6.90 8.45 9.13 8.78 7.50 5.59 3.62 2.20 1.64 Max. 2.87 4.15 5.89 7.78 9.17 9.79 9.47 8.31 6.56 4.64 3.16 2.54

2 With shading effect (kWh/m /day) M 2.30 3.51 5.24 7.17 8.62 9.23 8.91 7.72 5.93 4.01 2.58 1.99 SD 0.27 0.28 0.25 0.18 0.13 0.11 0.12 0.16 0.22 0.26 0.27 0.26 RSD 11.77 7.89 4.72 2.55 1.49 1.20 1.31 2.02 3.73 6.57 10.52 13.15 Min. 0.58 0.72 0.89 1.02 1.11 2.06 1.12 1.05 0.93 0.77 0.61 0.54 Max. 2.87 4.15 5.89 7.78 9.17 9.79 9.47 8.31 6.56 4.64 3.16 2.54 041807-12 Ciprian Nemes J. Renewable Sustainable Energy 5, 041807 (2013)

this value is not uniformly distributed over the whole territory, the higher amount of losses being located in the zones with a complex topography of terrain, especially the mountains zones.

VI. CONCLUSION In this paper, a software application based on the ESRA model and using a DEM database has been developed in order to calculate the irradiance as well as to build the maps of irradiation over different time intervals, in the assumption of the clear-sky conditions and in horizontal surface, for a case study of Romanian territory. The software results have been validated against the monthly average daily irradiation from SoDa and PVGIS databases, obtaining relative RBE and RMSE under 5%, the relative errors indicating that the software application has a good performance. This application has been involved in order to build and analyze the maps of monthly average of daily clear-sky irradiation in horizontal surface, over whole Romanian territory. This application represents an opportunity to estimate clear-sky solar radiation for any location inside the study area, considering also the shading effect, especially in zones with a complex topography features. The application presented in the paper uses a simple solar radiation model, which can be easily integrated with a DEM for a relative accurate and fast estimation of solar irradiance over an analyzed territory, taking also in account the shading effect of topographic features. The main advantage of the application, compared with other solar radiation estimation tools, is the estimation of database in a three-dimensional format, for any moment time, being easily analyzed in both graphical and numerical formats.

ACKNOWLEDGMENTS This paper was supported by the project PERFORM-ERA “Postdoctoral Performance for Integration in the European Research Area” (ID-57649), ﬁnanced by the European Social Fund and the Romanian Government.

1T. Muneer, M. S. Gul, and J. Kubie, J. Sol. Energy Eng. 122, 146 (2000). 2N. C. Coops, R. H. Waring, and J. B. Moncrieff, Int. J. Biometeorol. 44, 204 (2000). 3Y. Kun, T. Koike, and Y. Baisheng, Agricultural and Forest Meteorology 137, 43 (2006). 4R. A. Ball, L. C. Purcell, and S. K. Carey, Agron. J. 96, 391 (2004). 5F. R. Martins, E. B. Pereira, and S. L. Abreu, Sol. Energy 81, 517 (2007). 6L. Luo, D. Hamilton, and B. Han, Sol. Energy 84, 501 (2010). 7L. Diabate, P. Blanc, and L. Wald, Sol. Energy 76, 733 (2004). 8K. Scharmer and J. Greif, The European Solar Radiation Atlas: Fundamentals and Maps (Les Presses de l’Ecole des Mines Paris, 2000), Vol. 1. 9C. Rigollier, O. Bauer, and L. Wald, Sol. Energy 68, 33 (2000). 10See http://www2.jpl.nasa.gov/srtm for “shuttle radar topography mission.” 11M. Paulescu, Algoritmi de Estimare a Energiei Solare (Matrix ROM, Bucharest, 2005) (in Romanian). 12J. A. Dufﬁe and W. A. Beckman, Solar Engineering of Thermal Processes (John Wiley and Sons, New York, 1991). 13M. Gilbert, Renewable and Efﬁcient Electric Power Systems (John Wiley and Sons, New Jersey, 2004). 14S. Zekai, Solar Energy Fundamentals and Modelling Techniques (Springer, 2008). 15M. Lefevre, M. Albuisson, and L. Wald, Integration and Exploitation of Networked Solar Radiation Databases for Environment Monitoring Project; Joint Report on Interpolation Scheme ‘Meteosat’ and Database ‘Climatology’ I (Report for the European Commission, 2002). 16See http://www.soda-is.com/eng/index.html for “solar radiation database.” 17See http://www.helioclim.org/index.html for “HelioClim solar radiation.” 18See http://re.jrc.ec.europa.eu/pvgis/apps4/pvest.php for “photovoltaic geographic information system.” 19A. Coste and E. Eftimie, Environ. Eng. Manage. J. 10, 277 (2011). 20F. Kasten, Sol. Energy 56, 239 (1996). 21F. Kasten and A. T. Young, Appl. Opt. 28, 4735 (1989). 22J. Polo, L. F. Zarzalejo, L. Martin, A. A. Navarro, and R. Marchante, Sol. Energy 83, 1177 (2009). 23V. Badescu, C. A. Gueymard, S. Cheval, C. Oprea, M. Baciu, A. Dumitrescu, F. Iacobescu, I. Milos, and C. Rada, Renewable Sustainable Energy Rev. 16, 1636 (2012). 24See http://dds.cr.usgs.gov/srtm/ for “USGS EROS database website.” 25M. Suri and J. Horierka, Trans. GIS 8, 175 (2004). 26M. Suri, T. Huld, E. Dunlop, and H. Ossenbrink, Sol. Energy 81, 1295 (2007). 27M. Paulescu, E. Paulescu, and N. Stefu, Int. J. Energy Res. 35, 520 (2011). 28F. Mavromatakis and Y. Franghiadakis, Sol. Energy 81, 896 (2007). International Journal of Energy Engineering (IJEE) Dec. 2013, Vol. 3 Iss. 6, PP. 261-268

Statistical Analysis of Wind Speed Profile: A Case Study from Iasi Region, Romania Ciprian-Mircea Nemeş Faculty of Electrical Engineering, “Gheorghe Asachi” Technical University Iasi, Bd. Mangeron 21-23, Iasi, Romania [email protected]

Abstract- The increased integration of wind power into the electric power systems brings new challenges for effective planning and operation. The Weibull distribution is a widely used distribution, especially for modelling the random variable of wind speed. In the paper, the author presents a comparative analysis of some methods for estimating the Weibull parameters. These methods require historical wind speed data, collected over a certain time interval, to establish the parameters of the wind speed distribution for a particular location. Results for a real-world database, collected from the north-east area of Romania, are presented in a study case. Keywords- Wind Speed Data; Wind Energy; Probability Density Function; Weibull Distribution

I. INTRODUCTION The integration of renewable energy in electric power systems is growing rapidly due to the concerns related to environment and the depleting sources of conventional power generation. Unlike other renewable energy sources, wind power has become competitive with conventional sources of power generation, thus the growth rate of wind power has registered the highest growth among other renewable sources. In accordance with [1], the annual average growth rate of wind power over the last 10 years is about 22%. At the end of 2012, the global installed wind power capacity reached 282.5 GW, from that about 44.8 MW was installed in 2012. In 2012, Romania has doubled its total installed wind power capacity, from 982 MW at end of 2011, to 1905 MW at end of 2012. Wind generation brings a great amount of benefits to power systems, such as the cheaper energy compared with the thermal generation, emission reduction, wind energy is available for large areas, and development of a wind power farm can be implemented relatively easily [2]. Wind generation brings a series of difficulties to traditional power systems, for example, uncontrollability of power generation, the wind generation depends on wind availability, irregularly fluctuating and intermittence of power generation, respectively a poor predictability of the wind generation [3]. The important properties in wind generation are the wind speed frequency and magnitude. The wind power output obtained from the wind is directly proportional to the cube of the wind speed. One of the main characteristics of wind is that it is highly variable and its properties vary from one location to another. Wind speed changes continuously and in order to estimate its speed and frequency values, the statistical approach could be viable method for these estimations. Wind speed probability density function plays an important role in electric power generation applications with wind turbines. The most important requirement for effective wind power planning and operation in power systems is an accurate estimation of wind speed distribution. Investigation of wind power generation should be carefully performed in accordance with the wind speed probabilistic character. This paper presents a statistical analysis of main characteristics of the wind speed from the region around Iasi, Romania. The wind speed data is measured as hourly average values, being statistically analyzed over one year period of time. The probability density distribution is derived from these values and their distributional parameters are evaluated with different statistical methods. The paper is organized as follows. In order to evaluate the wind speed distribution, the main issues about the wind speed probability density function, about the dependence of the height and also about the wind direction are presented in Section 2. In Section 3, the main statistical methods used to evaluate the parameters of probability density functions are reported. Furthermore, two statistical tests used in this paper to establish the accuracy of the methods are also reported. In Section 4 a numerical analysis is developed in order to evaluate the statistical parameters, based on the hourly wind speed database collected from the Iasi region. Finally, the main conclusions are given in Section 5.

II. THE WIND SPEED DISTRIBUTION The available wind energy depends on the wind speed, which is a random variable. The wind-speed values occurrences over a long period of time can be described by its probability distribution function.

A. Wind Speed Probability Density Function A number of studies have been published in scientific literature related to wind energy, which propose to use a variety of probability density functions (e.g. normal, lognormal, gamma, Rayleigh, Weibull) to describe wind speed distributions [4, 5]. The common conclusion of these studies is that the Weibull distribution with two parameters may be successfully utilized to

- 261 - International Journal of Energy Engineering (IJEE) Dec. 2013, Vol. 3 Iss. 6, PP. 261-268 describe the principle wind speed variation. The Weibull probability density is given by the following expression:

b -1 b b æ v ö é æ v ö ù . (1) fW (v) = ç ÷ exp ê- ç ÷ ú a è a ø ëê è a ø ûú The corresponding cumulative probability function of the Weibull distribution is:

b é æ v ö ù , (2) FW (v) = 1 - exp ê- ç ÷ ú ëê è a ø ûú where a (m/s) is the scale parameter and b (dimensionless) is the shape parameter of the Weibull distribution. The Weibull distribution is one of the most widely used distributions in many technical fields. This distribution has a particular property, namely it does not have a specific characteristic shape, taking the characteristics of other distributions, based on different values of shape parameter, as is shown in Fig. 1.

β=3

β=2

β=1 Weibull probability densityWeibull probability function

Wind speed Fig. 1 Weibull probability distribution function with different values of shape parameter The Weibull distribution becomes a hyperexponential distribution when shape parameter is less than unity and it is a well- known exponential distribution when the shape parameter is equal to 1. The Weibull distribution is a Rayleigh distribution when shape parameter is equal to 2 and becomes a normal distribution when this parameter is equal to 3.4, respectively an approximate normal distribution when b has a close value to 4. Generally, the scale parameter provides information about the average of the wind speed profile, while the shape parameter provides information about the deviation of the wind speed values around the mean as well as the feature of probability density function. It can be seen that Weibull distribution gets relatively narrower and higher as shape parameter increases. The peak of density function also moves in the direction of higher wind speeds as shape parameter increases. The shape and scale parameters are interconnected through analytical expressions of mean and variance of Weibull probability density function.

B. Variation of Wind Speed with the Height The wind blows faster at higher altitudes because the influences of the ground surface and the air density have lower values. The most common expressions for wind speed variation with the height use the wind profile power law, which is based on the ground friction coefficient, described by the following equation:

k v(z) v(zr ) = (z zr ) (3)

In (3), v(z) and v(zr) are the wind speeds at desired z and registered zr heights, while k is the friction coefficient, which depends on surface roughness and atmospheric stability [6]. Numerically, it ranges between 0.05 for smooth terrain, and 0.5 for rough terrain, with the most frequently adopted value around 0.14. In accordance with [7], between the parameters of Weibull distribution for different heights, there are the following relationships:

k a(z) = a(zr ) × (z zr ) ; b (z) = b (zr ) (4) Previous relationship is based on the expression of Weibull distribution that it has been changed in accordance with the relationship between the wind speed and height. As can be seen from Eq. (4), the shape parameter is a specific property of wind profile, while the scale parameter may be adjusted, in narrow range, by changing the desired height.

- 262 - International Journal of Energy Engineering (IJEE) Dec. 2013, Vol. 3 Iss. 6, PP. 261-268

C. Wind Direction Distribution Generally, the wind blows from different directions, the wind directions could be depicted by the wind rose diagram. A wind rose is a chart which gives a view of how the wind speeds and directions are distributed at a particular location over a specific period of time. It is a very useful representation because a large quantity of data can be summarised in a single plot. If terrain roughness in all directions is similar, it makes no difference of the wind speeds above to hub of wind turbine, because the turbine yaw system makes the rotor to follow the wind direction. If terrain around the turbine is significantly different with respect to roughness or obstacles, the wind speeds to hub of wind turbine have different values depending on the wind direction distribution, this leading to necessity of developing detailed calculations. Therefore, with exception in critical cases, the fact that the wind comes from different directions is not used, instead all wind is assumed to come from the same direction.

III. METHODS FOR THE PARAMETERS ESTIMATION The wind speed distribution is completely determined when its parameters are numerically established. The estimates of the parameters of the Weibull distribution can be found using different estimation methods, which can be classified in graphical methods or analytical methods. The most common analytical methods are maximum likelihood estimator, method of moments and least squares method [8, 9]. Each analytical method, discussed in this paper, has specific criteria which yields estimates that are most suitable in some situations. The Weibull parameters play a major role in developing a model of electric power wind generator, so it is important that different estimation methods are compared to fit parameters of the Weibull distribution from wind speed database. This paper attempts to find an answer to the question: which method gives the best Weibull parameters estimation? The performance of these methods with the same wind speed database will be analyzed. The relative mean bias error (RMBE) and relative root mean square error (RRMSE) will be used in statistical evaluation of the performance of Weibull parameters evaluation.

A. The Maximum Likelihood Estimator The Maximum Likelihood Estimator (MLE) is an analytical method, widely applied in engineering and mathematics problems. For our case, for Weibull distribution of wind speed, in accordance with MLE theory, the likelihood function is built as the joint density of the n random variables and is a function of the two unknown parameters:

b -1 b n n b æ v ö é æ v ö ù L(a,b ) = f (v ) = ç i ÷ expê- ç i ÷ ú , (5) Õ i Õ a a a i=1 i=1 è ø ëê è ø ûú where a and b values can be achieved by using iterative methods or limits method. Last method of parameters evaluation involves taking the partial derivatives of the likelihood function with respect to the parameters, setting the resulting equations equal to zero: ¶ln(L) n n 1 n ¶ln(L) n 1 n = + lnv - vb × lnv = 0 and = - - vb = 0 (6) å i å i i 2 å i ¶b b i=1 a i=1 ¶a a a i=1 The values of a and b result from simultaneously solving of both equations.

B. The Method of Moments The method of moments (MOM) is another analytical method to establish the distribution parameters. If the set of wind data is konwn, the moments of unknown parameters that depend by the two-parameter Weibull distribution will be equalized with the empirical moments. The analytical expression of mean and the variance of Weibull distributions can be directly calculated from the following equations: M (v) = α ×G(1+1 β) and D2 (v) = a 2 × [G(1 + 2 b ) - (G(1 +1 b ))2 ] (7) where G( ) is the gamma function, while the empirical moments are calculated with following equations:

n n vi 2 1 2 v = å and s = å(vi - v) (8) i=1 n n i=1 The b parameter can be got from the coefficient of variation (by dividing the variance on the square mean) and, after that, the a parameter can be established based on first expression from Eq. (7).

- 263 - International Journal of Energy Engineering (IJEE) Dec. 2013, Vol. 3 Iss. 6, PP. 261-268

C. Least Squares Method For the estimation of Weibull parameters, the least-squares method (LSM) is extensively used in engineering problems. The method provides a linear relation between the two parameters having as start point the twice logarithms of Weibull cumulative distribution function, as follows:

é 1 ù ln lnê1- ú = b ln(v)- b ln(a ) (9) ë Fw (v)û This relationship represents a straight line, expressed as: Y = a × X + b , where:

é 1 ù Y = ln lnê1- ú , X = ln (v), and a = b , b = -b ln(a ) (10) ë Fw (v)û Performing rank regression on Y requires that a straight line to be fitted to a set of data points so that the sum of the squares of the deviations from the points to the line is minimized. Both a and b parameters can be evaluated from coefficients of polynomial linear fitting, using a simple linear regression.

D. Statistical Tests Analysis The relative mean bias error (RMBE) and the relative root mean square error (RRMSE) have been used in statistical evaluation of the performance of the Weibull distribution. These statistical tests are based on the following expressions: 1 N 1 N * * (11) RMBE(%) = å(vi - vi ) å(vi ) ×100 N i=1 N i=1

N N 1 * 2 1 * RRMSE(%) = å(vi - vi ) å(vi ) ×100 (12) N i=1 N i=1

th * th where vi is the i actual data, vi is the i predicted data with the Weibull distribution, N is the number of observations. The RMBE and RRMSE provide information about the model’s performance, lower values being desirable [9, 10]. Therefore, the best distribution function can be selected according to the lowest values of RMBE and RRMSE.

IV. CASE STUDY FROM IASI REGION In the present case study, the wind potential of the region is statistically analysed based on one year hourly measurements. The wind speed database behind these studies has been recorded from the north-east area of Romania, namely from Iasi region. The region of interest is located at 47°10' north latitude, 27°37' east longitude and 80 m above the sea level. The recorded data covers the period between 1th January 2010 and 31th December 2010. The measurements available in the original database are characterized by one hour acquisition intervals, the hourly average value being recorded. The wind speed values collected at anemometer height (10 m above the ground) have been involved in an adjustment with the height of wind turbine. In the paper, the Eq. (3) has been used to evaluate the wind speed values at the hub wind turbine height (100 m), considering the same terrain roughness around to wind turbine and 0.14 as the friction coefficient. Fig. 2 shows the wind speed values available at hub wind turbine height (100 m).

Height = 100 m 25

10 Wind speed (m/s)Wind

0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (h) Fig. 2 The hourly speed database collected on Iasi region

- 264 - International Journal of Energy Engineering (IJEE) Dec. 2013, Vol. 3 Iss. 6, PP. 261-268

The variation of wind speed with the direction has also been studied. The wind rose is a graphical representation of the wind directions and speeds, over the period of time at the specific location. To create a wind rose, average wind direction and wind speed values have been sorted by wind direction so that the percentage of time that the wind was blowing from each direction has been determined. Typically, the wind direction data is sorted into twelve equal arc segments, 30° each segment, in preparation for plotting a circular graph in which the radius of each of the twelve segments represents the percentage of time that the wind blew from each of the twelve 30° direction segments. Fig. 3 shows the wind rose diagram of the wind speed database at 100 m height. As can be seen, the main direction of the wind follows the north-south direction. Considering the same roughness of terrain around wind turbine and taking into account that the rotor follows the wind direction, it can be stated that the wind speed values are not affected by the wind direction.

Wind rose

NORTH

15%

10%

5% 20 - 22 WEST EAST 18 - 20 16 - 18 14 - 16 12 - 14 10 - 12 8 - 10 6 - 8 4 - 6 2 - 4 SOUTH 0 - 2

Fig. 3 The wind rose associated to the wind database Based on these measurements, the parameters of Weibull distribution that approximate the real database of wind speed frequency have been estimated. In order to compare the methods described earlier, a Matlab® program has been developed to evaluate the Weibull parameters, based on previous methods and same wind speed database. To evaluate the performance of these methods, the relative mean bias error (RMBE) and relative root mean square error (RRMSE) have been used to evaluate the accuracy of estimated probability density function to real distribution. The RMBE and RRMSE are statistical tests widely used to evaluate the difference between values provided by an estimated probability density function and the real values of the database distribution. The Table 1 shows the Weibull parameters for analyzed database, the scale and shape parameters being determined using previous methods reported in Section 3.

TABLE 1 WEIBULL PARAMETERS FOR WHOLE YEAR

Parameters Method /statistical tests MLE MOM LSM

Scale parameter a 3.9788 3.9876 3.6466 Shape parameter b 1.8687 1.8767 1.6424 RMBE 0.3438 0.5310 -0.8379 RRMSE 1.0868 1.3991 1.6910

As can be seen, the parameters of Weibull distribution evaluated through above three methods are very close, the scale parameter lies between 3.6466 and 3.9876 m/s, while the shape parameter lies between 1.6424 and 1.8767. Likewise, in the last two lines of table are shown the values of RMBE and RRMSE for each used method. It is found that MLE is superior in accuracy and has a smaller error compared with the MOM and LMS methods. Furthermore, for this analysis, it seems that the LMS method is the least accurate method. Furthermore, in order to find if the diurnal variation of wind speed has a significant difference, the diurnal variations of the wind speed have been further studied. To this end, the wind speed database has been divided into two sets of wind speed values. For simplicity in this case, day-time is defined between 7 AM and 7 PM, and night-time is defined as 7 PM to 7 AM. These two databases have been separately analysed, a typical diurnal representation of the wind speed is shown in Figs. 4 and 5.

- 265 - International Journal of Energy Engineering (IJEE) Dec. 2013, Vol. 3 Iss. 6, PP. 261-268

Height = 100 m 25

Wind speed during the day (m/s)during speed Wind 5

0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Time (h) Fig. 4 The diurnal speed database (day-time)

Height = 100 m 16

4 Wind speed during the night (m/s)during speed Wind

0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Time (h) Fig. 5 The diurnal speed database (night-time) A similar analysis of the diurnal wind direction has been conducted, the wind rose diagrams for both sets of wind speed, over the day and night, being depicted in Figs. 6 and 7.

Wind rose during the day

NORTH

15%

10%

5% 20 - 22 WEST EAST 18 - 20 16 - 18 14 - 16 12 - 14 10 - 12 8 - 10 6 - 8 4 - 6 2 - 4 0 - 2 SOUTH Fig. 6 The wind rose associated to the diurnal wind database (day-time)

- 266 - International Journal of Energy Engineering (IJEE) Dec. 2013, Vol. 3 Iss. 6, PP. 261-268

Wind rose during the night

NORTH

15%

10%

WEST EAST

15 - 20 10 - 15 5 - 10 0 - 5 SOUTH Fig. 7 The wind rose associated to the diurnal wind database (night-time) The Weibull parameters analytically calculated for the available data using the methods given in Section 3 are presented in Table 2. It is seen from the table that the scale factor varies between 3.9183 and 4.1987 m/s, the shape factor ranges from 1.7631 to 1.9155 for the wind speeds during the day, while during the night, the scale factor varies between 3.5523 and 3.6997 m/s, and the shape factor ranges from 1.5542 to 1.7620.

TABLE 2 WEIBULL PARAMETERS FOR DIURNAL DATABASE

Parameters Method / statistical tests MLE MOM LSM

Scale parameter a 4.1288 4.1987 3.9183

time 1.8975 1.9155 1.7631

Shape parameter b - RMBE 0.6140 0.7268 -0.9390 ay

D RRMSE 1.1600 1.2660 1.6603 Scale parameter a 3.6793 3.6997 3.5523 time Shape parameter b 1.6796 1.7620 1.5542 -

RMBE 0.7629 0.9665 -1.2718 ight

N RRMSE 1.3682 1.2900 1.9136 In Table 2 the RMBE and RRMSE values are also calculated. As can be seen, the scale and shape parameters, from the whole database and from diurnal value, have the best estimation when the MLE is used. In order to compare the estimation methods using different sample sizes of database, the above methods have been applied for the wind speed values from the months from each of the four seasons. The average seasonal Weibull parameters are presented in Table 3.

TABLE 3 WEIBULL PARAMETERS FOR SEASONS

Season Parameters / Method statistical tests MLE MOM LSM Spring Scale parameter, a 3.6447 3.3856 3.6324 Shape parameter, b 1.8502 1.8367 1.8777 RMBE 0.1800 0.1953 -0.2083 RRMSE 1.3511 1.4322 1.8451 Summer Scale parameter, a 2.7708 2.8906 3.0591 Shape parameter, b 1.9701 1.9075 1.9158 RMBE 0.2276 0.2639 -0.2891 RRMSE 1.2514 1.4129 1.8633 Autumn Scale parameter, a 3.1094 3.1594 3.5774 Shape parameter, b 2.0348 1.9924 2.1058 RMBE 0.2402 0.2618 -0.2684 RRMSE 1.2214 1.3945 1.7412 Winter Scale parameter, a 4.4682 4.1037 4.5159 Shape parameter, b 2.1584 2.1605 2.0556 RMBE 0.1695 0.1884 -0.2458 RRMSE 1.1855 1.3868 1.7015

- 267 - International Journal of Energy Engineering (IJEE) Dec. 2013, Vol. 3 Iss. 6, PP. 261-268

As can been observed, the scale and shape parameters, from the whole database and from seasonal values, have the best estimation in case of MLE. Thus, it can be summarised that the MLE is the best method used to estimate the parameters for the two-parameter Weibull distributions taking into consideration the RMBE and RRMSE as measurements of comparison, while the LMS method is the least accurate method.

V. CONCLUSIONS In practice, it is very important to describe the variation of wind speeds for optimal design of the wind generation systems. The wind variation for a typical site is usually described using the Weibull distribution. Therefore, it is very important to know the best method for parameters evaluation, with minimal errors. This study has been developed to compare the results of three methods of parameters estimation, for the same database. In the present study, hourly wind speed data of Iasi has been statistically analyzed. The probability density distributions have been derived from this database and the distributional parameters have been evaluated. It has been shown, from computational results, that method which gives the lowest values of statistical tests is the MLE, in both cases, for whole year database and for diurnal values. However, from accuracy viewpoint, the MOM and LSM of fitting Weibull function were also good methods because the methods give the close parameters as MLE and, more over, Matlab package contains functions and tools that estimate the parameters and confidence intervals for Weibull data.

ACKNOWLEDGMENT This paper was supported by the project PERFORM-ERA ‘Postdoctoral Performance for Integration in the European Research Area’ (ID-57649), financed by the European Social Fund and the Romanian Government.

REFERENCES [1] GWEC. Global Wind Statistic; 2012. [2] Alroza Khaligh, and Omar G. Onar, Energy Harvesting: Solar, Wind and Ocean Energy Conversion Systems, Taylor and Francis Group, LLC, New York (2010), pp. 101-109. [3] Gary L. Johnson, Wind Energy Systems, KS, Manhattan (2006), pp. 100-105. [4] Villanueva D., and Feijoo A., “Wind power distributions: A review of their applications”, Renewable and Sustainable Energy Reviews 14, 2010, pp. 1490-1495. [5] A.N. Celik, “A statistical analysis of wind power density based on the Weibull and Rayleigh models at the southern region of Turkey”, Renew. Energy 29, 2003, pp.593-604. [6] Razali A.M., Salih A.A., and Mahdi A.A., “Estimation Accuracy of Weibull Distribution Parameters”. Journal of Applied Sciences Research 5, 2009, pp. 790-795. [7] Nemes, M. Istrate, “Effects of wind profile in the wind energy systems performance”, 6th International Workshop on Deregulated Electricity Market Issues in South-Eastern Europe, Bled, Slovenia, 2011, pp. 95-100. [8] Bain L., and Engelhardt M., Introduction to Probability and Mathematical Statistics. Duxbury Press, California (1992). [9] Isaac Y. F. Lun, and Joseph C. Lam, “A study of Weibull parameters using long-term wind observations”. Renewable Energy Journal, vol 20, iss. 2, June 2000, pp. 145-153. [10] Lange P. M., “On the uncertainty of wind power predictions – analysis of the forecast accuracy and statistical distributions of errors”, Journal of Solar Energy Engineering, vol. 127, 2005, pp. 177-184.

Ciprian Nemes was born in Romania, on May, 1975. He received the M.Sc degree in electrical engineering and Ph.D degree in reliability engineering, from Techincal University of Iasi, Romania, in 1998 and 2005, respectively. Since 1998, he has been with Faculty of Electrical Engineering, Technical University of Iasi, where currently he is a Senior Lecturer. He is now involved in a postdoctoral research project that focus on the integration of wind and photovoltaic sources into electrical distribution networks. His research interests cover the area of electric power reliability, adequacy assessment of power systems, power system planning based on risk assessment, renewable energy sources operation and their planning.

- 268 - ELECTRONICS AND ELECTRICAL ENGINEERING ISSN 1392 – 1215 2012. No. 4(120) ELEKTRONIKA IR ELEKTROTECHNIKA

ELECTRICAL ENGINEERING T 190 ───────────────────── ELEKTROS INŽINERIJA

Statistical Analysis of Wind Turbine’s Output Power

C. Nemes, M. Istrate Department of Electric Power Engineering, Faculty of Electrical Engineering, Technical University of Iasi, Bd. Mangeron 51-53, Iasi, Romania, phone: +40 232 278683, e-mail: [email protected]

http://dx.doi.org/10.5755/j01.eee.120.4.1447

Introduction of the output power may be used to relate the degree of the intermittent behaviour of the output power. During the last years, the concerns related to the In the paper there are developed two analytical growth of the dependence on the fossil fuels and models for the first two statistical moments of the output environmental issues have increased the interest for power variable, estimating their limits and the dependence renewable energy. Unlike other renewable energy sources, on the wind profile characteristics and the operational wind energy has the highest growth among other sources. parameters of the wind turbine. Even with the major benefits, the wind energy brings some challenges in electrical power systems operation, Wind speed statistical modelling related to the wind generation capacity unpredictability, intermittence and high variability. The stability of the The wind speed is continuously changing, making it electrical power systems is based on the reliable power desirable to be described using statistical models. It is generation that is permanently balanced by the load. The widely agreed that the random behaviour of the wind variability of power from wind turbine has several negative speeds can be described by two parameters Weibull effects on the power systems operation. The electrical distribution, characterised by the following probability power output of a wind turbine depends on many factors, density and cumulative distribution functions; including the wind profile associated to the placement’s  1    v    v   site and the operational parameters of the wind turbine vf )(    exp     , (1) W   generator. The variability of the generated power increases       due to the variability of the wind, but it can be estimated  and limited through a suitable choice of the operational   v   vF exp1)(     , (2) W  parameters. The influence of the wind power generation on     the power systems operation can be controlled by reducing the standard deviation of the wind turbines’ output power where  (m/s) is the scale parameter and  (dimensionless) [1]. Therefore, the power systems must be extended with is the shape parameter of the Weibull distribution. those wind turbines that best match to the wind speed Generally, the scale parameter provides information profile and to the power system units’ structure. about the average of the wind speed profile, while the Selection of the suitable wind turbine was discussed shape parameter provides information about the deviation differently in various papers. Two important aspects must of the wind speed values around the mean. The shape and be considered in the choice of the wind turbines, namely the scale parameters are interconnected through analytical the energy production and the power system’s stability. expressions of mean and variance of Weibull distributions, Many authors focus their research to select the most given by the following expressions: suitable wind turbine generators that maximize the annual energy production, as well as the capacity factor [2,3]. vM  11)(  , (3) Capacity factor is defined as the first statistical moment of 2 2 the normalized output power. The second moment of the vVar    1121)(  , (4) output power variable is the variance that relates the where  ( ) is the Euler’s gamma function. deviation of the output power to the mean, reflecting the variability of the wind turbines’ generated power. Wind turbine’s output power modelling Therefore, the mean of the output power may be used to estimate the average energy production, while the variance The output power of a wind turbine is a function of

31 the wind speed, the power curve of turbine giving the M(P)   P(v) f (v)dv , (9) relation between the wind speed and the electrical power 0 W output. These curves come available from the wind turbine where P(v) is the power curve function and fW(v) is the manufacturers or there are plotted using recorded wind Weibull distribution of the wind speed. speed and corresponding output power data. Considering the expression of the power curve from The power curve of a pitch-regulated wind turbine is eq. (3), the mean of the output power can be written as characterized by the following three speeds: the cut-in, the rated and the cut-off speed. The wind turbine starts M (P)  P  vr A v 2  A v  A f v dv  P  vco f v dv . (10) r v  2 1 0  W r v W  generating power when the speed exceeds the cut-in speed ci r (vci). The output power increases with the wind speed The first integral of eq. (10) can be written as a sum between the cut-in speed and the rated one (vr), after that of the three similar integrals Ik, with k=1,2,3. The integral the output power remains constant at the rated power level Ik,, having the following expression, can be easily solved  (Pr). The cut-off wind speed (vco) is the maximum wind by making the change in variables y ( v ) , speed at which the turbine allows power generation, it dy     ( v )1 and respectively v    v1  , as follows being usually limited by safety constraints. Different wind generators have different output 1    vr k vr k   v   v  power performances, thus the model used to describe the Ik   ( Ak v ) fW (v)dv   ( Ak v )   exp    dv  vci vci          performances is also different. A literature survey indicates   A yr ( v1/  )k exp( y)dy k A yr y (k /  1)1 exp( y)dy . that the most used models for output power are the linear,  k y     k y    (11) the quadratic and the cubic models. The output power of a ci ci wind turbine generator can be accurately modelled using The previous integral it is common used in the the quadratic model, as results from many papers [3, 4]: statistical analysis, being known as lower incomplete gamma function, having the following structure  2 Pr  A2v  A1v  A0 , for vci  v  vr ,  t a 1  (5)  y exp(  y)dy  (a)  P(t, a) , (12) Pv   Pr , for vr  v  vco , 0  0, otherwise ,  where ( ) and P( ) are the gamma and the lower incomplete gamma functions, respectively. where the coefficients A , A and A have the following 2 1 0 The second integral of the eq. (10) represents the expressions: Weibull cumulative distribution function, evaluated 3 between v and v limits. After substituting of integration 1   v  v   r co A  2  4 ci r   , (6) 2 2   limits and their reduction to the minimum number of v  v  2vr  ci r     terms, the result became as follows  3  1  vci  vr  , (7)      A  4v  v    3v  v   2  k   v  k   v  k  1 2 ci r   ci r M (P)  P A  k  1 P r  , 1  P in  , 1  vci  vr   2vr   r   k           k0                    3 1   v  v     (13) A  v v  v  4v v  ci r   . (8)  exp((vr /) )  exp((voff /) )  . 0 2 ci ci r ci r   vci  vr   2vr     It should be mentioned that the quantity inside the Equation (5) expresses the instantaneous value of the brackets of the eq. (13) is the so-called mean power electrical power output as a function of the instantaneous coefficient (MPC) or, better known, as the capacity factor. wind speed. To consider the effect of the wind speed The mean power coefficient is defined as the ratio of the variation, the next section will consider the probability average output power over a time period to the rated power distribution function of the wind speed. of wind turbine generator, eq. (14), being a very important parameter of a wind energy conversion since it determines Proposed models for statistical moments of power the total energy production. Likewise, it should be noticed that the MPC is independent of rated power The objective of this section is to combine the output 2        power dependence on the wind speed with the variation of k  k    vr  k   vci  k  MPC   Ak   1 P   , 1  P   , 1              the wind speed at a given site, in order to find the mean k0        and the variance of the output power from a given wind   (14)  exp((vr /) )  exp((vco /) ) . turbine located at the specified site. The mean and the variance are the first two statistical The variance of the output power from a wind turbine moments that provide information on the level and is the expected value of the squared difference between the dispersion of a set of output power values. The mean value output power values and the mean of electrical power of the electrical power output of a wind turbine can be output, integrated over all possible power values estimated from its probability density function, Var(P)   P(v)  M(P) 2  f (w)dv  M(P2)  M(P)2.(15) representing the generated power at each wind speed value, 0 w integrated over all possible power values [5]. In accordance with the statistical approach, the mean of the Considering the same mathematical technique, the electrical power output is calculated as follows variance power coefficient (VPC) is defined as the ratio of

32 the variance output power to the squared of the rated power coefficients. power, as follows Table 1. Mean Power Coefficient values from analytical model 4        (AM) and Monte Carlo Simulation (MCS) k  k    vr  k   vci  k  VPC   Bk   1 P   , 1  P   , 1              MPC (%) for 1.5 xle-GE (vci=3.5m/s, vr=11.5m/s, vco=20m/s) k0           =3 =4 =5 =6   2 (16)  exp((vr /) )  exp((vco /) )  MPC ,  AM MCS AM MCS AM MCS AM MCS

2 0.5 11.311 11.233 12.848 12.771 13.841 13.834 14.504 14.568 where the new coefficients are evaluated as: B4  A2 , 1 6.875 6.847 12.325 12.479 17.294 17.260 21.328 21.332 2 2 1.5 2.354 2.404 6.760 6.862 12.972 13.004 19.851 19.797 B3  2 A2 A1 , B2  A1  2 A2 A0 , B1  2 A1 A0 and B0  A0 . 2 1.008 1.016 3.942 3.934 9.226 9.280 16.44716.430 The (14) and (16) express the relationships between 2.5 0.529 0.519 2.720 2.765 7.188 7.168 14.00813.943 cut-in, rated and cut-off speeds parameters and the mean 3 0.310 0.308 2.102 2.081 6.103 6.074 12.51012.506 and variance power coefficients. For a given wind profile, with  and  parameters known, it can be established the Table 2. Variance Power Coefficient values from analytical values of the wind turbines’ operational parameters which model (AM) and Monte Carlo Simulation (MCS) lead to a maximal mean power coefficient and/or a VPC (%) for 1.5 xle-GE (vci=3.5m/s, vr=11.5m/s, vco=20m/s) minimal variance power coefficient. =3 =4 =5 =6  AM MCS AM MCS AM MCS AM MCS Models validation and numerical example 0.5 7.789 7.822 8.787 8.643 9.427 9.398 9.856 9.902 1 3.701 3.686 7.012 7.112 9.838 9.820 11.947 12.015 In order to validate the proposed models, their results 1.5 0.588 0.601 2.548 2.492 5.787 5.804 9.287 9.259 have been compared with those given by other evaluation 2 0.103 0.113 0.775 0.781 2.725 2.606 5.949 6.035 2.5 0.028 0.027 0.302 0.309 1.330 1.328 3.652 3.642 technique, namely from the Monte Carlo simulation. Both 3 0.010 0.010 0.153 0.149 0.760 0.771 2.328 2.311 evaluation techniques were applied on a real wind turbine and a real wind speed database. The considered wind Usually, the scale parameter lies between 3-6 m/s and turbine is the 1.5 XLE GE-Energy, having 1.5 MW rated the shape parameter lies between 1.5-3. It can be seen that power, 3.5m/s cut-in, 11.5 m/s rated and 20m/s cut-off the results obtained from both methods are very close. The wind speed parameters. The wind speed database was analytical models provide comparative results with Monte collected from the north-east of Romania, for the year Carlo simulation, these proving the accuracy of the 2009, the hourly average values being recorded and developed analytical models. adjusted to the hub wind turbine height (80m). The parameters of the Weibull distribution used to fitting Analysis and discussions database have been evaluated with Maximum Likelihood Method and they were founded as the scale parameter The proposed models can be used to assess the effect =4.83 m/s and the shape parameter =1.87. of optimal selection of wind turbine parameters, based on A Matlab program has been developed in order to the maximization of the mean power coefficient, on the evaluate the mean and variance power coefficients. The variability of the output power generated. program has been structured by two main functions. First Figure 1 shows the dependence of the mean and function is developed based on the eq. (14) and (16). variance power coefficients for various normalised turbine

Second function has been developed based on Monte Carlo rated speed (vr/), for a typical turbine with (vci/vr=0.275) simulations technique (MCS). and (vco/vr=1.75) [6]. As can be seen, a maximum value of The Monte Carlo technique generates different values mean power coefficient is achieved for a local minimum of the wind speed, in accordance with their Weibull value of the variance power coefficient. Therefore, the distribution, these values being used to calculate the output wind turbine design considering the maximum value of the power, based on the power curve of the wind turbine mean power coefficient involves a minimum variability of generator. The expected values of the mean and variance the output power. power coefficients may be observed from the average and variance of all output power values, over a long number of WTG (Vci/Vr=0.275) (Vco/Vr=1.75) samples. 0.9 The simulation is stopped when a specified degree of 0.8 beta=1.5 beta=2 confidence has been achieved. The number of the 0.7 beta=2.5 simulations results from the condition that the deviation of 0.6 beta=3 the coefficient of variation of MPC and VPC around the 0.5 expected value to be under a settled value. For a settled 0.4 value of 0.01%, the convergence process has required 0.3 about 10,000 necessary samples. beta=1.5 beta=2 0.2 The capacity factor values provided by analytical Variance Coefficientand Power Mean 0.1 model and sequential Monte Carlo simulation are given in beta=2.5 beta=3 Table 1 and Table 2. 0 0 0.5 1 1.5 2 2.5 3 3.5 4 For a better comparison between models, there were Vr/alfa considered two ranges of speed wind distribution Fig. 1. The mean and variance power coefficients from Monte parameters into the evaluation of the mean and variance Carlo simulation

33 The case study shows that the turbines having a high There are presented analytical models to evaluate the mean power coefficient generate power in a lower mean and variance power coefficients for a wind turbine, intermittent behaviour. Furthermore, the model allows the located in a specific site. The results were validated using calculation of the maximum value of the output power the Monte Carlo simulations providing that the proposed variation. For the considered case, the biggest variation of models give more accurate estimations. These analytical hourly electrical power output is evaluated to be about models can be used in a preliminary evaluation of the 25% (equivalent to a standard deviation by 5%). annual energy production and of the biggest variation of generated power, based on the characteristics of the wind 0.25 beta=1.5 profile and the operational parameters of the wind turbines. beta=2 beta=2.5 The important point is that the annual energy 0.2 beta=3 maximization design rule reduces the variance of the electrical power output to a local minimum value, the

0.15 decrease of the variance being more emphasised for the wind profile having higher value of the shape parameter.

0.1 Acknowledgements Variance Power Coefficient This paper was supported by the project PERFORM-ERA 0.05 ‘Postdoctoral Performance for Integration in the European Research Area’ (ID-57649), financed by the European Social Fund and the Romanian Government. 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mean Power Coefficient References Fig. 2. The variance power coefficients as function of mean power coefficient 1. Azubalis M., Azubalis V., Slusnys D. Estimation of the Feasible Wind Power in a Smapp Power System Energy // Figure 2 illustrates the dependence between the first Electronics and Electrical Engineering. – Kaunas: two moments of the output power. As can be seen, the sites Technologija, 2011. – No. 1(107). – P. 79–72. with a lower shape parameter are characterised by the 2. Kunicina N., Galkina A., Zhiravecka A., Chaiko Y., higher mean and variance power coefficients than those Ribickis L. Increasing Efficiency of Power Supply System sites with a larger shape parameter. This is true only if the for Small Manufactures in Rural Regions using Renewable Energy Resources // Electronics and Electrical Engineering. – average wind speed is the same at each site. However, if Kaunas: Technologija, 2009. – No. 8(96). – P. 19–22. two sites are characterised by the same average wind 3. Kaigui Xie, Roy Billinton. Energy and reliability benefits of speed, the site with the lower shape parameter will have a wind energy conversion systems // Renewable Energy, 2011. larger energy production, but also, a higher intermittent – Vol. 36. – P. 1983–1988. behaviour. 4. Albadi M. H., El–Saadany E. F. Optimum turbine–site matching // Energy, 2010. – Vol. 35. – P. 3593–3602. Conclusions 5. Nemes C., Munteanu F. Development of reliability model for wind farm power generation // Advances in Electrical and The mean and variance power coefficients are Computer Engineering, 2010. – Vol. 10. – P. 24–29. important factor into the analysis of the wind energy 6. EL–Shimy M. Optimal site matching of wind turbine potential and the generating behaviour of a wind turbine generator: Case study of the Gulf of Suez region in Egypt // Renewable Energy, 2010. – Vol. 35. – P. 1870–1878. located in a specific area.

Received 2011 06 08 Accepted after revision 2011 11 13

C. Nemes, M. Istrate. Statistical Analysis of Wind Turbine’s Output Power // Electronics and Electrical Engineering. – Kaunas: Technologija, 2012. – No. 4(120). – P. 31–34. In this paper, the authors develop the analytical models for the mean and variance of the output power based on the power curve of wind turbine and the Weibull distribution of wind speed at an investigated site. In order to validate the model, the results are compared with those given by Monte Carlo simulation technique. The analytical models have the advantage to put into the evidence the effects of the wind profile and of the wind turbines’ operational parameter on the values of the statistical moments. The models can be used in the preliminary planning of the electrical power system with new energy wind sources, to evaluate the amount of energy and the variability of the generated electrical power. Ill. 2, bibl. 6, tabl. 2 (in English; abstracts in English and Lithuanian).

C. Nemes, M. Istrate. Vėjo turbinos išėjimo galios statistinė analizė // Elektronika ir elektrotechnika. – Kaunas: Technologija, 2012. – Nr. 4(120). – P. 31–34. Remiantis galios kreive ir vėjo greičio Veibulo skirstiniu tiriamojoje vietovėje, sukurti vėjo turbinos išėjimo galios vidurkio ir dispersijos analitiniai modeliai,. Siekiant patikrinti modelį, rezultatai palyginti su Monte Karlo metodu gautais rezultatais. Analitiniai modeliai yra pranašesni, nes įvertina vėjo profilio efektus ir vėjo turbinų eksploatacinius parametrus. Modeliai gali būti panaudoti preliminariam elektros energijos sistemos su naujais vėjo energijos šaltiniais planavimui, skaičiuojant generuojamos energijos kiekį ir kintamumą. Il. 2, bibl. 6, lent. 2 (anglų kalba; santraukos anglų ir lietuvių k.).

View publication stats U.P.B. Sci. Bull., Series C, Vol. 74, Iss. 1, 2012 ISSN 1454-234x

OPERATIONAL PARAMETERS EVALUATION FOR OPTIMAL WIND ENERGY SYSTEMS DEVELOPMENT

Ciprian NEMEŞ1, Florin MUNTEANU2

A method based on maximization of normalised average power is proposed to establish the operational parameters of wind turbine. The method is a compromise between those operational parameters that provide the highest values of the capacity factor and the rated power. The method can be used in developing stage of wind energy system, being a useful tool to make an adequate choice of the wind turbine with optimal operational parameters, which yield the highest amount of energy.

Keywords: Wind energy, capacity factor, normalized average power.

1. Introduction

In the last decades, a growing interest in renewable energy resources has been observed. Unlike other renewable energy sources, wind energy has become competitive with conventional power generation sources and therefore the application of wind turbine generators has the highest growth among other sources. Wind is one of fastest growing energy source and it is considered as an important alternative to conventional power generating sources. The energy production from a wind energy system depends on many factors. These factors include the characteristics of the wind profile, and most importantly, the operational parameters of the wind turbine generator itself. The suitability of a wind turbine to a site involves designing of wind turbine’s operational parameters exactly according to the wind characteristics of the site. One more applicable method is to match a specified site with a suitable wind turbine among existing ones. Numerous criteria have been proposed for the suitability procedure. Some authors determined the wind turbine generator’s parameters at maximum capacity factor or at maximum density rated output power [1,2], between these two methods there is a complementarity. Thus, if a higher capacity factor is desired, then turbine’s rated speed cannot be chosen too high. In this case, even if the wind turbine operates at rated power for most of the time, not too much of energy in higher speed wind will be captured. On the other hand, if the capacity factor is

1 Lecturer, Electrical Engineering Faculty, Technical University of Iasi, Romania 2 Prof., Electrical Engineering Faculty, Technical University of Iasi, Romania 224 Ciprian Nemes, Florin Munteanu

chosen too low, then although the wind turbine can generate a higher amount of output power at rated speed, the turbine will seldom operate at its rated power and will lose much of energy of lower speed wind. A wind turbine should always extract the highest possible power from the wind. Therefore, in this paper is presented a method to identify optimal speed parameters of the wind turbines in order to yield maximum energy production. This means that the rated speed has to be selected such that the turbine delivers higher average power.

2. Wind speed modelling

Wind is a turbulent movement mass of air resulted from the differential pressure at different locations on the earth surface. The wind speed is continuously changing, making it desirable to be described by the statistical models. It is widely agreed that the randomly behaviour of wind speeds can be described by two parameters Weibull distribution. For Weibull distribution, the probability density function and the cumulative distribution function are given by:

β −1 ⎡ β ⎤ ⎡ β ⎤ β ⎛ v ⎞ ⎛ v ⎞ , ⎛ v ⎞ (1) fW (v) = ⎜ ⎟ exp ⎢− ⎜ ⎟ ⎥ F (v) = 1 − exp ⎢− ⎜ ⎟ ⎥ α α α W α ⎝ ⎠ ⎣⎢ ⎝ ⎠ ⎦⎥ ⎣⎢ ⎝ ⎠ ⎦⎥ where α (m/s) is the scale parameter and β (dimensionless) is the shape parameter of the Weibull distribution. The shape and the scale parameters are related to the average wind speed by M (v) = αΓ(1+1 β) and to the variance of wind speed by Var(v) = α2 [Γ(1+ 2 β)− (Γ(1+1 β))2 ]. In many studies, the Rayleigh distribution is often used, the shape parameter being chosen to 2. The estimated parameters of the Weibull distribution can be found using different estimation methods [3,4].

3. Wind turbine’s output power modelling

The output power of a wind turbine is a function of the wind speed, the power curve of turbine gives the relation between the wind speed and the electrical output power. These curves come available from the wind turbine manufacturers or there are plotted using recorded wind speed and corresponding output power data. The power curve of a pitch-regulated wind turbine is characterized by the following three speeds: the cut-in, the rated and the cut-off speed. The wind turbine starts generating power when the speed exceeds the cut-in speed (vci). The output power increases with the wind speed between the cut-in speed and the rated one (vr), after that the output power remains constant at the rated power level (Pr). The cut-off wind speed (vco) is the maximum wind speed at which the turbine Operational parameters evaluation for optimal wind energy systems development 225

allows power generation, being usually limited by safety constraints. Thus, the power curve of a wind turbine can be modelled by four specific parameters, namely the cut-in speed, the rated speed, the cut-off speed and the rated power. Different wind generators have different output power performances, thus the model used to describe the performances is also different. A literature survey indicates that the most used models for output power are the linear, the quadratic and the cubic models. The electric output power of a wind turbine generator can be accurately modelled using quadratic model, originally proposed in [5] and used in many other papers [1,6]:

⎧ 2 Pr ⋅ (A2 v + A1v + A0 ) for vci < v < vr ⎪ , (2) P()v = ⎨ Pr for vr < v < vco ⎪ 0 otherwise ⎩⎪ where the coefficients A2, A1 and A0 were first time calculated in [7], having the following expressions:

3 ⎡ ⎛ v v ⎞ ⎤ 1 ci + r , A2 = ⎢2 − 4⎜ ⎟ ⎥ 2 ⎢ ⎜ 2v ⎟ ⎥ ()vci − vr ⎣ ⎝ r ⎠ ⎦

3 ⎡ ⎛ v v ⎞ ⎤ 1 ci + r , A1 = ⎢4()vci + vr ⎜ ⎟ − 3()vci + vr ⎥ 2 ⎢ ⎜ 2v ⎟ ⎥ ()vci − vr ⎣ ⎝ r ⎠ ⎦ 3 ⎡ ⎛ v + v ⎞ ⎤ 1 ci r . A0 = ⎢vci ()vci + vr − 4vci vr ⎜ ⎟ ⎥ 2 ⎢ ⎜ 2v ⎟ ⎥ ()vci − vr ⎣ ⎝ r ⎠ ⎦ Equation (2) expresses the instantaneous value of the electrical output power as a function of the instantaneous wind speed.

4. Wind turbine’s performance indicators

A measure of the wind turbine performance is given by the average power value, being a very important parameter of wind energy conversion, determining the total energy production. The average power is related to rated output power by capacity factor. The capacity factor is defined as the ratio of the energy generated in real operation, over a time period, to the energy that could have been generated at constant rated power operation during the same period. Considering the capacity factor definition, the average of output power may be expressed as product between rated power (Pr) and capacity factor (CF) value: Pavg = Pr ⋅CF (3) 226 Ciprian Nemes, Florin Munteanu

The rated power output at rated wind speed vr is given by:

1 P = ⋅η ⋅ ρ ⋅ A⋅v3 (4) r 2 r r

3 where ηr is the rated overall efficiency at rated power, ρ (kg/m ) is the air density, 2 and A (m ) is the turbine swept area. The rated overall efficiency, ηt=Cpr⋅ηmr⋅ηgr, includes the coefficient of performance, the mechanical transmission efficiency and the generator efficiency, all evaluated at rated power. The capacity factor can be calculated by integrating the normalised power curve multiplied by the wind speed distribution, over all wind speed values. Thus, the capacity factor can be derived based on the power curve of wind turbine generator and the parameters of wind speed distribution. Considering the quadratic model of power curve and Weibull distribution of wind speed, the authors have been developed an analytical model for capacity factor evaluation, expressed by the following equation:

2 ⎡ ⎛ β ⎞ ⎛ β ⎞⎤ k ⎛ k ⎞ ⎜⎛ vr ⎞ k ⎟ ⎜⎛ vci ⎞ k ⎟ β β CF = ∑ Ak ⋅α ⋅Γ⎜ +1⎟⋅⎢P ⎜ ⎟ , +1 − P ⎜ ⎟ , +1 ⎥ + exp(−(vr /α) ) − exp(−(vco /α) ) β ⎢ ⎜⎝ α ⎠ β ⎟ ⎜ α β ⎟⎥ k =0 ⎝ ⎠ ⎣ ⎝ ⎠ ⎝⎝ ⎠ ⎠⎦ (5) where Γ( ) and P( ) are the gamma and the lower incomplete gamma functions. The details of the capacity factor evaluation can be founded in [8]. Therefore, the average electrical output power can be expressed considering the above relationship:

1 P = ⋅η ⋅ ρ ⋅ A⋅v3 ⋅CF (6) avg 2 r r

In equation (6), the choice of rated wind speed, which maximize the average power, will not depend on the rated overall efficiency, the air density, or the cross sectional area of wind swept by turbine blades. Thus, these quantities can be normalized out. Also, since the capacity factor is expressed in normalized wind speeds, it is convenient to do likewise for average power by dividing to α in 3 order to get the term (vr/α) . Therefore, a new indicator is obtained. This indicator, namely the normalised average power (PN), takes both capacity factor and rated power into account, being utilized in this study to estimate the optimal rated speed parameter [9,10]. This indicator is defined as the normalised average power:

Operational parameters evaluation for optimal wind energy systems development 227

3 Pavg ⎛ v ⎞ P = = CF ⋅ r (7) N 3 ⎜ ⎟ 1/ 2⋅ηr ⋅ ρ ⋅ A⋅α ⎝ α ⎠

The equation (7) expresses the effect of the cut-in, the rated and the cut-off speeds, respectively the wind distribution parameters, on the normalised average power values. Thus, for a given wind profile, with α and β parameters known, the values of the wind turbines’ operational parameters which lead to the maximum normalised average power, can be established. From the three speed operational parameters, the rated speed is the most important for the wind turbine design. The cut-in speed is that fraction of the rated speed, which contains enough power to cover all the wind turbine’s strengths. On the other hand, the choice of the cut-off speed depends on the capacity of the turbine control system to maintain a constant power output for that wind speed values over the rated speed [11,12]. Thus, it has been shown that the rated wind speed has a significant effect on the other speed parameters of the wind turbine, and also, in the normalised average power value.

Pr Power Outpu

Wind speed

Fig. 1. Effect of rated wind speed increasing on rated power value

If the rated wind speed is chosen too low, too much energy generated under higher wind speeds will be lost. Instead, if the rated speed is too high, the turbine seldom operates at its rated power and will lose too much energy at lower speed winds. Thence, the average power output of wind turbine can reach a maximum value at a specific rated wind speed, as is depicted in figure 1.

5. Matching wind turbine generators with a wind profile. A study case

In this section, a study for optimal operational parameters’ evaluation is developed, based on maximization of the normalized average power. In this study, the rated speed vr is chosen to maximize PN with the prespecified vci and vco 228 Ciprian Nemes, Florin Munteanu

values. The typical commercial wind turbines have the values of vci between 3 and 4 m/s and vco between 20 and 25 m/s [13]. For a numerical analysis, a real wind speed database is used. The wind speed database was collected from the north-east of Romania, for the year 2008, the hourly average values being recorded [14]. The parameters of the Weibull distribution, used for fitting the database, have been evaluated with Maximum Likelihood Method and they were founded as the scale parameter α=4.9399m/s and the shape parameter β=1.8656. With specified vci=3.5 m/s and vco=20 m/s, the capacity factor and rated power curves are calculated and shown in figure 2. It can be seen that the capacity factor and the rated power curves reach their peaks at different rated speeds, involving that a compromise should be made to achieve the best result. Since normalised average power curve takes capacity factor and rated power into account, its peak can be used to specify the optimal rated speed of the wind turbine.

CF 1.8 Prat PN 1.6

1.4 WT2 WT1 1.2 X: 2.22 1 Y: 1.219

0.8 X: 0.7 Y: 0.5922 0.6 X: 0.7 Y: 0.343 0.4 X: 2.22 X: 2.63 Y: 0.1114 0.2 Y: 0.06466 X: 0.7 X: 1.82 Y: 0.2031 0 Y: 0.1901 0 0.5 1 1.5 2 2.5 3 3.5 Vrat/alfa Fig. 2. The capacity factor, rated power and normalised average power curves

To determine the optimum wind turbine that best matches the analysed wind profile, the capacity factor and the normalised power are calculated by varying normalised power rated (vr/α) and the interest points from curves are summarized in Table 1. Table 1 Estimated performance indicators Vr/α Vr Vci Vco PN Prat CF At CF max 0.7 3.5 3.5 20 0.2031 0.343 0.5922 At PN max 2.22 10.966 3.5 20 1.219 10.941 0.1114

Operational parameters evaluation for optimal wind energy systems development 229

From Table 1, it can be seen that the normalized power is the greatest at vr/α= 2.22, therefore, the optimum rated wind speed is then 10.97 m/s. To evaluate the accuracy of the optimum rated wind speed, in the following, it will be evaluated the average power for two wind turbines with different rated speeds, for which an optimization for matching to the analysed wind profile is required. It is considered that both wind turbines have the 1 MW rated power, 3.5 m/s cut-in speed, 20 m/s cut-out speed, 100 m hub’s height, but the first one has the 9 m/s rated speed and the second one has the 13 m/s rated speed. The average power values for unmodified wind turbines for analysed wind profile depend on their rated power and the capacity factor. As can be seen in figure 2, for the first wind turbine, the capacity factor is CF1=0.1901 and for the second one is CF2=0.0646. Considering these values for capacity factor, the averages power for unmodified wind turbines are PavgWT1=1000×0.1901=190.1 W and PavgWT2=1000×0.0646=64.6 kW, respectively. Considering the optimization of wind turbines, the rated speed and the rated power will be adjusted to optimal values. Therefore, for first wind turbine, the rated power, assuming all efficiencies remains the same, will just be in the 3 ratio of the cube of the wind speeds, namely Prat1O=1000×(10.97/9) = 1810.9 kW. For optimum rated speed wind speed, the capacity factor is CF1O=0.1114. These values for rated power and capacity factor are used to calculate the turbine average power output, PavgWT1O=1805.9×0.1114=201.73 kW The same computations have been done for the second wind turbine. Thus, the rated power, assuming all efficiencies remains the same, it is 3 Prat2O=1000×(10.97/13) =600.9 kW, but it will function with a capacity factor equal with 0.1114. Under these conditions the turbine average power output is PavgWT2O=66.93 kW. Thus, as can be seen, for both wind turbines under consideration, the highest average power is determined for optimal rated speed.

6. Conclusions

A method of site matching of wind turbine generator problem based on normalized average power maximization is presented in this paper. The method is proposed to determine the operational parameters of the wind turbine, considering the maximization of energy production. The method is a useful tool to make an adequate choice of a wind turbine generator with optimum speed parameters to give higher energy. A real wind data base has been evaluated to estimate the optimal wind turbine generator power curve and optimal operational parameters in order to yield the highest energy production. 230 Ciprian Nemes, Florin Munteanu

It is found that the optimal rated speed of a wind turbine, that will be placed in site under consideration must, have about 11 m/s. thus, a wind turbine with the 3.5 m/s cut-in speed, 11 m/s rated speed and 20 m/s cut-off speed will allow yielding highest energy production.

Acknowledgement

This paper was supported by the project PERFORM-ERA "Postdoctoral Performance for Integration in the European Research Area" (ID-57649), financed by the European Social Fund and the Romanian Government.

REFERENCES

[1]. M.H. Albadia, E.F. El-Saadanyb, “New method for estimating CF of pitch-regulated wind turbines” Electric Power Systems Research, vol. 80, 2010, pp. 1182–1188. [2]. Ahmed R. Abul’Wafa, “Matching wind turbine generators with wind regime in Egypt”, Electric Power Systems Research, vol. 81, 2011, pp. 894–898. [3]. Ahmad Mahir Razali, Ali A. Salih, Asaad A. Mahdi, “Estimation Accuracy of Weibull Distribution Parameters,” Journal of Applied Sciences Research, vol. 5, 2009, pp. 790-795. [4]. Seyit A. Akdag, Ali Dinler, “A new method to estimate Weibull parameters for wind energy applications”, Energy Conversion and Management, vol. 50, 2009, pp. 1761–1766. [5]. C.G. Justus, W.R. Hargraves, A. Yalcin, “Nationwide assessment of potential output from wind- powered generators”, Journal Appl. Meteorol., vol. 15, 1976, pp. 673– 678. [6]. M.H. Albadi, E.F. El-Saadany, “Optimum turbine-site matching” Energy, vol. 35, 2010, pp. 3593– 3602. [7]. Giorsetto P, Utsurogi KF., “Development of a new procedure for reliability modeling of wind turbine generators”. IEEE Trans. Power App. Syst., vol. 102, 1983, pp. 134-143. [8]. C. Nemes, F. Munteanu “A probabilistic approach of the wind energy system performance”, 12th WSEAS International Conference on Mathematical Methods and Computational Techniques in Electrical Engineering Conference, Timişoara, Romania, 2010, pp. 161-121. [9]. M. EL-Shimy, “Optimal site matching of wind turbine generator: Case study of the Gulf of Suez region in Egypt”, Renewable Energy, vol. 35, 2010, pp. 1870–1878. [10]. R. D. Prasad, ”A case study for energy output using a single wind turbine and a hybrid system for Vadravadra Site in Fiji Islands” The Online Journal on Power and Energy Engineering (OJPEE), vol.1, no.1, January 2010, pp. 22-25. [11]. Fawzi A.L. Jowder, “Wind power analysis and site matching of wind turbine generators in Kingdom of Bahrain”, Applied Energy, vol. 86, 2009, pp. 538–545. [12]. M.Q. Lee, C.N. Lu, H.S. Huang, “Reliability and cost analyses of electricity collection systems of a marine current farm. - A Taiwanese case study” Renewable and Sustainable Energy Reviews, vol. 13, 2009, pp. 2012–2021. [13]. Tai-Her Yeh, Li Wang, Senior, “Benefit analysis of wind turbine generators using different economic-cost methods”, The 14th International Conference on Intelligent System Applications to Power Systems, Kaohsiung, Taiwan, Nov. 2007, pp. 359-364. [14]. Ciprian Nemeş, ”A comparative analysis of wind speed distribution evaluation”, Bulletin of the Polytechnic Institute of Iaşi, tome LVII (LXI), fasc.2, 2011, pp. 145-151.

View publication stats 2012 International Conference and Exposition on Electrical and Power Engineering (EPE 2012), 25-27 October, Iasi, Romania Adequacy Indices to Evaluate the Impact of Photovoltaic Generation on Balancing and Reserve Ancillary Service Markets

Giuseppe Marco Tina Carmelo Brunetto Ciprian Nemes Technical University of Catania, ENEL Roma, Technical University of Iasi, Catania, Italy Roma, Italy Iasi, Romania [email protected] [email protected] [email protected]

Abstract—Ancillary services associated to a power system are those demanded load and generating capacity to be characterized by services that support the Transmission System Operator to ensure uncertainties, especially due to the variability of renewable the requirements concerning the safe, security and reliability of the sources. Therefore, the integration of renewable energy adds a power system. Real-time behaviour of these services can be assessed new dimension in the approach of generation adequacy. using the adequacy indices of generation system. The main objective of this paper is to analyse the evolution in time of hourly Taking into account that the generation power from values of tertiary reserve margin and hourly values of adequacy renewable sources and demanded load are characterized by a indices associated to a hybrid system that contain a conventional higher degree of volatility, it should be noted that the reserve unit and a photovoltaic system. The hybrid system is involved in a margin as well as the adequacy indices are also volatile in time. study in order to supply differed load profiles, characterised by the In this order, these parameters should be evaluated at the same amount of demanded energy. system level, but due to the complexity of generation system, they can be analysed separately to subsystems levels. Thus, this Keywords- solar energy; hourly clearness index; adequacy indices study is conducted in order to evaluate and analyse the dependence between the hourly values of reserve margin and I. INTRODUCTION hourly adequacy indices, associated to a hybrid system The reliability of power system is one of the three main composed of a programmable conventional unit and a non- requirements of the energy market liberalization, besides the programmable photovoltaic system, designed to supply competition and sustainability. In general, the reliability different load profiles. associated with a power system is a measure of the ability of The study conducted in this paper is presented according to the system to generate and supply electrical energy. The the following sections: Section II presents a brief review about reliability requires a continuous and almost instantaneous operating reserve requirement in power systems. Section III is balance between the generated power and the demanded load focused on main adequacy indices and also of main probability of system. This balance can be achieved based on the real time density functions associated to electrical output power and market coordination and also on the ancillary services, by demanded load involved to evaluate the adequacy indices. In keeping the generation power above to the expected load. This section IV, a solar measurement database is analyzed over balance could be ensured in the short-term through a proper different time frame intervals, in order to establish the upper assessment of the operating reserve margin and, in the and average values of hourly clearness index, these values medium-long term, also through the generation adequacy being used to establish the probabilistic model of photovoltaic assessment. Generation adequacy is not a new issue, but in the system’s output power. Then, the hourly margin of reserve and last decades since a growing interest in renewable energy hourly values of LOLP based on three scenarios related to load resources has been observed, it is differently managed. Even if profile are evaluated and analyzed in section V. Finally, the electric energy from renewable energy sources brings various main conclusions of this paper are given in section VI. benefits to power system, due to the variable and intermittent II. OPERATING RESERVE REQUIREMENTS IN POWER behaviour of many renewable resources, their integration into SYSTEMS electric grid leads to new challenges in the system operation. If in traditionally generating systems composed only of In Europe according to the “Policy 1” of Entsoe-E [1], conventional energy sources, only the demanded load has been TSOs have to respect different requirements in order to ensure known to be characterized by the uncertainties, while the the balance between electricity production and demand during generating capacity was characterized by a known and power delivery. In particular the power/frequency reserve is foreseeable reliability of generation units; nowadays, the divided into three different cluster depending on the time integration of renewable energy sources leads as both response, as it has been reported in Fig.1.

level and the time needed to reach it. Therefore generators have to satisfy minimum requirements both for time response and for power margin. A typical value of tertiary reserve demand is about 7-8% of the system daily peak-load. The correct amount of tertiary reserve is determined taking into account both a probabilistic and a deterministic approach. The last one is considered especially when the TSO has to bear an outage of a power plant or a transmission line in a certain area (so-called N-1criterium). It is quite simple because the amount of reserve is equal to the maximum power of the power plant (or the most used interconnector) expected in operation. The probabilistic approach is mainly adopted in order to assess, for each relevant period (i.e. typically 1 hour), the contribution to the tertiary reserve margin (TRM) of uncertainty in load forecasting, as it has been reported in the Figure 1. Frequency reserve (primary, secondary and tertiary). relation 1.

The primary reserve is necessary to provide the first ⋅⋅= LOADFTRML (1) automatic response to the deviation of frequency in order to where: maintain the electrical system balanced. The generators are : Standard deviation of the peak-load forecast; activated immediately after the frequency perturbation and F: Cumulative function of the normal standard distribution their contribution expired after 30 sec. They have to provide to with a certain probability (e.g. = 0.997 in Italy); the TSO a certain power margin both for up and down LOAD: forecasted system load. regulation (in Italy this margin is at least equal to 1.5% of rated power, except for islands where this percentage is equal to 10%) With a high penetration of intermittent renewable [2]. generation also their effect has to be taken into account. TSOs The secondary power/frequency regulation tries to recover can adopt a similar approach to the load. In Italy for example the system frequency to the rated value keeping also the cross also for intermittent generation (e.g. wind, PV, etc.) is border power flows to their scheduled value. The generators considered an error corresponding to 99,7 % of probability. are activated automatically by a TSO regulator that sends them An important issue is that the contributions to the tertiary a certain power set-point. The secondary reserve contribution reserve of load forecast and of intermittent generation are expires after 15 min. According to the Entso-E considered totally independent. Among different intermittent recommendations, the minimum amount of secondary reserve power sources the Italian TSO [2] uses different approach in is depending on the electrical load as it has been reported in Fig. forecasting between large wind power plant (i.e. neural 2 (in Italy the minimum margin accepted for secondary reserve network approach combined with weather parameters forecast) is equal to 6% of rated power, but it can be higher depending and small wind with other power sources. on the generator dynamic performances) [2]. These different methodologies lead to an overestimate of tertiary reserve demand. Therefore an approach with a more sophisticated probabilistic technique (e.g. convolution) can allow an efficient assessment of reserve reducing the amount of ancillary services bought in the market.

III. RELIABILITY INDICES AND CONVOLUTION TECHNIQUES The generating system adequacy evaluation process focuses on the balance between generating capacity and demanded load. The most common index used for generating system adequacy is the Loss of Load Probability (LOLP) [3,4]. LOLP (%) indicates the probability that the demanded load has to exceed the available generating capacity for a given period of time. Figure 2. Secondary regulation demand as a function of area load. A. Adequancy indicese evaluation technique The tertiary reserve is essential to recover in a stable way The analytical method used in this paper for assessing the the system balance for example after a shutdown of a power generation system adequacy, is based on convolution technique. plant or owing to a wrong load forecast. Moreover it allows to The basic approach to evaluate the adequacy of a generating restore the margin of secondary reserve in order to keep the system consists in development of two probabilistic models, system safe for another perturbation. Power plants are activated one for generating capacity and another one for demanded load. by TSO with suitable messages which establish a certain power Then, the capacity and load models are convolved to create the

946 system’s reserve margin variable, defined as the difference C. Probabilistic model of load profile between the available generating capacity and demanded load. The demanded load in power system is variable in time. Adequacy indices can be obtained by observing the available There is no one unique profile or mathematical equation that system’s reserve margin variable. It is clear that the system can be adopted to represent the load characteristic curve. Thus, encountered several values of generating capacity which are three different models related to load profile will be adopted in not enough to satisfy the demanded load, obtaining a negative this paper. One of them will have a smooth profile, having the domain of system reserve margin. A positive reserve denotes same values of load for all hours of day, the second one will be that the system generation is sufficient to meet the system load, considered to have a smooth profile but only for daily hours, while a negative margin implies that the system load is not and the last one will have a peak profile, with a maximum served. value on the midday. For hourly load values, a normal B. Probabilistic model of photovoltaic system’s output power distribution function will be adopted, with a given standard deviation around to average value of the hourly load. Thus, the The electrical output power of a photovoltaic system L depends on many factors, especially by the environmental probability density function of the hourly load, j , can be conditions associated to the placement’s site. An important expressed as follows: factor that affects the value of electrical output power is the C S 1 − LL jj clearness index value. This index provides information Lf = D− T j )( expD T (3) concerning the real amount of solar radiation falling on the 2 E 2 2 U earth surface compared with the available extraterrestrial solar radiation. Actually, the estimation of this index requires the where L j is the hourly average load and σ is the standard solar data measurements. deviation of load. Thus, the probability density function of the photovoltaic electrical output power can be derived knowing the clearness IV. SOLAR MEASUREMENTS DATABASE index distribution, the maximum available radiation, and The solar data behind these studies has been recorded over technical data about photovoltaic panel, respectively. The 10 minutes interval on the “Systèmes Physiques de present study is conducted based on the analytical expression l’Environnement” Laboratory, Ajaccio (Corsica, France), the of probability density function of electrical output power recorded data covering the period between 1999 and 2003. The generated by a photovoltaic system, developed based on the meteorological station is located on Mediterranean seaside, at probability density function of clearness index proposed by 41°55' north latitude, 8°48' east longitude and 75 m above the Holland and Huget [5]. In Holland and Huget distribution the sea level [9,10]. This section focuses on an analysis of hourly variables that affect the probability density function values are clearness index values recorded over hourly time frame interval. only upper and average values of hourly clearness index. In [6, The measurements available in the original database are 7, 8] detailed analytical approaches are developed based on the characterized by 10 minutes acquisition intervals, thus the analytical relation between the photovoltaic output power and original database has collected and recorded six values for each the irradiation level. Thus, the probability density function of hour between 4 and 20 o’clock, every day. Thus, in the paper, monthly hourly output power of a photovoltaic system can be 37230 data of clearness index have been used for each year expressed as follows: over the whole five years period. The clearness index values over 10 minutes time frame have been drawn out and involved I Δ± C ,, jmjm S in a database transformation. Thus, an hourly frame database L DkC − T jm , D tujm T , () Δ± L E 2 U ,, jmjm has been evaluated as the average value of measured values ⋅ e 2 , L ⋅⋅⋅ TAk ′ Δ⋅ over one hour frame interval. Then, the database has been L ,,, jmjmjmctu (2) ()Pf jmP = J analysed in order to evaluate the upper and average values over , jm , ∈ []()kPPfor L , jm ,0 tu L the whole period under study. The average values of clearness L0 index are the same for both 10 minutes and hourly time frame KL otherwise databases, while the upper values seem obvious to have the higher values for the 10 minutes time frame database than ⋅ P T jm 2 4 , jm , where: ()P α −=Δ ; α jm = ; those from hourly database. Based on hourly time frame ,, jmjm , jm η ⋅ ′ ⋅ AT , T ′ ,, cjmjm , jm database, the upper and average values of the clearness index have been calculated for every hour of the month, covering the , TT ′ jmjm are some parameters used in solar irradiation ,, whole five-year period. An example is shown in Fig. 3, where C λ evaluation; , jm and , jm are the Holland and Huget the upper and average values of clearness index over 10 and 60 minutes time frames have been drawn in same diagram, for distribution parameters; plus/minus signs is used for T > 0 , jm summer and winter solstices months (June and December) and T′ < T > T′ ≥ and , jm 0 , respectively , jm 0 and , jm 0 . More for spring and autumn equinoxes months (March and September). These values indicate the maximum values of details and analytical expressions of parameters involved in the clearness index measured for a given hour, for all days of each probability density function are reported in [6,7]. month, considering the whole five years period.

947

March June hours (annual), the value of power being around 39.5 W 1 1 and a load factor equal with 100%; − the second load profile has been considered to have also a smooth profile, but only for daily hours (i.e. between 6 and 0.5 0.5 18 o’clock); for this profile, the hourly average values are around 72.94 W, with a load factor 54.16%; 0 0 − the third load profile has been considered to be composed 0 5 10 15 20 0 5 10 15 20 Hour Hour from a base smooth profile for the entire day, with about September December 10% from daily average load, and the rest of energy has 1 1 been distributed so as to reach a peak of load at midday, the peak values reaching 75.06 W with a load factor

0.5 0.5 52.63%. For all profiles, the values of the demanded load have been Upper and average values of hourly clearness index thought to be normally distributed in an interval with 5% 0 0 0 5 10 15 20 0 5 10 15 20 standard deviation. Thus, the distribution frequencies of load Hour Hour are evaluated for each load profile, in accordance with relation

(3). Fig. 4 depicts the daily load features for all three cases of Figure 3. Upper values of hourly clearness index (:for 10 min. time frame interval, : for 60 min. time frame interval ) and average values load profiles, the daily as well as the annual demanded energy (*) of hourly clearness index. being the same.

In order to evaluate the probability density function of 80 photovoltaic output power, there is required the linear 40 correlation between the hourly diffuse fraction and hourly (W) Load1 0 clearness index, this correlation has been found to be gave by 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Hour = 52110391 ⋅ k.-.k t . More analysis of this solar database and also the numerical values of hourly clearness index from 80 measurement database are reported in [11]. 40 Load2 (W) Load2 0 V. NUMERICAL ANALYSIS 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Hour This section focuses on a numerical analysis of hourly tertiary reserve margins and hourly values of adequacy index 80 associated to a hybrid system composed by a conventional 40

programmable unit and a non-programmable photovoltaic (W) Load3 0 system, both sources supplying same amount of energy on a 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 load profile. In first part of this section, the tertiary reserve Hour margin values are calculated based on probabilistic approach Figure 4. Daily load profiles: hourly average values (bar graph) and variation reported in section II, while in the second part of the section, intervals (dotted lines). the hourly values of LOLP index are calculated by convolving the probability density functions of hourly output power of The probability density functions of the photovoltaic hybrid system and of hourly demanded load, respectively. electrical output power, based on relation (2) have been In order to define the daily load profiles, the annual amount adjusted with upper and average values of clearness index from of global solar radiation on horizontal surface has been hourly frame database, thus a set of hourly monthly calculated as sum of all values from the original database, and distributions associated to output power have been drawn and it has been found 1441.9 kWh/m2. This amount of irradiation, involved in study. It should be noted that the probability falling on one square meter of a photovoltaic panel with density functions of hourly output power are affected only by efficiency around 12%, could produce an annual energy around the upper and average values of hourly clearness index, which 173.03 kWh. Same amount of energy is also produced by the means that different probability density functions are obtained conventional unit. The annual amount of energy produced by for different hours and different months, respectively. the hybrid system (346.06 kWh) is intended to supply different Based on the above probability density functions, the load profiles. Thus, three types of load profiles with the same tertiary reserve margin has been evaluated as the hourly annual energy have been adopted. For each load profile, the monthly values. The hourly values of reserve margin have been computed using the probabilistic approach, based on the load factor (kL) defined as the ratio between the average power and the peak power over a period of time has been calculated. uncertainties of demanded load and generated power. Thus, for The three load profiles are presented below: each previous load profiles, the hourly values of tertiary − the first load profile has been considered to be a smooth reserve margin have been computed based on the cumulative load profile, with same hourly average values for all 8760 function of the normal standard distribution for = 0.997, on standard deviation of load equal with 5%, and also on hourly

948

values of load. For instance, for May to 9:00 A.M., for profile, to 72.94 W for second profile and to 57.28 W for third previous load profiles, the load values are 39.5 W, 72.93 W and load profile, respectively, with a 5% standard deviation. 57.28 W respectively, all values being normally distributed with 5% standard deviation. Therefore, based on relation (1), Interference of power and load pdf-s, for May 9:00 AM the tertiary reserve margins for May 9:00 A.M. have been 0.2 Load 2 pdf found TRML1=5.37 W, TRML2=9.89 W and TRML3=7.76 W, 0.18 Hourly power pdf Load 1 pdf respectively. As in the previous case, for each hourly monthly 0.16 probability density function of generated power, the mean and Load 3 pdf standard deviation of generated power from photovoltaic 0.14 system has been computed. For May 9:00 A.M., the probability 0.12

density function based on hourly frame database has the mean 0.1 of 69.38 W and standard deviation of 20.48%. Based on similar relation as that for load, the tertiary reserve margin for 0.08 0.06

generated power has been found TRMP60=39.04 W. The Probab ility desity functions

numerical values of tertiary reserve margin associated to hybrid 0.04 system have been evaluated as probabilistic requirement level, 0.02 the hourly values for previous load profiles are TRM1=44.41 W 0 for first profile, TRM2=48.93 W for second profile, and 0 20 40 60 80 100 120 TRM3= 46.80 W for third load profile. Power (W) Similar calculations like those presented above have been Figure 5. Common view of generate power and demanded load pdf-s, conducted every month and every hour, between sunrise and for May, 9:00 AM. sunset, for each previous load profile. However, in order to not exceed the paper limitations, the numerical results of tertiary For instance, Fig. 6 indicates the values of LOLP index, for reserve margin are reported in Table I only for first load profile. May 9:00, based on cumulative distribution functions of above three load profiles and on cumulative distribution function of TABLE I. HOURLY MONTHLY VALUES OF TERTIARY RESERVE MARGIN (W), generated power, considering the upper and average values of FOR LOAD PROFILE 1 clearness index from the hourly frame database. The hourly Month values of LOLP are LOLP =0.0077 for first profile, Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1 Hour LOLP =0.1039 for second profile, and LOLP = 0.0394 for 6 25.12 25.12 25.12 25.12 25.12 25.12 25.12 25.12 25.12 25.12 25.12 25.12 2 3 7 25.12 25.12 25.12 25.12 29.90 12.79 25.12 25.12 25.12 25.12 25.12 25.12 third load profile, respectively. 8 25.12 25.12 33.74 28.87 10.48 0.00 0.00 0.00 25.12 25.12 25.12 25.12 9 25.12 37.38 21.73 23.00 0.00 0.00 0.00 0.00 6.23 29.68 25.12 25.12 Interference of power and load cdf-s, for May 9.00 AM 10 39.17 33.55 14.05 14.43 0.00 0.00 0.00 0.00 0.00 31.86 37.87 37.43 1 11 44.05 39.04 5.95 1.27 0.00 0.00 0.00 0.00 0.00 18.46 40.59 39.13 12 40.14 30.50 2.58 0.00 0.00 0.00 0.00 0.00 0.00 15.06 41.75 40.86 0.9 L1 L3 L2 13 39.51 40.51 14.32 2.28 0.00 0.00 0.00 0.00 0.00 18.75 39.56 39.66 0.8 14 38.80 41.31 30.71 15.43 0.00 0.00 0.00 0.00 9.92 25.35 37.84 38.08 15 25.12 35.98 33.87 30.36 0.00 0.00 0.00 0.00 14.39 35.04 25.12 25.12 0.7 16 25.12 25.12 31.38 38.10 23.26 12.39 28.52 8.14 25.12 25.12 25.12 25.12 17 25.12 25.12 25.12 25.12 39.07 28.19 25.12 25.12 25.12 25.12 25.12 25.12 0.6 18 25.12 25.12 25.12 25.12 25.12 25.12 25.12 25.12 25.12 25.12 25.12 25.12 0.5

In second part of this section, the hourly values of LOLP 0.4 index have been evaluated based on analysis of convolution 0.3

® Cumulative distribution functions technique. In order to calculate these values, a Matlab 0.2 program has been developed and used in this order, more 0.1 details about this technique being reported in [12]. The LOLP 0 index results from analysis of probability density function of 0 20 40 60 80 100 120 reserve margin, which is derived from the combination of load Power (W)

and generation probability density functions considering the Figure 6. Interference between cdf-s of generate power and demended load, hypothesis of independent (not correlated) distribution for May, 9:00 AM. functions. Fig. 5 shows in same diagram a suggestive view of probability density functions associated to generated power and Similar calculations have been conducted every hour, every load profiles, for May to 9:00 A.M. month, between sunrise and sunset, the hourly values of LOLP For this case, the numerical values of LOLP index have for first load profile and for hourly frame database are been evaluated from the intersections of cumulative tabulated in Table II. As expected, the LOLP values feature distribution functions of generated power and the load follows the sun moving across the sky, reaching a minimum demanded to 9:00 o’clock. For each load profile, the demanded value at midday. load being normally distributed around to 39.5 W for first

949

TABLE II. HOURLY MONTHLY VALUES OF LOLP INDEX FOR LOAD PROFILE 1 VI. CONCLUSIONS Month This study has been developed in order to evaluate and to Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Hour analyse the correlation between hourly values of tertiary 6 100 100 100 100 100 100 100 100 100 100 100 100 7 100 100 100 100 35.43 6.93 100 100 100 100 100 100 reserve margin and hourly values of adequacy indices 8 100 100 51.81 20.02 3.55 0.19 1.21 0.92 100 100 100 100 associated to a hybrid system designed to supply the three 9 100 50.56 9.95 6.65 0.77 0.15 0.16 0.35 2.34 27.45 100 100 different load profiles. The hourly reserve margin as well as the 10 62.29 20.93 3.86 3.27 0.42 0.08 0.01 0.07 0.70 15.19 45.88 70.47 adequacy indices provides the system capacity to follow the 11 53.67 18.24 2.02 1.41 0.62 0.06 0.02 0.03 0.74 5.58 37.33 52.04 12 43.69 12.57 1.56 1.00 0.55 0.03 0.01 0.02 0.57 4.34 35.38 47.18 load features and to be able to cover the random events 13 50.91 26.11 3.43 1.50 0.27 0.05 0.01 0.00 0.51 6.00 37.02 50.69 associated to load and photovoltaic power generation. 14 71.78 42.26 10.84 3.35 0.26 0.05 0.06 0.13 2.77 12.47 51.72 65.67 Based on values of correlation coefficients, it can be stated 15 100 66.51 26.14 11.27 1.22 0.33 0.14 0.04 5.50 41.28 100 100 that between hourly tertiary reserve margin and hourly values 16 100 100 66.93 33.56 10.77 4.17 10.96 3.16 100 100 100 100 17 100 100 100 100 53.18 33.88 100 100 100 100 100 100 of LOLP index there is a straight correlation. Consequently, in 18 100 100 100 100 100 100 100 100 100 100 100 100 order to establish the behaviour in time of hourly reserve margin, the analysis of adequacy indices of system could be a The previous numerical results have been involved in a proper and a faster method. correlation analysis, thus, the hourly correlation coefficients between hourly values of tertiary reserve margin and hourly ACKNOWLEDGMENT values of LOLP index have been calculated and presented in This paper was supported by the project PERFORM-ERA Fig. 7. ‘Postdoctoral Performance for Integration in the European Research Area’ (ID-57649), financed by the European Social 1 Fund and the Romanian Government.

0.8 REFERENCES 0.6 [1] Operation Handbook, Entsoe, “P1, Policy 1: Load-Frequency Control 0.4 and Performance” available www.entsoe.eu [2] Codice di Rete, Terna, “Allegato 22: Procedura per la selezione delle 0.2 risorse per la fase di programmazione del MSD” available www.terna.it

0 [3] Billinton R., Allan R. Reliability evaluation of power systems, 2nd Edition, Plenum Press, NewYork, 1996. -0.2

Correlation coeficientsCorrelation [4] Endrenyi J. Reliability modelling in electric power systems, Wiley and Sons, NewYork, 1978. -0.4 [5] Hollands T., Huget G. A Probability Density Function for the clearness Load Profil 1 index, with applications. Solar Energy vol. 30, 1983, pp. 195-209. -0.6 Load Profil 2 Load Profil 3 [6] Tina G., Gagliano S., Raiti S. Hybrid solar/wind power system -0.8 probabilistic modelling for long-term performance assessment Solar 6 8 10 12 14 16 18 Hours Energy, vol. 80, 2006, pp. 578–588. [7] Tina G. M., Gagliano S. Probabilistic modelling of hybrid solar/wind Figure 7. Hourly values of correlation coeficients between reserve margins power system with solar tracking system, Renewable Energy vol. 36, and LOLP index values. 2011, pp. 1719-1727. [8] Conti S., Crimi T., Raiti S., Tina G., Vagliasindi U. Probabilistic As can be seen, the correlation coefficients, with few approach to assess the performance of grid-connected PV systems, exceptions, have positive values for all three profiles. Proceedings of 7th International Conference on Probabilistic Methods Applied to Power Systems, Naples, Italy, September 2002. Moreover, the correlation has the highest value on midday, [9] Notton G., Poggi P., Cristofari C. Predicting hourly solar irradiations on when the hybrid system is characterised by a lowest value of inclined surfaces based on the horizontal measurements: Performances LOLP and also by a lowest value of reserve margin. Taking of the association of well-known mathematical models, Energy into account that the values of correlation coefficients are Conversion and Management, vol. 47, 2006, pp. 1816–1829. positive, it can be concluded that between tertiary reserve [10] Notton G., Cristofari C., Poggi P. Performance evaluation of various margin and LOLP index there are a straight dependence. It hourly slope irradiation models using Mediterranean experimental data of Ajaccio, Energy Conversion and Management, vol. 47, 2006, pp. seem obvious that lower values of LOLP index to involve the 147–173. narrower domain of tertiary reserve margin, while a higher [11] Tina G.M., Notton G., Nemes C., Time frame measurements impact on value of LOLP to lead to a larger domain of reserve margin. probabilistic behaviour of photovoltaic systems, 9th World Energy In previous analysis, there are registered two exceptions, System Conference, June 28-30 2012 Suceava, Romania. Published in AGIR Bulletin, World Energy Systems. Towards Sustainable and namely to sunrise and sunset, when the hourly correlation Integrated Energy Systems, Tome XVII, issue 2/2012, pp.1-9, ISSN-L coefficients have negative values. These observations are 1224-7928, Online: ISSN 2247-3548. argued by the fact that for those moments characterized by a [12] Neme C., Munteanu F. A probabilistic model for power generation lower solar radiation, it is difficult to cover the peak of load adequacy evaluation, Revue Roumaine des Sciences Techniques, Série from the above profiles, the hourly values of reserve margin Électrotechnique et Énergétique, Tome 56, Issue 1, 2011, pp. 36-46. and hourly values of LOLP index having constant values over different months.

950 A PROBABILISTIC MODEL FOR POWER GENERATION ADEQUACY EVALUATION

CIPRIAN NEMEŞ , FLORIN MUNTEANU

Key words: Power generating system, Interference model, Monte Carlo simulation. The basic function of a modern electric power system is to supply the system load requirements as economically as possible and with a reasonable degree of reliability and quality. In power system generation planning, many models and techniques have been developed to evaluate the reliability performance. The objective of this paper is to develop a probabilistic model for power generating system reliability performance evaluation, based on the convolution technique.

1. INTRODUCTION

The quality and availability required of electrical energy is directly related to the power system reliability concept. The reliability associated with a power system, in a general sense, is a measure of the overall ability of the system to generate and supply electrical energy. Due to the complexity of the electric power system, it is divided into functional areas namely generation, transmission and distribution. Reliability of each functional area is usually analyzed separately for an easier evaluation and eventually combined to assess the system reliability. The electricity has a characteristic due to the fact that it can’t be stored in the system for long time and important quantity. As a result, electricity must be used when it is produced and generated there and then when there is demand. Therefore, generating systems are designed and operated in order to meet the demand load of the system with a certain reliability level. Generating system reliability is used to evaluate the capacity of the generating power system to satisfy the total system load. The reliability assessment of a power generating system can be divided into two main aspects: system adequacy, which is related to the existence of sufficient facilities to satisfy system load demand, and system security, which is related to the ability of the system to respond to dynamic or transient disturbances. Load demand can exceed the generating capacity for two main reasons. First, if there is a very high load peak demand that exceeds the installed capacity of the

“Gh. Asachi” Technical University of Iaşi, Bd. D. Mangeron 53, 700050 Iaşi, România; E-mail: [email protected]

Rev. Roum. Sci. Techn.– Électrotechn. et Énerg., 56, 1, p. 36–46, Bucarest, 2009 2 A probabilistic model for power generation adequacy evaluation 37 system, the system cannot supply the load peak. Second, if some generating capacity units are out of service because of failures or periodic maintenance, a high load peak demand that does not exceed the installed capacity of the system can exceed the available capacity at that moment. In this paper, the authors focus on an evaluating model for the generating system adequacy, having in view the behaviour of generating units. The developed model is based on the combination of random variables that describe the generating capacity and system load demand and, finally compared with the results of the other models or simulation techniques.

2. GENERATING SYSTEM ADEQUACY INDICES

The adequacy associated to a generating power system, in a general sense, is a measure of the ability of system generating capacity to satisfy the total system load. Generally, the generating power system evaluation process does not consider the transmission and distribution systems, only concentrates on the balance between generating capacity and load demand. The approach to generating adequacy evaluation is to develop a capacity model for all the capacity from the system and to join this with an established load model. The most popular indices used in generating power system are the Loss of Load Expectation and the Loss of Energy Expectation, which have to base the Loss of Load Probability (LOLP) [1, 2]. The Loss of Load Expectation (LOLE) [hours/year] indicates the average number of hours in a given period (usually one year) in which the load is expected to exceed the available generating capacity. It is obtained by calculating the loss of load duration in hours for that daily peak demand exceeding the available capacity for each day and adding these times for all the days for a number of sample years. The Loss of Energy Expectation (LOEE) [MWh/year], sometime known as the Expected Energy Not Supplied (EENS), specifies the expected energy that will not be supplied by the generation system due to those occasions when the load demanded exceeds the available generating capacity. Other indices can be defined as generating system reliability performance measures, such as: Expected Loss of Load Frequency [occurrences/year], Expected Duration of Loss of Load [hours/occurrence], Load Not Supplied per Interruption [MWh/occurrence], Expected Energy Not Supplied per Interruption [MWh/occurrence], Energy Index of Reliability [%], and others. Frequency and duration of loss of load are a basic extension of the LOLE index, in that they identify the average frequency and the average duration of the occurrence, for a certain period (usually one year).

38 Ciprian Nemeş, Florin Munteanu 3

3. A PROBABILISTIC MODEL FOR GENERATING ADEQUACY

The basic approach to evaluate the adequacy of an electrical power generating system consists of two parts: capacity model and load model. The capacity and load models are joined using a probabilistic model which evaluates the reliability performance of the generating system in terms of adequacy indices. This problem can be analyzed like the problem of interference of the stress- strength random variable model, described in many areas of reliability [3, 4]. Mathematically, the interference model conducts to a joined variable of the two random variables of generating capacity and load demand, as is showed in Fig. 1.

Fig. 1 – The interference model of the generating capacity and load demand.

The interference model depends on the nature of random variables involved in model, continuous or discrete. The stress-strength interference model and its practical evaluations were developed, examined and presented by authors in many others papers [5, 6]. In this paper, both variables have multiple power [MW] levels, with each level having a probability of occurrence, so that these variables can be modelled as discrete random variables. The generation system adequacy indices can be obtained by observing the joined variable called the available capacity margin, and defined like the difference between available generating capacity C and load demand L variables [7]. A positive margin denotes that the system generation is sufficient to meet the system load, while a negative margin implies that the system load is not served. The LOLP is a widely used indicator, as a criterion for generation system reliability, because it indicates the probability of the load to exceed the generating capacity of a proposed generation power system, during a given time interval.

4 A probabilistic model for power generation adequacy evaluation 39

NL NC LOLP =<=Pr()CL∑∑ Pr () Lijij ⋅ Pr() C ⋅ I, (1) ij==11 where C is a discrete random variable representing the available generating capacity and L is a discrete random variable of the load demand. In equation (1), Pr(Li) is the th probability of the i load level (Li) and NL the number of the load levels in the load th probability function. Pr(Cj) is the probability of the j generation capacity level (Cj) and NC the number of the generation capacity levels in the generation capacity probability function. Iij is an indicator function defined as

ILCLCij=≤{ 0, if i j ; 1, if i > j } . The LOLP index is assessed only for the area in which load exceeds capacity, such the previous relationship can be rewritten as:

N LOLP =Pr() M<= 0∑ Pr ()() MkM = F 0 , (2) k =1 where M=C–L is a discrete random variable representing the available capacity th margin. The Pr(Mk) is the probability of the k available capacity margin level (Mk) and N is the number of the available capacity margin levels from negative margin area. The FM(M) is the cumulative distribution function of M variable. A relationship between the probability function, cumulative distribution function of the available capacity margin variable, respectively the LOLP index is presented in Fig. 1. In power generating system, sometimes others adequacy indices are necessary, beside the LOLP. It is important to know how many times the load exceeds the available generating capacity, or how much energy has been lost due to interruption. The LOLE and LOEE indices can be expressed using the LOLP and expected values of available capacity margin in negative margin area, for a given period using the following relationships:

N LOLE = LOPT ⋅ T and LOEE = ∑ MMTkk⋅⋅Pr(), (3) k =1 where T is the total number of hours from analysed period, Mi and Pr(Mi) have been defined for equation (2). In the following, is presented a procedure for determining the available capacity probability function for a various number of capacity units, respectively to establish a load probability function, that combined to estimate the adequacy indices. The quantitative adequacy evaluation invariably leads to a data required supporting such studies, so, for our study we will use the IEEE Reliability Test Systems (IEEE-RTS), developed by the Subcommittee on the Application of Probability Methods in the IEEE Power Society, to provide a common test system which could be used for comparing the results obtained from different methods, available to [8].

40 Ciprian Nemeş, Florin Munteanu 5

The IEEE-RTS generation system is composed by a combination of 32 generation units ranging from 12 MW to 400 MW. Generation system data contains the type of generation units, rated capacity [MW] and number of each units, respectively the reliability data as mean time to failure (MTTF ) and mean time to repair (MTTR), in hours. The capacities of each generation units can be modelled like a discrete random variable. Let Ck be a discrete random variable representing the capacity that can be supplied by the kth generation unit, with k = 1, …, 32. Is assumed that each unit has two possible working states, down and up. The random variable can be described by its associated probability function that contains all the capacity states, in an ascending order, and its probabilities:

{0× qk , CU k × pk }, (4) where: th − CUk is the rated capacity of k generation Unit (it is used CUk, not to be confused with Ck, the generation capacity level of discrete random variable C);

MTTRk th − q=k = Pr() 0 is probability that available capacity of k MTTFkk+ MTTR generation unit to be zero;

MTTRk − pCU=kk= Pr() is probability that available capacity of MTTFkk+ MTTR th k generation unit to be rated capacity, CUk [MW]. From the probability theory, it is known that the sum of two random variables (C1 + C2) is a new random variable, with probability that both unit 1 and unit 2 to be simultaneously in operation during a specified period of time. New random variable can be obtained by convolving the both random variables:

C = conv (C1, C2), (5) described by its associated probability function:

{}0 × q1 ⋅ q2 , CU1 × p1 ⋅ q2 , CU 2 × q1 ⋅ p2 , (CU1 + CU 2 ) × p1 ⋅ p2 . (6) So, all the capacity generation units can be modelled as discrete random variables and the capacity generation system for the whole system is a new random variable described by:

C = conv (C1, C2,…, C32 ). (7) The capacity probability function may be computed using the gradual convolution for all units of system. After convolving all 32 discrete random

6 A probabilistic model for power generation adequacy evaluation 41 variables of the generation units, it is obtained the following random variable for the IEEE-RTS generating system (Fig. 2).

Fig. 2 – IEEE-RTS available capacity probability function.

The load demand in power systems is variable in time. There is no one unique profile or mathematical equation that can be adopted to represent the load characteristic curve. In this paper, the load characteristic curve has been modelled using the load model of IEEE-RTS. The annual load curve, included in IEEE-RTS, considers the seasonal, weekly, daily and hourly peak of load. The hourly load is percentage of daily peak, the daily peak load is percentage of weekly peak and weekly peak load is percentage of annual peak for three seasons, so the hourly load dependents by three indices (i,j,k). The index i represents a certain hour of the day, the index j represents a certain day of the week, and the index k represents a certain week of the year, respectively. The hourly load level HL(i,j,k) can be obtained from multiplication the three percentages of the peak of load, as: HL(ijk , , )=×××[ pr_hour( i ) pr_day( j ) pr_week( k )] annual_peak _ of_load , (8) where: pr_hour(i)(%) is the percentage of ith hour from the daily peak of load, i = 1,...,24; pr_day(j)(%) is the percent of jth day from the weekly peak of load, j = 1,..,7; pr_week(k)(%) is the percent of kth week from the yearly peak of load, k = 1,...,52.

42 Ciprian Nemeş, Florin Munteanu 7

The final hourly load for one year HL, can be described like a discrete random variable, with 8,736 hourly load levels (24 hours/day × 7 days/week × 52 weeks/year), having the following probability function:

{HL(ijk , , )×⋅⋅( Pr()()() hour i of day Pr day j of week Pr week k of year )} (9)

Taking into account that some hourly load levels may have the same values, the probability of any value of HL variable, can be evaluated using the count number of the discrete random variable levels with the same values. For that, the authors have been developed a Matlab-Simulink function that allowed determination of the probability function associated of the discrete random variable of IEEE-RTS hourly load, as is shown in Fig. 3.

Fig. 3 – IEEE-RTS hourly load probability function.

Capacity margin is the amount by which generating capacity exceeds system load demand, expressed as a discrete random variable. The available capacity margin variable results from difference of generating capacity and load demand variables, and its probability function arises from the load-capacity probability functions interference. The probability function of capacity margin variable is presented in the Fig. 4.

8 A probabilistic model for power generation adequacy evaluation 43

Fig. 4 – The available capacity margin probability function.

4. STUDY RESULTS

The Matlab program based on available capacity margin variable has been developed to compute the LOLP, LOLE and LOEE indices using the relations (2–3). The program results of previous model have been compared with results from other model. Sequential Monte Carlo simulation has been used to provide information relates to the average values of the adequacy indices. This technique generates a chronological operating cycles for each generation unit. The simulation is done by using the MTTF and MTTR parameters to produce two sequence of up and down times, necessary to estimate the chronological operating unit's cycles. The generating system operating model can then be obtained by combining the operating cycles of all units. The required adequacy indices may be observed from the overlapping generation operating model to a chronological load curve, over a long time period. The simulation can be stopped when a specified degree of confidence has been achieved. Figure 5 shows an example of LOLE and LOEE indices evaluation, using the sequential Monte Carlo simulation, for 10,000 sample years, necessary to ensure the convergence process. The IEEE-RTS generating system adequacy indices provided by probabilistic model and sequential Monte Carlo simulation are shown in Table 1. For a better comparison between models, three levels of annual peak of load were considered in the adequacy indices evaluation. It can be seen that the results obtained from both methods are very close. The probabilistic method provides comparative results

44 Ciprian Nemeş, Florin Munteanu 9 with Monte Carlo simulation, but the computing time is considerably less than the simulation technique, requiring fewer CPU resources.

Fig. 5 – Results from sequential Monte Carlo simulation: a) LOLE [103 h/yr]; b) LOEE [MWh/yr].

The proposed model may be used in a generation source planning, to analyze different adding new generation unit scenarios and select the suitable sources as the generation system to meet a load growth, maintaining the same level of adequacy.

Table 1

IEEE-RTS Generating system adequacy indices Index LOLP [%] LOLE [h/yr] LOEE [MWh/yr] Annual peak 2750 2850 2950 2750 2850 2950 2750 2850 2950 load [MW] Probabilistic 55.841 109.421 201.765 4.878 9.559 17.626 566.950 1171.50 2331.60 model × 10-3 × 10-3 × 10-3 Monte Carlo 55.058 108.219 199.823 4.810 9.454 17.457 559.620 1158.80 2311.30 Simulation × 10-3 × 10-3 × 10-3

For IEEE-RTS case study, if the system with 2,850 MW annual peak of load is considered having an acceptable adequacy level, the developed model may be used to establish the best solution for generating system expansion, assuming an annual peak load growth. It is assumed that the annual peak load of the IEEE-RTS system is increased with 100 MW, from 2,850 MW to 2,950 MW. So, the LOLE and LOEE indices increase from 9.559 h/yr and 1171.5 MWh/yr for base case (with 2,850 MW peak load) to 17.626 h/yr, respectively 2,331.6 MWh/yr for the new annual peak of load (2,950 MW). The 100 MW growth of peak of load may be supplied from different adding new generation unit scenarios, until the LOLE and

10 A probabilistic model for power generation adequacy evaluation 45

LOEE indices decrease under the acceptable adequacy level, as is presented in Table 2. It shows the adequacy indices for all possible adding generation unit scenarios, to supply the increased peak of load, having in view various number and type of generation units available in IEEE-RTS. As shown Table 2, the scenario of adding a 1×50 MW, 1×20 MW and 2×12 MW generation units, leads to a minimum added capacity (94 MW).

Table 2

Reliability indices for various expansions of IEEE-RTS Added LOLE LOEE Cases System capacity [MW] [h/yr] [MWh/yr] 1 IEEE-RTS with 8×12 MW 96 9.0120 1 123.3 2 IEEE-RTS with 5×20 MW 100 9.3306 1 166.2 3 IEEE-RTS with 2×50 MW 100 8.6911 1 080.8 4 IEEE-RTS with 1×100 MW 100 8.9763 1 120.8 5 IEEE-RTS with 1×20+7×12 MW 104 8.6396 1 069.3 6 IEEE-RTS with 2×20+6×12 MW 112 8.2240 1 017.6 7 IEEE-RTS with 3×20+3×12 MW 96 9.3615 1 170.7 8 IEEE-RTS with 4×20+2×12 MW 104 8.9712 1 114.6 9 IEEE-RTS with 1×50+1×20+2×12 MW 94 9.1828 1 152.8 10 IEEE-RTS with 1×50+2×20+1×12 MW 102 8.8513 1 097.6

5. CONCLUSIONS

In this paper, a probabilistic model for the power generating capacity adequacy evaluation is presented. The main aspect of the probabilistic model is to evaluate a random variable that describes the difference between capacity of generation units and load demand and it evaluating in the negative domain. The model enables the evaluation of the most popular indices in generating capacity planning, namely the loss of load probability, loss of load expectation, and the loss of energy expectation, respectively. The results were validate using the sequential Monte Carlo simulation and found that provided results are very close. The model has the advantage that can be easily implemented in computer programs and require a computing time considerably less than in the case of simulation methods. The developed model can be used not only to evaluate the previous mentioned indices, but also may be used in a generation source planning, to select the suitable sources as the generation system to meet a load growth, maintaining the same level of adequacy level. The probabilistic model can provide more significant information for system planning, since they consider probabilistic aspects of generating units. The future work to be done will use the probabilistic

46 Ciprian Nemeş, Florin Munteanu 11

model presented in this paper in a generation source planning with integrated wind energy sources.

Received on June 22, 2009

REFERENCES

1. R. Billinton, R. Allan, Reliability Evaluation of Power Systems, Plenum Press, 2th edition, 1996. 2. Wenyuan Li, Risk Assessment of Power Systems Models, Methods and Applications, John Wiley and Sons, 2005. 3. J. Endrenyi, Reliability modeling in electric power systems, John Wiley and Sons, New York, 1978. 4. Joel A. Nachlas, Reliability engineering: probabilistic models and maintenance methods, CRC Press, Taylor and Francis Group, 2005. 5. D. Ivas, Fl. Munteanu, E. Voinea, C. Nemeş, Ingineria fiabilităţii sistemelor complexe, Edit. Perfect, Bucureşti, 2001. 6. C. Nemes, Fl, Munteanu Tehnici moderne de analiză a disponibilităţii elementelor şi sistemelor Edit. Politehnium, Iaşi, 2008. 7. R. Billinton, B. Bagen, Incorporating reliability index distributions in small isolated generating system reliability performance assessment, IEE Proceedings Generation, Transmission and Distribution, 151, 4, pp. 469–476 (2004). 8. IEEE Commmitte, IEEE Reliability Test System (Report), IEEE Transactions on Power Apparatus and Systems, 98, 6, pp. 2047-2054 (1979).

View publication stats INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES

The wind energy system performance overview:

capacity factor vs. technical efficiency

Ciprian Nemes 1, Florin Munteanu

suitable wind turbine for a specific wind profile. Abstract—The main objective of the paper is to develop a A measure of the suitability of wind turbine to a specific probabilistic model for capacity factor and technical efficiency location is given by the capacity factor and efficiency values. estimation for a wind turbine located in a specific area, model based The capacity factor is defined as the ratio of the expected on the output power distribution of wind turbine. This model was output power over a period of time to the rated power of wind applied for a wind turbine located to a region in the North-East of Romania, the model results being validated by results from Monte- turbine generator. All power plants have capacity factors, and Carlo simulation. Finally, the model was used to evaluate the effects they vary depending on resource, technology and purpose. of wind turbine generator parameters, for a given wind profile, on the Typical wind power capacity factors are 20-40%, with values capacity factor and technical efficiency values. at the upper end of the range in particularly favourable areas [4]. The capacity factor is not an indicator of efficiency. A Keywords— wind energy, Weibull distribution, output power measure of turbine efficiency is the power coefficient. This distribution, capacity factor, technical efficiency. coefficient indicates how efficiently a turbine converts the wind energy into electricity. This coefficient varies with the I. INTRODUCTION wind speed [2]. Efficiency is the expected power coefficient, NVIRONMENTAL factors such as global warming and over a period of time, and is defined as ratio of the useful Epollution have heightened the need to introduce into the output energy to the input wind energy. generation mix a greater percentage of renewable sources. In In literature are presented various approaches for capacity the last time, wind power has drawn much attention as a factor and efficiency estimations, mostly obtained from promising renewable energy resource, which has shown some simulations techniques based on wind speed data [5,6] and prospects in curtailing fuel consumption and reducing the sometimes from computational models [1,7], that need emission of pollutants into the atmosphere. Unlike other numerical integration techniques or some approximations. renewable energy sources, wind energy has become Having in view the stochastic nature of the primary energy, competitive with conventional power generation sources and the probabilistic methods can be proper solutions for capacity therefore application of wind turbine generators has the most factor and efficiency evaluations. In the paper, a probabilistic growth among other sources. Wind is one of fastest growing model is developed to evaluate these values and to analyze the energy source and is considered as an important alternative to dependence of the wind turbine generator characteristics. The conventional power generating sources. proposed model is based on probability density function of The energy production from a wind turbine or a wind park, output power generated by the wind turbine. In order to in a specific location, depends by many factors. The main validate the model, the results model were compared with the factors include the wind speed conditions from the area, and results of the other model, namely with Monte Carlo most importantly, the characteristics of the wind turbine technique. The model has the advantage that can be easily generator itself, particularly the cut-in, rated and cut-off wind implemented in computer programs and require a computing speed parameters. The output power of a wind turbine time considerably less than in the case of simulation or generator does not vary linearly with the wind speed. The numerical methods. output power increases with the wind speed between the cut-in Selection of the optimal wind turbine was discussed in speed and the rated wind speed, after that the power output different manner in various papers, among which the remains constant at the rated power level, until the cut-out maximization of capacity factor and/or efficiency [5,6,7]. The speed, when the turbine is stopped for safety reasons [1,2]. choice of turbine involves choosing parameters that lead to Different types of wind turbines are commercially available on maximizing these factors. The turbines must be chosen with the market. It is therefore desirable to select a wind turbine the parameters that match those of wind profile area. Based on which is best suited for a particular location in order to obtain these issues, in the paper, a numerical analysis is realized to a maximum power from power transposed by the wind. These have a comparison between effects of different parameters of important aspects bring suitability concerns regarded by the the wind turbine generator on capacity factor and efficiency energy potential of a specific location and the selection of the values, analyzing the importance and weight of each parameter of those values. 1 Corresponding author: Tel +40232278683, email [email protected]

Issue 1, Volume 5, 2011 159 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES

II. WIND AND OUTPUT POWER WIND TURBINE PROBABILISTIC To establish the parameters of probability density CHARACTERISTICS distribution is necessary an accurate dataset of wind speed. The output power from a wind turbine depends by the The wind speed database can be obtained from meteorological availability of the energy source, namely the wind speed and station, where, usually, the measurement point (anemometer) height above the surface may be to 10 m or 50 m. Depending the power-wind characteristics of the wind turbine generator. by the wind measurement level, the speed data must be A. Probabilistic model of wind speed adjusted for the change in height desired according to a Wind is a turbulent movement mass of air resulting from the logarithmic profile previously mentioned. differential pressure at different locations on the earth surface. B. Wind speed-power relationship One of the main characteristics of wind is that it is highly The power transported by an air stream flowing with a given variable in time and space, the variation of wind exists from speed, v, can be calculated according to [1] using the following instantaneous, hourly, daily to seasonal, and its properties vary simple expression: from one location to another. The wind property of interest in 3 the power generation problems is the wind speed probabilistic V 21 AP ⋅ρ⋅= v (3) model. where ρ is the air density and A the area of the air stream, The speed of the wind is continuously changing, making it measured in a perpendicular plane to the direction of the wind desirable to be described by the probabilistic models. The speed. probability density function of wind speed is important in The calculation of the mechanical power that can be numerous wind energy applications. A large number of studies extracted by the rotor of a wind turbine, requires Betz’ law to have been published in scientific literature related to wind be taken into account. This law specifies that only a maximum energy, which propose the use of a variety of functions to 16/27 of the wind energy can be converted into mechanical describe wind speed frequency distributions [9],[10]. The power. This value is known as the Betz limit. In practice, the conclusion of these studies is that the Weibull distribution of collection efficiency of a rotor is not as high as 59%. A more two parameters may be successfully utilized to describe the typical efficiency is 35% to 45%. principle wind speed variation. The Weibull probability The mechanical power is converted in electrical power by density and cumulative distribution function are given by: generator, so, the output electric power of a wind turbine is a

β−1 β β function of the wind speed. The power curve gives a relation β  v    v   ;   v   (1) f W (v) =   exp −    FW v exp1)( −−=    between the wind speed and the output electric power, a  αα    α     α   typical curve of the wind turbine generator is nonlinear related The scale parameter α (m/s) and a shape parameter β to the wind speed. However, the assumption of the linear (dimensionless) of the Weibull distribution can be found using characteristic of power with the wind speed, brings a different estimation methods [2],[4]. Each method has a significantly simplifies of calculations, without roughly errors. criterion, which yields estimates that are best in some The power output characteristic can be assumed in such way: situations. Different results are produced based on that − it starts generating power when the speed wind exceeds criterion. The most commonly methods are Maximum the minimum wind speed, namely cut-in speed (vcut-in); Likelihood Estimator, Method of Moments, Least Squares − the power output increases with the wind speed, when Method or Regression Method. The Maximum Likelihood wind varies between cut-in and rated speed wind (vrated), Estimator is so commonly applied in engineering and value for that the power achieves the rated power (Prated). mathematics problems [6], so, this method is used in this paper − the rated power of a wind turbine, generally the maximum to establish the parameters of wind speed distribution. power output of a generator at highest efficiency, is In many studies, the shape parameter is often chose to 2 produced when the speed lies between rated and cut-off and therefore a Rayleigh distribution can be used, with a same wind speed (vcut-off). accuracy and with a simpler model. − cut-off wind speed is the maximum wind speed at which The wind blows faster at higher altitudes because of the the turbine is allowed to produce power, usually limited reduced influence of drag of the surface and lower air by engineering design and safety constraints. viscosity. The effect of the altitudes in the wind speed is most Thus, the electric power PE may be calculated from the wind dramatic near the surface and is affected by topography, speed as follows: surface roughness, and wind obstacles such as trees or  ( −⋅ vvP ) buildings. The most common expression for the variation of rated −incut << vvvfor  − vv cut −in rated wind speed with hub height is the power law having the  ( rated −incut ) (4) ()vP = P vvfor << v following logarithmic profile model [8],[9]. E  rated rated −offcut  0 other else ⋅= ( ( ) lnln)()( (zzzzvzv z )) (2)  r 0 r 0  where v(z) and v(zr) are the wind speeds at a desired z and This curve comes available from the wind turbine registered zr height, and z0 is the surface roughness length, a manufacturer or plotted using recorded wind speed and characterization of a ground terrain. corresponding output power data, a typical curve of the wind

Issue 1, Volume 5, 2011 160 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES

β turbine generator is shown in the figure 1. −1 2β  2P  3  β  Electric power Pf )( = ⋅  V  exp AP ρα−⋅ 3 3 (6) VPV 3  3   ()V  (MW) 3Aρα  Aρα    Prated Therefore the power transported by the wind can be represented by a Weibull distribution, with α’ and β’ parameters given by: 21' A 3 β=βα⋅ρ⋅⋅=α 3', (7)

Wind speed where ρ is the air density, A the area of the wind turbine rotor, (m/s) and α,β are the wind distribution parameters. vcut-in vrated v cut-off The output power probabilistic model for a wind turbine and Fig. 1 The power curve of a wind turbine its practical evaluations were developed and evaluated by authors in [11],[12]. Similar results has also been obtained by Equations (3) and (4) express the instantaneous values of others authors in [ 13],[14]. wind power and electric output power, as a function of the The possible values of FWT(P) may be roughly classified in instantaneous wind speed. However, the wind speed may vary 0, Prated and in the interval that lies between mentioned values, during a period of time. To consider this effect, we are going respectively. Each possible value has been evaluated, having to work with the wind speed probability distribution function. in view the probability to achieve that value. The cumulative C. Probabilistic model of wind power and wind turbine distribution function of the output power of the wind turbine output power is:

The probability distribution functions of wind power and of  − [ −offcutW − vFvF −incutW )()(1 ] Pfor E = 0  (8) the wind turbine output power can be obtained using the PF EPE )( =  WT W −+ −incutW 0)()()0( E << PPforvFWFF rated  analytical dependence between wind speed, wind power, and  1 E = PPfor rated the output power respectively, operating a change of variables. The probability density function results from differential of Lets assume that v is a continuous random variable with cumulative distribution function: cumulative distribution function, FV(v) and that P=J(v) defines  ℜ1 Pfor E = 0 a one-to-one transformation from a region of the wind-space,   ( rated − vv cut−in ) (9) to a region of the power-space, with inverse transformation Pf EPE )( =   ⋅ W () 0 PforWf E << Prated -1 P v=J (P), the cumulative distribution function of power can be  rated   ℜ2 = PPfor calculated according [16],[17]:  E rated where: = < = < pXJpPPF ))(Pr()Pr()( = (5) −1 −1 − −=ℜ [ vF −offcutW − F v cutW −in ]= FPE )0()()(11 , represents Pr( <= = W pJFpJX ))(())( In order to obtain the probability density function, a the value of output power cumulative distribution function differential of cumulative distribution function must be in the 0 point, − , operated. In figure 2 is presented the intuitive process of the ℜ = vF −offcutW − FW v rated −0 −= PE PF rated − 0 )(1)()(2 random variables transformation, the wind speed variables represents the increase value of the cumulative being transformed in wind power variable PW, on the right, distribution function in the Prated value, and respectively in the output electrical power variable PE, on the  P  −  E  . left.  (vW rated v −incut ) +⋅−= v cut−in   Prated  PE,PV PE PV P =J (v) P =J (v) E 1 V 2 III. CAPACITY FACTOR AND EFFICIENCY EVALUATIONS Capacity factor and efficiency depend both on the wind speed distributions in the area and the turbine parameters. To illustrate the effect of wind turbine parameters on capacity v fPE(PE) fPV(PV) factor and efficiency values, a probabilistic model for these fV(v) indicators is developed. The capacity factor (CF) of wind v turbine is the ratio of expected output power over a period of

Fig. 2 Transformation of the wind speed variable time to rated power. The efficiency (EF) of wind turbine is the ratio of useful output energy to the input wind energy, or If (5) is applied for the wind power relationship (3), having expected power output from wind machine to expected power in view the Weibull distribution, the probability density available in wind over a period of time. The expected output function of the wind power may be expressed as a function of power is used in both on the capacity factor and efficiency the variable PV: evaluations, so, it will be firstly evaluated.

Issue 1, Volume 5, 2011 161 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES

A. The expected output power of a wind turbine of wind turbine generator. α  1  The expected output power of a wind turbine depends on Γ      β   β  the output power values and the probability to achieve that PE E )( β  β   vrated  1  v −incut  1 CF == ⋅ P  ,  − P  ,  power, described by the probability distribution functions of − vvP  α β   α β  rated rated −incut        output power. The expected output power (E(P )) from a wind E v α−− β ))/(exp( (15) turbine generator can be estimated from its power-wind and −offcut wind characteristics, being represented by the probability The efficiency is a measurement of how much energy from distribution of wind speed. This is given by [3] as: the wind is converted in electrical power energy. For that, the ratedP expected output power must be divided to the expected wind E(PE ) ⋅= EPEE )( dPPfP E (10) power input to measure how technically efficient is the wind ∫0 turbine. where PE is the output power function variable and fPE(PE ) is P α  1  probability density function of output power from wind rated ⋅ Γ      β   β  PE E )( rated − vv −incut β β  vrated  1  v −incut  1 turbine. EF == ⋅ P  ,  − P  ,      Having in view the expression of output power distribution PE V )(  1    α  β  α  β 1' +Γ⋅α    function, from (9), the expected output power can be obtained  β' by: Prated β (16) − −⋅ v −offcut α ))/(exp( Pr ated  1  PE E 10)( +ℜ⋅= ⋅ )( EEPEE PdPPfP rated 2 =ℜ⋅+ ∫0 1' +Γ⋅α  (11)  β'  v rat − vv −incut = Prated ⋅ W () Pdvvf rated ℜ⋅+⋅⋅ . The equations (15) and (16) show the results of authors’ ∫v cut−in v − v rated −incut research based on laborious calculations and detailed analysis where f (v) is a probability density function of wind speed. W of statistical distributions. This relationship represents an Taking into account that , the equation W W )) ⋅= dv(vf(vdF equation which shows the effects of cut-in, rated, and cut-off (11) may be written as: speeds parameters on the capacity factor value. For a given β  vrated v α−− ))/(exp(1  wind regime, with known α and β parameters, we can select E )( rated ⋅=  ()vFPPE −offcutW − dv  = ∫v  −incut rated − vv −incut  that values of vcut-in, vrated and vcut-off that maximize the

v β expected output power, and thereby maximize the capacity rated − v α ))/(exp( β (12) Prated ⋅= dv Prated v −offcut α−⋅− ))/(exp( ∫v factor. cut−in rated − vv −incut The integration can be accomplished by making the change IV. MODEL VALIDATION AND NUMERICAL EXAMPLE in variable vy α= )/( β , and therefore αβ= −β 1 vdvdy α)/()/( . For validation of probabilistic model, its results have been After substitution of integration limits and their reduction to compared with results from other model. The Monte Carlo the minimum number of terms, the result is: simulation has been used to provide information related to the α  1  P ⋅ Γ  average values of the capacity factor. A Matlab program has rated     β   β  β  β   vrated  1  v −incut  1 PE )( = ⋅ P  ,  − P  ,  (13) been developed to validate the probabilistic model. E − vv  α β   α β  rated −incut        The program has been structured by two main functions. β First function is based on a probabilistic model previously Prated v −offcut α−⋅− ))/(exp( developed and modelled with (15) and (16), respectively. where Γ ( ) and P( ) are the gamma and the lower incomplete Second function has been developed based on Monte Carlo gamma functions, respectively [15]. simulations technique. This technique generates different B. The expected output power model of a wind turbine values of wind speed, in accordance with their Weibull distribution (with the shape and scale parameters estimated As it has been presented in (6), considering the wind power from the real data base) and these wind values are used to having a Weibull distribution, the expected value of the wind generate the output power, having in view the characteristics power can be expressed as a function of the parameters α’, β’ of the wind turbine generator. The expected output power from and the Gamma function: a wind turbine is the power produced at each wind speed  1  (14) sample, integrated over all possible wind speeds. The required PE V 1'][ +Γ⋅α=   β' capacity factor values may be observed from the average of all output power values, over a long number of samples. The C. The probabilistic model for capacity factor and efficiency is observed from average of all output power values, technical efficiency and also from all power transported by any values of wind The capacity factor of a wind turbine means its energy speed. The simulation can be stopped when a specified degree output divided by the theoretical maximum output, if the wind of confidence has been achieved. turbine generator were running at its rated (maximum) power The methodology presented in this paper was applied to a during all the time. So, the capacity factor is the ratio of the real wind turbine and for a real wind speed database, to expected output power over a period of time to the rated power validate the probabilistic model and to evaluate the influence

Issue 1, Volume 5, 2011 162 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES

of main parameters of wind turbine generator to capacity The probability density function (PDF) and cumulative factor and efficiency value. The wind speed database was distribution function (CDF) of the output power for a 1.5 XLE collected from the north-east area of Romania, for a wind turbine, considering a Weibull distribution with measurement interval to one hour for the year 2008. The mentioned parameters, are presented in the figure 5.a,b. figures 3.a,b present the wind speed collected from wind station height (10m) and adjusted to the hub wind turbine Probability density function for 1.5 XLE Wind Turbine (GE) 0.9 height (80m). 0.8 Wind speed to measured height (10m) 15 0.7

10 0.6

0.5 5 pdf(Pel) wind speed (m/s) speed wind 0.4 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0.3 time (hr) Wind speed to 80 m height 0.2 30

0.1 20 0 0 0.5 1 1.5 10 Pel[MW] wind speed (m/s) speed wind Cumulative distribution function fot 1.5 XLE Wind Turbine (GE) 0 1 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 time (hr) 0.9 Fig. 3.a,b Wind speed data base from the north-east area of 0.8

Romania, for 10m, respectively 80 m height 0.7

0.6 The parameters of the Weibull distribution have been 0.5

estimated using the hourly wind data base. The probability cdf(Pel) density and the density function fitted for different wind speed 0.4 values in 1 m/s steps are presented in figure 4. The probability 0.3 distribution function used to fit is a Weibull distribution with 0.2 scale parameter α=4.82253 m/s and a shape parameter 0.1

0 β=1.8656. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Pel[MW]

Wind speed data Fig.5.a,b The PDF and CDF of output power wind turbine 0.16 Weibull fit 0.14 Considering the ρ=1.225 kg/m3 for the air density (at 15°C) 0.12 and the rotor are A=5346 m2, in the previous equation (7), the 0.1 scale parameter and shape parameter of Weibull distribution 5 0.08 Dens ity parameters are obtained as α’=3.6725×10 m/s and β’=0.6219, 0.06 respectively. The probability density function and cumulative

0.04 distribution function of the wind power are in accordance with

0.02 Weibull distribution with mentioned parameters. -4 x 10 0 0 5 10 15 20 Wind speed data (m/s) Fig. 4 The wind data base associated distribution

The wind turbine chose for analyze is an active blade pitch 2 control wind turbine, namely 1.5 XLE GE-Energy, manufactured by GE Energy [18], with their technical

specifications presented in table 1: pdf(Pw)

1 Table 1 Technical specifications of wind turbine GE Energy Rated Cut-in Rated Cut-off Rotor Hub Turbine power speed speed speed diameter Heights Model (MW) (m/s) (m/s) (m/s) (m) (m) 1.5xle - GE 0 1.5 3.5 11.5 20 82.5 80 0 0.5 1 1.5 Energy Wind Power Pw [MW]

Issue 1, Volume 5, 2011 163 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES

1 0.4

0.9 0.35

0.8 0.3 0.7 0.25 0.6

0.5 0.2 cdf(Pw)

0.4 Efficiency (%) 0.15 WTG 1.5xle - GE Energy 0.3 Vin=3.5 m/s 0.1 Vrat=11.5 m/s 0.2 Voff=20 m/s 0.05 0.1 Wbl(a=4.82253;b=1.8656)

0 0 0 0.5 1 1.5 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Wind Power Pw [MW] Number of samples

Fig.6.a,b The PDF and CDF of wind power Fig. 8 Efficiency result from Monte Carlo simulation

Using the output power and wind power distribution The capacity factor values provided by probabilistic model functions in the probabilistic model previously developed, the and sequential Monte Carlo simulation are shown in Table 2. capacity factor and efficiency values were evaluated and For a better comparison between models, three ranges of speed presented in table 2 and 3. For validation, in the following is parameters of wind turbine were considered in capacity factor presented an example of capacity factor and efficiency evaluation. Commercial wind turbines typically have cut-in evaluation, for a 1.5 XLE GE-Energy wind turbine, using the speeds between 2.5 and 4.5m/s, a rated wind speeds between Monte Carlo Simulation technique (MCS). This technique 10 and 15 m/s and a cut-off speeds between 20 and 25 m/s. creates a fluctuating convergence coefficient of variation range for various numbers of samples for capacity factor (in figure 7) Table 2. Capacity factor values from probabilistic model (PM) and and for efficiency (in figure 8). The coefficient of variation of Monte Carlo simulation (MCS) the wind speed generated range can be used to improve the Capacity Factor Capacity Factor Capacity Factor vrat=11.5m/s vcut-in=3.5m/s vcut-in=3.5m/s effectiveness of MCS, this being often used as the convergence vcut-off=20m/s vcut-off=20m/s vrat=11.5m/s criterion in simulation techniques. vcutin PM MCS vrat PM MCS vcutoff PM MCS Number of simulations results from condition that the 2.5 22.3301 22.2012 10 20.4575 20.3350 20 16.8492 16.7547 deviation of the coefficient of variation of CF and EF ranges 3 19.5020 19.6152 11 17.9203 17.9318 21 16.8492 16.9726 3.5 16.8492 16.6911 12 15.8886 15.6188 22 16.8493 16.7824 to expected value to be under a settled value. Using 4 14.4048 14.8282 13 14.2455 14.1276 23 16.8493 16.7034 simulations techniques, for a settled value (0,01%) is obtained 4.5 12.1901 11.9975 14 12.8995 12.8572 24 16.8493 16.8065 about 10.000 necessary samples, the convergence process of 5 10.2157 10.3042 15 11.7815 11.7896 25 16.8493 16.7920 CF being presented in the figure 7, and for EF in figure 8. As described above, table 3 shows the efficiency values for 24 different values of cut-in, rated and cut-off turbine speeds. WTG 1.5xle - GE Energy Vin=3.5 m/s 22 Vrat=11.5 m/s Table 3. Efficiency values from probabilistic model (PM) and Voff=20 m/s Monte Carlo simulation (MCS) Efficiency Efficiency Efficiency 20 Wbl(a=4.82253;b=1.8656) vrat=11.5m/s vcut-in=3.5m/s vcut-in=3.5m/s vcut-off=20m/s vcut-off=20m/s vrat=11.5m/s 18 vcutin PM MCS vrat PM MCS vcutoff PM MCS 2.5 37.5769 37.3589 10 32.1705 32.2894 20 28.3533 28.4966 3 32.8178 32.6990 11 30.1561 29.7850 21 28.3535 27.9834 Capacity Factor (%) 16 3.5 28.3536 27.7772 12 28.3536 27.7068 22 28.3536 29.1203 4 24.2402 24.2985 13 26.7372 26.9143 23 28.3537 28.5333 4.5 20.5134 20.8663 14 25.2835 25.4662 24 28.3537 27.9599 14 5 17.1909 16.3235 15 23.9722 23.5458 25 28.3537 28.2550

12 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 It can be seen that the results obtained from both methods Number of samples are very close. The probabilistic method provides comparative Fig. 7 Capacity factor result from Monte Carlo simulation results with Monte Carlo simulation, these proving the accuracy of probabilistic model, developed in equations (15) and (16). The analytical expressions developed in (15) and (16) were

Issue 1, Volume 5, 2011 164 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES

used to study the effect of mean wind speed on both 30 coefficients. Graphic representation of dependence is shown in Vcut-in (3.5 m/s) affected by wind speed steps figure 9, with solid line for efficiency and dashed line for Vrated (11.5 m/s) affected by wind speed steps Vcut-off (20 m/s) affected by wind speed steps capacity factor. 25

0.7 Capaciy factor 20 0.6 Efficiency

0.5 15 Capacity Factor (%)

0.4 10 0.3

5 0.2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Capacity factor efficiencyand wind speed step (m/s)

0.1 Fig. 10 Effect of wind turbine parameters on capacity factor

0 0 5 10 15 20 0.5 Mean wind spead (m/s) Vcut-in (3.5m/s) 0.45 Vrated (11.5 m/s) Fig. 9 Effect of mean wind speed on CF and EF Vcut-off (20.5 m/s)

0.4 As it can be seen, the wind turbine efficiency is largest (in this case 50%) at a relatively low wind speeds, around some 4 0.35 m/s. But, at low wind speeds, efficiency is not so important, because there is not much energy to be converted. At higher 0.3 wind speeds, the turbine can not convert the excess energy that Efficiency (%) exceeds the limits of the generator. So, the efficiency is not the 0.25 best indicator for evaluating the suitability of wind turbine to a 0.2 specific location. For suitability evaluation, the capacity factor is a better indicator. If the wind turbine is located in an area -1.5 -1 -0.5 0 0.5 1 1.5 with an average speed around some 10m/s, the maximum wind speed step (m/s) output power will be generated, even if the efficiency is low. Fig. 11 Effect of wind turbine parameters on efficiency values So, it is not an aim in itself to have a high technical efficiency of a wind turbine. Since the fuel is free, the technical As it can be seen from the figure 9, a certain value of efficiency is not important, the wind energy can be used or will capacity factor or efficiency can be achieved by changing the be lost. What really matters is the amount of generated energy, two parameters of wind turbine generator. Most important even with a lower efficiency parameter and providing the greatest degree of freedom is cut- Also, the proposed model may be used to analyze the effects in wind speed. It has been shown that the cut-in wind speed of different cut-in, rated and cut-off wind speeds on the has a significant effect on the capacity factor and efficiency capacity factor value. Using a 1.5-XLE GE Energy wind values. Their values decrease approximately linearly as the turbine, placed in Iasi location, with previously wind profile, cut-in wind speed increases. the capacity factor will be 16.85% and the efficiency, 28.35%. The second parameter of wind turbine generator with effect for this wind profile, from table 2, it can see, a wind turbine on the capacity factor and efficiency is the rated wind speed. It generator can be expected to operate with a maximum capacity has been shown that the rated wind speed has a relatively small factor of 22.33% and a maximum efficiency of 37.57% for a effect on these values. The rated wind speed growth leads to wind turbine generator characterised by a wind speed the capacity factor and efficiency values decrease, but this parameters set to vcut-in=2.5m/s, vrated=11.5m/s and vcut- effect is less significant than that of the cut-in wind speed off=20m/s, respectively. A capacity factor of 22.33% from a It has been shown that the cut-off wind speed has no effect 1.5kW wind generator means a mean output power of 0.335 on either the capacity factor nor efficiency values. The cut-off kW or an annual power output of 2934.6 kWh. wind speed is a safety parameter and is usually large. For The effects of the wind turbine generator parameters on the relatively few times the instantaneous wind speed at a capacity factor and efficiency are shown in figure 10, and particular area will be greater than the cut-off speed. The figure 11, respectively. In the same system coordinates is selection of the cut-off speed parameter is therefore less shown the dependence of capacity factor and efficiency for important than that of the cut-in and the rated wind speed various wind speeds values around of wind turbine generator parameters. parameters (cut-in, rated, cut-off speeds).

Issue 1, Volume 5, 2011 165 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES

V. CONCLUSION [14] D. Villanueva, A. Feijoo, “Wind power distributions: A review of their applications”, Renewable and Sustainable Energy Reviews 14 (2010) Integration of wind energy is an important activity in the 1490–1495 developing process of the electric power system. Knowing the [15] MathWorks Products. Statistics Toolbox. Function Reference, Gamma capacity factor values is a key factor when examining wind Distribution. [16] L. Bain, M. Engelhardt, Introduction to Probability and Mathematical energy potential for a wind turbine located in a specific area. Statistics, Duxbury Press, California, 1992. The probabilistic methods are the recommended solution for [17] A. Papoulis, Probability, Random Variables and Stochastic Processes, wind integration analysis, since they can take into account the McGraw-Hill, New York, 1984. [18] http://www.ge-energy.com/prod_serv/products /wind_turbines/ wind power uncertainty. This paper presents a probabilistic model to evaluate the capacity factor and technical efficiency of a wind turbine Ciprian Nemes was born in Turda, Romania, on based on the output power distribution. The results were May, 1975. He graduated from “Gh. Asachi” Technical University of Iasi and received the MSc validated using the Monte Carlo simulations, and the analysis degree in electrical engineering and PhD degree in demonstrates that the probabilistic model results are very reliability engineering. accurate. The model has the advantage that can be easily He is currently a Senior Lecturer and research interests are in the area of power equipment implemented in computer programs and require a computing reliability, power system planning based on risk time considerably less than in the case of simulation methods. assessment, renewable energy sources operation and The electric energy output of a wind turbine for a specific planning. area depends on many factors. These factors include the wind Florin Munteanu was born in Campina, Romania, speed conditions at the area, and the characteristics of the wind on April 10, 1954. He graduated from “Gh. Asachi” turbine generator. Technical University of Iasi and he received the MSc The case studies show that turbine cut-in wind speed has a degree in Power Engineering in 1979 and PhD degree in Reliability Engineering in 1995. Starting significant effect on the both capacity factor and technical with 1984 he is with “Gh. Asachi” Technical efficiency values while the cut-off wind speed has almost no University of Iasi where, from 1999, he is holding a effect. Significant benefits can be obtained by selecting full time professor position and from 2008 he is also the head of Power Engineering Department. The suitable wind turbine parameters for the specific wind profile. main fields of interest included transients of power systems, power quality and reliability. REFERENCES [1] Tony Burton, David Sharpe, Nick Jenkins, Ervin Bossanyi, Wind Energy Handbook, John Wiley & Sons 2001. [2] Thomas Ackermann, Wind Power in Power Systems, Ed. John Wiley & Sons 2005. [3] R. Billinton, W. Y. Li, Reliability assessment of electrical power systems using Monte Carlo method, Plenum Press, New York, 1994. [4] Nicolas Boccard, Capacity Factor of Wind Power, Realized Values vs. Estimates, Raport project SEJ2007-60671. Depart. d’Economia, Universitat de Girona, Spain. October 2008. [5] Z. M. Salameh, I. Safari, Optimum windmill –site matching, IEEE Trans. Energy Convers., vol. 4, no. 7, pp. 669-675, 1992. [6] Tsang-Jung Changa, Yi-Long Tua, Evaluation of monthly capacity factor of WECS using chronological and probabilistic wind speed data: A case study of Taiwan, Renewable Energy 32, 2007, pp.1999–2010. [7] E. Kavak Akpinar, S. Akpinar, “An assessment on seasonal analysis of wind energy characteristics and wind turbine characteristics” Energy Conversion and Management 46, 2005, pp. 1848–1867. [8] Isaac Y. F. Lun, Joseph C. Lam, “A study of Weibull parameters using long-term wind observations” Renewable Energy Journal, Volume 20, Issue 2, June 2000, pages 145-153. [9] J.A. Carta, P. Ramırez, “A review of wind speed probability distributions used in wind energy analysis. Case studies in the Canary Islands” Renewable and Sustainable Energy Reviews, volume 13, Issue 5, June 2009, pp. 933-955. [10] Mohammad A. Al-Fawzan, “Algorithms for Estimating the Parameters of the Weibull Distribution” Interstat Journal, October 2000. http://interstat.statjournals.net/YEAR/2000/articles/0010001.pdf [11] Nemes C., Munteanu F., “Optimal Selection of Wind Turbine for a Specific Area”, OPTIM 2010, Brasov, Romania. 978-1-4244-7020- 4/10/26.00 '2010 IEEE, pp. 1224-1229. [12] Nemes, C., Munteanu, F., “Development of Reliability Model for Wind Farm Power Generation”, Advances in Electrical and Computer Engineering, ISSN 1582-7445, e-ISSN 1844-7600, vol. 10, no. 2, 2010pp. 24-29. [13] M.H. Albadia, E.F. El-Saadanyb “New method for estimating CF of pitch-regulated wind turbines”, Electric Power Systems Research 80 (2010) 1182–1188.

Issue 1, Volume 5, 2011 166 [Downloaded from www.aece.ro on Wednesday, October 10, 2018 at 04:58:47 (UTC) by 81.180.218.239. Redistribution subject to AECE license or copyright.]

Advances in Electrical and Computer Engineering Volume 10, Number 2, 2010 Development of Reliability Model for Wind Farm Power Generation

Ciprian NEMES, Florin MUNTEANU "Gh. Asachi" Technical University of Iasi Bd. D. Mangeron 53, RO-700050 Iasi, Romania [email protected]

Abstract—The objective of the present paper is to estimate modeling, is that each WT will not have an independent the electric power distribution of a wind farm using the output output power distribution, due to the dependence of the power distribution and the probability of simultaneously individual turbine by the same energy source, the wind. The running of wind turbines. In this paper, a methodology for the output power from a WF is determinate by the electric electric power distribution of a wind turbine, using the characteristic output power – wind and the Weibull output power from each wind turbines and the number of distribution of wind speed was given first. The methodology simultaneously running wind turbines. was applied to a region from north-east of Romania, and, For the reliability evaluation, a WF output power can be finally, the analytic distributions were compared to Monte- modeled as a continuous random variable that reflects the Carlo simulation. fluctuating characteristics of wind speed and aspects of wind turbines reliability. Index Terms—Electric power system, Probability, Wind In the paper, an analytical approach is developed to power generation analyze the probabilistic model of an output power of a WF. I. INTRODUCTION The purpose is to establish a probabilistic function for total power generated by a WF, having in view the probabilistic A growing interest in renewable energy resources has output power of a WT and the probability that a certain been observed in last time. Unlike other renewable energy number of whole WTs, to be simultaneously running. sources, wind energy has become competitive with The output power random variable of WF can be conventional power generation sources and therefore calculated using variety methods. The two main approaches application of wind turbine generators has the most growth will be considered: among other sources. Wind is one of fastest growing energy • analytical method, where the WF is represented by sources, and is regarded as an important alternative to mathematical models of whole WTs, and every traditional power generating sources. output power value is direct analytically evaluated, Wind generation brings a great amount of benefit to respectively; power systems, such as cheap energy comparing with the • Monte Carlo simulation (MCS), which estimates thermal generation, emission reduction, and development of the possible values of output power, by simulating a wind power farm can be implemented much easier and the behaviour of WF with random wind speed faster than building a thermal or hydro plant, wind energy values. available for large areas, etc. Meantime, wind generation A numerical application is presented to validate the brings a series of difficulties to the traditional power developed model, the analytical results being compared with systems, as follows: uncontrollability of power generation, MCS. the wind generation depends on wind availability, intermittence of power generation, the wind generation II. WIND TURBINE MODELING shows irregularly fluctuating and intermittent behaviour, respectively a poor predictability of the wind generation The output power from WT depends by the availability of [1,2]. the energy source, namely by the speed of the wind, and by The integration of a large wind generation will have a the power-wind characteristics of the WT’s generator. significant impact on the reliability indices of the electric A. Probabilistic model of wind speed power system. In order to include wind generation into Wind is a turbulent movement mass of air resulting from power system reliability assessment, it is necessary to the differential pressure at different locations on the earth develop models and suitable techniques that take into surface. One of the main characteristics of wind is that it is account the main features of this energy source: the highly variable in time and space, the variation of wind availability of its components (that is a typical aspect in exists from instantaneous, hourly, daily to seasonal, and its reliability) and the variability and randomness of its output properties vary from one location to another. The wind power. property of interest when it comes to power generation is the A probabilistic predictability of the output power of wind wind speed probabilistic model. turbine (WT) is possible to establish having in view the The wind blows faster at higher altitudes because of the probabilistic distribution of the energy source (wind speed). reduced influence of drag of the surface and lower air For a wind farm (WF), constituted from a large number of viscosity. The increase in velocity with altitude is most WTs, the main characteristic and a major difficulty in dramatic near the surface and is affected by topography,

Digital Object Identifier 10.4316/AECE.2010.02004 24 1582-7445 © 2010 AECE [Downloaded from www.aece.ro on Wednesday, October 10, 2018 at 04:58:47 (UTC) by 81.180.218.239. Redistribution subject to AECE license or copyright.]

Advances in Electrical and Computer Engineering Volume 10, Number 2, 2010

surface roughness, and upwind obstacles such as trees or and rated speed wind (vrated), value for that the buildings. Typically, the increase of wind speeds with power achieves the rated power (Prated). increasing height follows a logarithmic profile model [3], • the rated power of a wind turbine, generally that matches the wind speeds at a desired and registered the maximum power output of a generator at height, having in view the surface roughness length, a highest efficiency, is produced when the speed characterization of a ground terrain. lies between rated and cut-out wind speed (vcut- To establish the parameters of probability density out). distribution is necessary an accurate dataset of wind speed. • cut-out wind speed is the maximum wind speed The wind speed database can be obtained from at which the turbine is allowed to produce meteorological station, where the measurement point power, usually limited by engineering design (anemometer) height above the surface may be to 10 m or and safety constraints. 50 m. Depending by the wind measurement level, the speed Electric power data must be adjusted for the change in height desired (MW) according to a logarithmic profile previously mentioned. Prated The probability density function (pdf) of wind speed is important in numerous wind energy applications. A large number of studies have been published in scientific literature related to wind energy, which propose the use of a variety of pdf-s to describe wind speed frequency distributions [4,5]. The conclusion of these studies is that the Wind speed Weibull distribution of two parameters may be successfully v v (m/s) utilized to describe the principle wind speed variation. cut-in rated vcut-out For account the variability of wind speed, it is assumed to Figure 1. The power curve of a wind turbine. be characterized by a Weibull distribution with a scale This curve comes available from the WT manufacturer or parameter  (m/s) and a shape parameter  (dimensionless). plotted using recorded wind speed and corresponding power The Weibull probability density and cumulative distribution output data. Thus, the electric power PWT may be calculated functions are given by: from the wind speed v as follows: P v  v   1       rated cutin for v  v  v   v   v  ;  v  (1)  cutin rated fw (v)    exp    Fw (v)  1 exp    vrated  vcutin              (2)     PWT v   Prated for vrated  v  vcutout  0 other else  In many studies, the Weibull factor is often chose to 2 and  therefore a Rayleigh distribution can be used, with a less accurate, but simpler model. C. Probabilistic model of WT output power The estimates of the parameters of the Weibull Lets assume that the wind speed v is a continuous random distribution can be found using different estimation methods variable with cumulative distribution function Fw(v) and that [2,4]. Each method has a criterion, which yields estimates PWT=J(v) defines a one-to-one transformation from a region that are best in some situations. Different results are of the wind-space, to a region of the electric power-space, -1 produced based on that criterion. The most commonly with inverse transformation v=J (PWT) [7]. methods are Maximum Likelihood Estimator, Method of T W

P PWT Moments, Least Squares Method or Regression Method. PWT=J (v) The Maximum Likelihood Estimator is so commonly applied in engineering and mathematics problems [6], so, the MLE method to establish the parameters of wind speed distribution is used in this paper. )

T v W FW(v) B. The output power model of a wind turbine P ( P

The output electric power of a WT is a function of the F wind speed. The power curve gives a relation between the v wind speed and the output electric power, a typical curve of the WT generator is nonlinear related to the wind speed. Figure 2. Transformation of the wind speed variable. However, the assumption of the linear characteristic of The cumulative distribution function of output power of power with the wind speed, brings a significantly simplifies WT is given by: of calculations, without roughly errors. The power output F (P)  Pr(P  p)  Pr(J(X )  p)  characteristic can be assumed in such way: WT (3) 1 1 • it start generating power when the speed wind  Pr(X  J ( p))  FW (J ( p)) exceeds the minimum wind speed, namely cut- The possible values of F (P) may be roughly classified in speed (v ); WT cut-in in 0 and P , respectively, in the interval that lies between the power output increase linearly with the rated • mentioned values. Each possible value has been evaluated, wind speed, when wind varies between cut-in

25 [Downloaded from www.aece.ro on Wednesday, October 10, 2018 at 04:58:47 (UTC) by 81.180.218.239. Redistribution subject to AECE license or copyright.]

Advances in Electrical and Computer Engineering Volume 10, Number 2, 2010 having in view the probability to achieve that value. The unavailability of WT may be associated with a simple two probability density function results from differential of state model (up and down state) [#]. cumulative distribution function [8]:   1 for P  0 UP DOWN   vrated  vcutin  (4) fWT (P)     fW W  for 0  P  Prated  Prated  Figure 4. A two state model for a WT.  2 for P  Prated The unavailability of WT may be modelled by its forced where: outage rate (FOR) and availability as: 1 1  F (v )  F (v ) F )0( ; W cutout W in WT FOR   (  ) and AV 1  FOR   (  ) (5) 2  F (v )  F (v ) 1  F (P ) , and W cutout W rated WT rated where  and  are the expected failure and repair rates and  P  considering the exponential distribution of the mean time to   W  vrated  vcut in   vcut in  . repair and of the mean time to failure.  Prated  The new technology has achieved such a level of quality, The probability density function (pdf) and cumulative that wind turbines obtain a technical availability of 95-98 distribution function (cdf) of the output power of a 1.5xle percent [10]. wind turbine manufactured by GE Energy [9], for a Weibull distribution with a scale parameter =4.9939 m/s and a III. WIND FARM MODELING shape parameter =1.8656 are presented in the figure 3.a,b. A WF is a group of WTs in the same location used for Probability density function for 1.5 XLE Wind Turbine (GE) production of electric power. Individual turbines are 0.9 interconnected between them and connected to electric 0.8 power network. WFs have many turbines and each extracts

0.7 some of the energy of the wind. The wind park effect refers to the loss of wind energy due to mutual interference 0.6 between turbines is not consider in the paper. )

l 0.5 e P

( A. Probabilistic model of output power for N f d p 0.4 simultaneously available wind turbines 0.3 Let assume a WF that contains N WTs. For whole N WTs,

0.2 simultaneously running, the output power from WF is sum of the output power of each WT. 0.1 N 0 0 0.5 1 1.5 P  P (6) WF  WTk Pel[MW] k 1

Cumulative distribution function fot 1.5 XLE Wind Turbine (GE) 1 Generally, is not necessary to assume that all the turbines in a WF are alike, this assumption is valid in practice [11]. 0.9 Having in view, that all WTs running in same time, turned 0.8 by the same gust of wind, with the same wind speed, it 0.7 results that the output power density function from WF is 0.6 density power of a random variable obtained from output ) l e P

( 0.5 power variable multiply with N. f d c 0.4 PWF  N  PWT (7) 0.3 The, probabilistic density function of the output power 0.2 from a WF constituted by N wind turbines, simultaneously 0.1 running, is: 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Pel[MW]  1 for P  0 Figure 3. a,b-The PDF and CDF of output power wind turbine.  vrated  vcutin  (8) f k WT (P)     fW Wk  for 0  P  k  Prated D. Availability of a wind turbine  k  Prated  Availability is the most general concept including the  2 for P  k  Prated reliability, and it refers to the percentage of time that a WT  P  is ready to generate (that is, not out of service). Availability   where Wk  vrated  vcutin   vcutin  . includes downtime for any reasons (faults, repairs,  k  Prated  maintenance, upgrades). The duration of downtimes, caused by malfunctions, are dependent on necessary repair work, on Because in the following analytical model, the probability the availability of replacement parts and on the personnel of zero WTs simultaneously running is discussed (is a capacity of service teams. The concept of availability and theoretical assessment), the probability density function of

26 [Downloaded from www.aece.ro on Wednesday, October 10, 2018 at 04:58:47 (UTC) by 81.180.218.239. Redistribution subject to AECE license or copyright.]

Advances in Electrical and Computer Engineering Volume 10, Number 2, 2010 output power from a WF with zero wind turbines is output power for different number of available WTs of the evaluated: farm. If the events Ek , with k = 1,2,3... , is an accountably partition of event space, with the assumption that the events 0 1 for P  0 fWT (P)   (9) are mutually exclusive, then for any event P from the event 0 for P  0 space we have: The following figure shows the probability density N function and cumulative distribution function for output Pr(P)   Pr(P / Ek )  Pr(Ek ) (11) power from a wind farm with N=1,2,3 wind turbines, k0 simultaneously running. where:

Probability density function for N x 1.5XLE Wind Turbine (GE) 0.9 • Pr(P) is probability of events that the output power

0.8 to be equal with PWF;

0.7 • Pr(P / Ek ) is the conditional probability that the N=1 WT

0.6 output power to be PWF, when k wind turbines running: 0.5 ) l e P ( f PWF 0 d p 0.4 Pr (P  PWT / Ek )  f k WT (P)dP (12) P 0 WF 0.3 N=2 WT • Pr (E ) is the probability of event that k from N WTs 0.2 k to simultaneously running. This probability is 0.1 N=3 WT modelled using Binomial distribution:

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Pel [MW] Pr (Ek )  k / N, AV  (13)

Cumulative distribution function for N x 1.5XLE Wind Turbine (GE) 1 Appling the total probability theory to mentioned events, it is obtained: 0.9 N=1 WT N=2 WT N P 0.8  WF 0  N=3 WT   Pr(P  PWF )   f k WT (P)dP  k / N, AV  (14)   P  0.7 k0  WF 0   0.6 For each interval of the output power, in according with ) l e P

( 0.5 f

d the eq. 8, the pdf of output power for whole domain is: c 0.4  N  /0 N, AV  1 k / N, AV 0.3        k1 0.2  for PWF  ,0 N 0.1  (15) N  f (P ) k / N, AV fWF (PWF )    k WT WF   0 k1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5  Pel[MW]  for (k  )1  Prated  PWF  k  Prated , Figure 5. The PDF and CDF of output power for N wind turbines  2k / N, AV  simultaneously running.   for PWF  k  Prated . B. Availability of the k/N wind turbines The number of simultaneously running WTs depends by where: whole number of turbines from WF and by its reliability.  vrated vcutin   vrated  vcutin  P  Taking into account that the availability of any WT is alike, f k WT (PWF )    fW   vcutin   k  P   k  P  the each outage of a WT from the farm may be counted using  rated   rated  the Binomial distribution. The probability to have k identical turbines simultaneously running, from whole N turbines, is As can see, the pdf of output power is a continuous calculated according to the Binomial distribution. function on the (k  )1  Prated  k  Prated intervals, and with k k N k discrete values in the points where the power is equal to k / N, AV = C N AV (FOR)  (10) multiples of the rated power. where AV/FOR is availability/unavailability of any wind turbine. IV. NUMERICAL EXAMPLE In following section, the binomial distribution is applicable for counting the probability of outage of k wind The methodology presented in this paper was applied to a turbines from whole N number wind turbines of farm. wind farm composed by N=1,2,3 wind turbines. The wind turbines chose for analyze are manufactured by GE Energy C. Probabilistic model of WF output power [5], and their technical specifications are presented in table The output power of a WF with k units depends by the 1: pdf-s of a power outage and by the probability to have k WTs running in same time. TABLE 1. TECHNICAL DATA OF WIND TURBINE GE ENERGY Rated Cut-in Rated Cut-off Hub Turbine Model Total theory probability may be used to obtain certain power speed speed speed Heights

27 [Downloaded from www.aece.ro on Wednesday, October 10, 2018 at 04:58:47 (UTC) by 81.180.218.239. Redistribution subject to AECE license or copyright.]

Advances in Electrical and Computer Engineering Volume 10, Number 2, 2010

(MW) (m/s) (m/s) (m/s) (m) techniques. It has been found that the coefficient of variation 1.5xle - GE 1.5 3.5 11.5 20 80 has the lowest rate of convergence [12]. Energy MCS creates a fluctuating convergence coefficient of A real wind speed measurement is utilized in this paper. variation range for various numbers of samples. This The wind speed database is collected from the north-east coefficient cannot be reduced to be zero and therefore it is area of Romania, for a measurement interval to one hour for always necessary to utilize a reasonable and sufficiently the year 2008. The figures 7.a,b present the wind speed large number of samples. Number of simulations results from collected to wind station height (10m) and adjusted to the condition that the deviation of the coefficient of variation of hub wind turbine height (80m). wind speed range to theoretical value to be under a settled value. Using simulations techniques, for a settled value Wind speed to measured height (10m) 15 (0,1%) is obtaining about 10000 necessary samples, the ) s

/ convergence process of coefficient being presented in figure m

( 10

d 9. e e p s 5 d 1 n i w

0 0.9 e

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 g n a

time (hr) r

d 0.8 Wind speed to 80 m height e e

30 p s

d ) n

i 0.7 s / w

r m ( 20 o

e 0.6 e c e n p a i s r 10 a d v n

0.5 i f o w

t n e

0 i

c 0.4 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 i f f

time (hr) e o

C 0.3 Figure 7. a,b - Wind speed data base from the north-east area of Romania, 0.2 for 10m, respectively 80 m height. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Number of samples The parameters of the Weibull distribution have been estimated using the hourly wind data base. The probability Figure 9. Convergence process of coefficient of variation. density and the density function fitted for different wind For each wind speed value, a random number from a speed values in 1 m/s steps are presented in figure 8. The Binomial distribution is generated, establishing the number probability distribution function used to fitting is a Weibull k of WTs of farm that simultaneously running. With the distribution with scale parameter =4.82253 m/s and a wind speed value and k number of WTs simultaneously shape parameter =1.8656. running, the output power of WF is calculate. These operations are repeated for whole 10000 generated values of

Wind speed data wind speed. The output power range, calculated from 0.16 Weibull fit mentioned method, was sort in ascending order. Each output 0.14 power value has in correspondence a number of

0.12 observations, which divided to the number of simulations and output power step give the probability density function,

y 0.1 t i

s as is presented in figure 10. n e 0.08 D Pdf from Monte Carlo Simulation with FOR=3% and MC=25000 0.45 0.06

0.4 0.04

0.02 0.35

0 0.3 0 5 10 15 20 Wind speed data (m/s) ) 0.25 P ( f d Figure 8. The wind data base associated distribution. p 0.2

The Matlab program has been developed to validate the 0.15 analytical model previously developed. The program has 0.1 been structured by two main functions. First function is developed based on analytical model, that has been 0.05 developed and modelled with eq.15. 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Second function has been developed based on simulations P[MW] techniques. Using MCS, a number MC wind speed values was Figure 10. Output power simulation pdf for a wind farm, with 3× 1,5 XLE generated, in accordance with the Weibull distribution (with and FOR=0,3. the shape and scale parameters estimated from the real data base). The coefficient of variation of the wind speed generated range can be used to improve the effectiveness of MCS, this being often used as the convergence criterion in simulation

28 [Downloaded from www.aece.ro on Wednesday, October 10, 2018 at 04:58:47 (UTC) by 81.180.218.239. Redistribution subject to AECE license or copyright.]

Advances in Electrical and Computer Engineering Volume 10, Number 2, 2010

Probability density function 3x1.5XLE (GE) with FOR=1%,3%,5% 3.1 0.9517 0.9529 0.9545 0.9532 0.9573 0.9565 0.45 … 0.22 FOR=1% 0.4 FOR=3% 4.4 0.9899 0.9910 0.9905 0.9904 0.9910 0.9910 0.2 FOR=5% 4.5 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.35 0.18

0.08 0.3 0.16 Analyzing the previously results, it can be observed a 0.07 1.4 1.5 1.6 good similitude between analytical and simulation methods, ) 0.25 0.06 P ( f

d these proving the accuracy of analytic model. p 0.05 0.2 0.015 2.95 3 3.05 0.15 0.01 V. CONCLUSION 0.1 0.005 4.46 4.48 4.5 Knowing the power domain is an important factor when 0.05 examining wind energy conversion systems. The energy 0 wind production must be maximized, but, in the other hand, 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 P[MW] it is necessary to consider the security of power system Figure 11. Output power analytical pdf for a wind farm, with 3× 1,5 XLE owing to add wind energy with an unforeseeable output and FOR=1%-3%. power. Figure 11 shows the analytical pdf of output power The primary energy source, wind is not always available generate by a WF with N=1,2,3 WTs, considering FOR=0.1- and the power output characteristic of any wind turbine 0.3. This graphic is presented to have a comparison with the generator (WTG) is different compared to the traditional simulations results. generation units Because the pdf values of output power, from MCS, are These require a good assessment of the probabilistic very fluctuating, is very difficult to have a comparison distribution of electric power injected in the electric power between analytical and simulations results for pdf of output network. power. For that, is necessary to generate the cdf of output This paper presents an analytical method to evaluate the power, from analytical and simulations methods. The generated power of a wind farm based on the wind speed cumulative distribution function results from divided distribution, the characteristic power- wind speed and the corresponding number of each output power values from reliability of a wind turbine, respectively. The results were generated range to the number of simulations. validated using the Monte Carlo simulations, and the final Cumulative distribution function 3x1.5XLE (GE) with FOR=1%,5% results provided the probabilistic functions are very 1 accurate.

0.9 An analysis of the pdf of output power of WF shows a relative grouping of the more probable output power in left 0.8 side of the power domain. In other words, that means the maximal values of the output power appear with a smaller ) P (

f 0.7 d

c probability that their minimal values.

0.6 REFERENCES FOR=1% Analytical method FOR=1% Monte Carlo Simul. 0.5 [1] Thomas Ackermann Wind Power in Power Systems, Ed. John Wiley FOR=5% Analytical method & Sons 2005. FOR=5% Monte Carlo Simul. [2] J. Endrenyi, Reliability modeling in electric power systems, Wiley 0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 and Sons, New York, 1978. P[MW] [3] Manwell J.F., McGowan J.G, Rogers A.L, Wind energy explained, Figure 12. The output power cdfs from the analytical model and Monte Ed. Wiley & Sons 2002. Carlo Simulations. [4] Isaac Y. F. Lun, Joseph C. Lam, A study of Weibull parameters using long-term wind observations Renewable Energy Journal, Volume 20, Figure 12 shows the cumulative distribution function of Issue 2, June 2000, pages 145-153. output power of WF, for various values of FOR, using the [5] J.A. Carta, P. Ramırez. A review of wind speed probability distributions used in wind energy analysis. Case studies in the Canary analytical model, respectively MCS. Differences between Islands. Renewable and Sustainable Energy Reviews, Volume 13, analytical and simulation results are very small, as is Issue 5, June 2009, Pages 933-955. www.elsevier.com/locate/rser. presented in table 2, where are calculated cdf values for [6] Mohammad A. Al-Fawzan Algorithms for Estimating the Parameters various output power and various FOR, difference being of the Weibull Distribution. Interstat journal. October 2000. http://interstat.statjournals.net/YEAR/2000/articles/0010001.pdf caused by the samples number of the MCS. [7] C. Nemes, F Munteanu, New techniques for availability analysis, Ed. Politehnium Iaşi, sept.2008; TABLE 2. CDF VALUES FROM ANALYTICAL AND MCS [8] G. Tina, S. Gagliano, S. Raiti Hybrid solar/wind power system r

e FOR=1% FOR=3% FOR=5% probabilistic modeling for long-term performance assessment.

w Elsevier Ltd. Solar Energy, Volume 80, Issue 5, May 2006, Pages o Monte Monte Monte

P Analytic Analytic Analytic Carlo Carlo Carlo 578-588. 0 0.4026 0.4031 0.4027 0.4036 0.4027 0.4036 [9] http://www.ge-energy.com/prod_serv/products/wind_turbines/ 0.1 0.4322 0.4332 0.4331 0.4344 0.4341 0.4343 [10] Holmstrøm, N. Barberis Negra1, Survey of Reliability of Large … Offshore Wind Farms European Wind Energy Conference, Milan, 1.4 0.7575 0.7584 0.7641 0.7632 0.7708 0.7694 Italy, 7-10 May 2007. [11] M. Fotuhi-Firuzabad, A. Salehi Dobakhshari Reliability-based 1.5 0.7759 0.7761 0.7825 0.7817 0.7890 0.7889 Selection of Wind Turbines for Large-Scale Wind Farms. World 1.6 0.7934 0.7942 0.7999 0.7995 0.8063 0.8053 Academy of Science, Engineering and Technology … volume 49, January 2009, pp 734-740. 2.9 0.9398 0.9406 0.9427 0.9425 0.9456 0.9444 [12] Roy Billinton, Wenyuan Li Reliability assessment of electric power 3 0.9461 0.9469 0.9493 0.9487 0.9524 0.9507 systems using Monte Carlo methods, published by Springer, 1994.