Impedance Modeling for Probabilistic Node Analysis

CHAPTER 9 (chapter 9)

R. Langella, A. Testa – Second University of Naples (Italy)

In the previous chapter, models for probabilistic network analysis have been considered assuming, for simplicity sake, the network component impedances remain constant. This assumption can be justified by the complexity in taking into account the time variance of the network configuration and the existing correlations present; the assumption is near to be verified in short term scenarios. In this chapter, the single node analysis is considered in a long term scenarios so that the dimensions of the problem are reduced and impedance variability cannot be ignored. The probabilistic aspects of harmonic impedances are introduced with reference to a case-study.

1 Introduction

A gGreat attention has been devoted to probabilistic approaches for the prediction of the harmonic currents injected by loads and of the corresponding nodal harmonic voltages 1-3. In this framework, minor importance has been devoted to the variability of the system impedances 4-6.

The network component parameter uncertainty and/or the network, supply and load configuration changing may cause significant variations of amplitude and phase of the nodal equivalent impedances. Parameter uncertainty is a consequence f.i. of their variability in time and with temperature or of the inherent difficulty in forecasting actual values in the designing stage for a new plant. Changing is a consequence of obvious operational needs; in particular, the effect of supply or load changing is relevant mainly when resonances are present. Under these conditions, a voltage distortion evaluation 7-8, based on the distorted load current injection, even if probabilistically modeled, but referring to a single reference condition for impedances, may produce a result that is incorrect and sometimes unacceptable.

1 In what follows, the effects of correlations among currents, impedances and voltages actually existing in a system are discussed. Then, the system impedance probabilistic model developed in 6 and taking into account parameter uncertainty, and load and supply configuration variability is recalled. Finally, the effects of modeling accuracy are discussed with reference to three methods for evaluating voltage distortion: the first does not take into account impedance variability; the second neglects the actual correlation effects between currents and impedances; the third takes into account the actual behaviour of currents and impedances.

2 General considerations

International standards suggest to the study ofthe frequency dependent behaviour of the supply system impedances in opportune conditions. The most critical conditions are related to the presence of strong parallel resonances causing very high values of the impedances around given frequencies. Resonance condition characteristics are determined in distribution networks by different factors among which the power factor correction capacitors, together with the load demand, play a very relevant role.

Thus, assuming invariant reference conditions to study the behaviour of the supply impedance means not to take into account the variability of the relevant quantities during time, which reflects on both resonance frequencies and impedance amplitude at resonant frequencies. On the other hand, it is well known that load variability in time requires tuning of the capacitors for the power factor correction.

Then, assuming impedance variability uncorrelated with current and voltage variability means not to take into account dependences which couple the behaviour of all the involved quantities in time. Having again in mind the power factor correction, the higher the load demand is the higher the capacitance value, and the lower the resonance frequencies, moreover, the higher the dumping effects of the loads are, acting as R-X elements in parallel to supplying impedances, are. Furthermore, the higher the whole load demand, at a given node, is, the higher the probability of having higher injected distorted currents and higher voltage back ground distortion.

The main links existing among the values assumed by the final quantity of interest (the harmonic voltage Vh) and the values of the quantities of direct influence and by quantities of indirect influence are reported in ??. The main direct influence quantities are: the harmonic h h h current I , the supply harmonic impedance Z and back ground voltage V BG; the main indirect influence quantities are: the total active and reactive load current I1 at fundamental frequency, the supply fundamental voltage V1 ….. and time t. Of course, in real cases further links do exist and their importance depends on the specific situation to analyze.

2 To have an idea also of the quantitative influence on the statistic parameters of interest, different correlation conditions can be analyzed with reference to a simple example. Both impedance and current are assumed uniformly distributed in a range of values between a h h h h minimum and a maximum value, Z min = 0, Z max = 1 and I min = 0 and I max =1 p.u., respectively; the absence of back ground voltage distortion is also hypothesized. The harmonic voltage behaviour can be obtained simply by applying the Ohm’s law and utilizing the Zh and Ih determinations obtained by means of a Monte Carlo simulation routine.

Vh h h h V BG I Z

1 V I1

Figure 9.1 Main links existing among the harmonic voltage Vh, and quantities of direct influence ( harmonic current Ih, supply h h 1 1 impedance Z , back ground voltage V BG) and of indirect influence (V , I , …. and time t).

?? reports the distributions of the harmonic voltage obtained in four different reference conditions:

 constant value for the impedance, which means f.i.:

h h Z max = Z = 1 p.u.;

 independence, which means that:

P( Zh | Ih ) = P( Zh ) and P( Ih | Zh ) = P( Ih ),

for each determination of Zh and Ih;

 “direct” dependence given by:

Zh  Zh Ih  Ih h h min min P( Z | I ) = 1 if h h  h h , otherwise = 0, Zmax  Zmin Imax  Imin

that means the determination of Zh has a distance from its minimum value, which is proportional to the distance of the corresponding determination of I h from its minimum value;

3  “cross” dependence given by:

Zh  Zh Ih  Ih h h min max P( Z | I ) = 1 if h h  h h , otherwise = 0, Zmax  Zmin Imax  Imin

that means the determination of Zh has a distance from its minimum value, which is proportional to the distance of the corresponding determination of Ih from its maximum value.

The observations that can be made are trivial and the differences among the shapes are very evident; 95% percentile values are also very different (the distances from the minimum values are respectively : 0.95, 0.70, 0.90 and 0.25 times p.u.).

Finally, the importance in principle of taking into account not only impedance variability but also the actual correlation among impedances and currents results is evident. Of course, the higher the variations of the impedance are, the higher their effects, hence the importance of taking them into account. An idea of impedance variability in a realistic condition is given in what follows. a) b) c) d)

f f f f f f f f d d d d p p p p p c p c p c p c

h h h h V 95% V 95% V 95% V 95%

h h h h h h h h V min V max V min V max V min V max V min V max Harmonic Voltage Harmonic Voltage Harmonic Voltage Harmonic Voltage

Figure 9.2 Pdf and Cdf of Harmonic Voltage for harmonic current and impedance uniformly distributed assuming: a) non variability of the impedance, b) Zh and Ih independence, c) Zh and Ih direct correlation and d) Zh and Ih cross correlation. 3 Impedance Variability

In 9, with reference to the IEEE industrial test system proposed by the IEEE Task Force on Harmonic modeling and simulation in 10, the impedance variability has been analyzed. The causes of impedance variability are: the system component parameter uncertainty, the load time variability and supply time variability.

The considered Industrial system consists of 13 buses and is representative of a medium-sized industrial plant. Due to the balanced nature of the system, only the unifilar scheme is presented (??); the positive sequence rated data are provided in 6 and are referred to 60 Hz fundamental frequency and to the 13.8 kV voltage.

G1 G2 50:GEN-1 100:UTIL-69

01:69-1 51:AUX

03:MILL-1

05:FDR F 26:FDR G 06:FDR H

49:RECT 29:T11 SEC 39:T3 SEC 11:T4 SEC 19:T7 SEC ASD

Figure 9.3 The IEEE industrial test system.

The plant is fed from a utility supply at 69 kV. Moreover, a local generator is directly connected to the local plant distribution system operating at 13.8 kV. The automatic tuning of the power factor correction capacitor is hypostasized.

Explicit reference is made to the self-impedance of the bus 49:RECT supplying the unique non- linear load. Here and in the following sections, the impedance values are all referred to the 480V level. The behavior of this impedance, obviously, can give an idea of what can happen to all the network buses except for the generation buses.

Adequate equivalents to represent the system components for harmonic and interharmonic penetration analyses are needed. The authors refer to classical models adopted in literature [9] and fully described in 6 for supply, lines and cables, transformers and loads.

3.1 Parameter uncertainty

The system component parameters L and C value uncertainty has been modeled in a Gaussian scenario. In particular, the majority of determinations (99.57%) of L and C fall in the intervals

(Lr-5%Lr,Lr+5%Lr) and (Cr-5%Cr,Cr+5%Cr), respectively, being Lr and Cr the rated values. The effects on the bus 49:RECT impedance magnitude are reported in ??, for different values of L and C parameters versus frequency (zoom from 240 to 680Hz).

5 0.16 -5% +5% 0.14

[ 0.1 e d u 0.08 t i n g

a 0.06 M 0.04

0.02 Lr,Cr 0 300 400 500 600 Frequency [Hz] Figure 9.4System impedance magnitude versus frequency (zoom from 240 to 680Hz), at node 49:RECT, for different values of L and C.

3.2 Load time variability

In order to take into account the effects of the load variations, all the loads present in the system have been modelled by means of a unique time varying equivalent circuit as described in 6. The parameter values of the equivalent load are set according to the values of power consumption. The capacitor varies its capacitance exactly compensating the reactive power requested by the equivalent load.

For each time interval of a typical working day, ?? reports the active power consumption, the power factor and their corresponding frequencies of occurrence, which can be considered as probability of occurrence in a generic instant of a working day.

Table 9.1 active power consumption, power factor and probability for different time intervals

     

P [pu] 0.2 0.5 0.8 0.7 1.05 0.6 cos 0.83 0.82 0.77 0.83 0.75 0.80

P(Ti) 0.459 0.125 0.125 0.083 0.125 0.083

?? shows the effects of the different power consumption on the system equivalent impedance as seen from the bus 49:RECT.

It can be observed that:

 the resonance frequency varies in a wide range from a minimum value (450 Hz) to a maximum value (1120 Hz); such a variability depends basically on the adjustments of the capacitance combining with the equivalent inductance at the node 03:MILL-1 practically fixed by the generator equivalent inductances;

 the impedance amplitude peak grows, as expected, as the load power consumption decreases;

6  the load variability strongly influences the system impedance at a given frequency, in a wide range of frequencies (f.i. from 500 to 800 Hz).

0.3 __T1

]  [ 0.25 __T4 __T6 e

d __T3 __T2

u 0.2 t i __T5 n g

a 0.15 M

0 0 500 1000 1500 F r e q u e n c y [ H z ]

Figure 9.5 System impedance magnitude versus frequency, at node 49:RECT, for different power consumption values.

3.3 Supply time variability

Supply characteristics and configuration may vary in time, f.i. according to duty cycles depending on different economic and technical reasons. For the sake of simplicity outages are here considered as the only cause of variability. ?? contains the local generator and the utility supply reliability parameters 11, in terms of failure rate, , unavailability, U, and of probability of outage in a generic time instant, P(GiOFF).

Table 9.2 supply reliability parameters

failures/year] U [h/year] P(GiOFF) -3 G1 1.64 2.4 410 -3 G2 4.5 32 0.3 10 ?? shows the effects of the different supply configurations on the system equivalent impedance as seen from the bus 49:RECT.

It can be observed how the supply configuration variability influences the equivalent system impedance much more than the other phenomena. Anyway, it must be considered that the probability of occurrence of the events G1OFFG2ON and G1ONG2OFF are very small, so, in practice, their mean influence is not relevant.

7 0.45

0.4 __

G1ONG2OFF ] 0.35  [ 0.3 e d u

t 0.25 i

n G1OFFG2ON g

M __ 0.15 G1ONG2ON

0 0 100 200 300 400 500 600 700 F r e q u e n c y [ H z ]

Figure 9.6 System impedance magnitude versus frequency, at node 49:RECT, for different supply configurations.

3.4 Model

In order to obtain the probabilistic model of the system impedance, Z, all the impedance variability causes, analysed in subheadings 3.1, 3.2 and 3.3, must be taken into account. A possibility is to take advantage of the Bayes formula 12 as described in 6.

In 6 probabilistic modelling was made for different time scenarios: i) the entire week (week), ii) the working day (day) and iii) the working-hours (w-hours) of the working day. For the weekend, the power consumption was assumed equal to that of time interval T1 of a working day.

In the following pdf diagrams, the abscissa axis is divided in a finite number of equal classes from 0 to 100% of the maximum value, for each impedance.

?? shows the probability density function (pdf) and cumulative probability function (cpf) of the system impedance magnitude for different frequencies referred to only working day scenarios. In particular ??a) and ??b) refer to harmonic frequencies (5th and 11th) while ??c) to the interharmonic frequency of 636Hz. It is possible to distinguish the contribution of the different time intervals, Ti, of the day: the color characterizing each time interval is chosen so that the darker the color is the lower the power load consumption. It is evident how the 5 th harmonic impedance has mono-mode behaviour, with a very low variance, due to its distance (left side) from the shifting resonance peak: also a deterministic modeling could give equivalent information.

The 11th harmonic impedance presents a quadri-mode behaviour characterized - obviously - by a great dispersion, and reaches the greatest values for medium load condition (T4); this can be explained (see ??) considering that the equivalent impedance resonance peak approaches and crosses the frequency of interest. The interharmonic impedance behaviour is similar to that out lighted for the 11th harmonic due to the short distance (24 Hz) between the two frequencies. 8 0.35 0.35 0.35 ZMAX=0.052  ZMAX=0.199  ZMAX=0.189  0.3 1 0.3 1 0.3 1

T1 T2 0.25 0.25 T1 0.25 T1 T6 0.75 T2 T2 T4 0.75 0.75 T6 T6

0.2 T3 f f f f f f T4 T d 4 p d d p T5 p p c p c p c T3 T3 0.15 0.15 0.5 T5 0.5 0.15 T5 0.5

0.1 0.1 0.1 0.25 0.25 0.25 0.05 0.05 0.05

0 0 0 0 0 0 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Impedance magnitude at 300 Hz [%] Impedance magnitude at 660 Hz [%] Impedance magnitude at 636 Hz [%]

(a) (b) (C)

Figure 9.7 Pdf and cpf of the system impedance magnitude, at bus 49:RECT, [%] of its maximum value during a working day:a) 5th harmonic, b) 11th harmonic and c) interharmonic at 636Hz.

?? reports some statistic parameters for the same aforementioned frequencies but with reference to a whole week. In the last column of the table also the rated value is reported, defined as that characterizing the IEEE test system in the condition considered in 10. The table confirms the considerations developed for ??.

Table 9.3 49:rect bus impedance (whole week)

Mean Max Z95% Rated

Z300Hz [] 0.048 0.052 0.049 0.049

Z660Hz [] 0.106 0.199 0.152 0.087

Z636Hz [] 0.102 0.189 0.131 0.083

4 EFFECTS ON DISTORTION MODELING RESULTS

The result of the previous paragraph together with the data characterizing the ASD, which is the harmonic source connected to the bus 49:RECT, are utilized to perform the analysis of the plant harmonic voltage distortion with methods assuming different representations of time variability and correlations among currents and impedances.

It has been assumed that the ASD changes its working point according to the same time intervals utilized for the plant power consumption reported in ??. The corresponding asynchronous motor frequencies and active power consumption vary as described in 6. This originates evident strong correlations between current and impedance.

4.1 Distortion Models

Three different methods were utilized to evaluate the harmonic and interharmonic voltages at bus 49:RECT:

9  the first applies the Ohm law to all the determinations of Z and I and utilizes Bayes relations to obtain the voltage probability, taking into account the actual behaviour of the involved quantities;

 the second is the simplified procedure which takes into account impedance variability but applies the product V = Z I to the homologous statistic parameters (mean, max, etc.), so ignoring the actual correlation effects;

 the third takes into account only current variability, assuming the rated value for the impedances.

4.2 Results

For the sake of brevity, the resulting voltages reported in this subsection once again refer only to 49:RECT bus. Anyway, the behaviour of the other bus voltages is very similar to this.

?? reports the pdf and cpf of harmonic and interharmonic voltage magnitude, in percentage of their maximum values, in a working day, obtained by means of the third procedure, that gives the actual values due to the absence of simplifying assumptions.

Pdf and cpf cannot be obtained by the second method.

?? can be still utilized for the results of the first method, considering that the voltages are obtained from the currents simply applying a constant scale factor.

0.05 0.05 0.05 MAX MAX MAX 0.137 V =13.55 V V =16.26 V 0.225 V =2.13 V 1 1 1 0.04 0.04 0.137 0.04

T1 T1 T1 T2 T2 0.75 T2 0.75 0.75 T6

0.03 0.03 0.03 T6 T6 f f f f f f d d d p T4 p T4 T4 p p c p c p c T3 T3 T3 T 0.02 T5 0.5 0.02 T5 0.5 0.02 5 0.5

0.01 0.25 0.01 0.25 0.01 0.25

0 0 0 0 0 0 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Voltage Magnitude at 300 Hz [%] Voltage Magnitude at 660 Hz [%] Voltage Magnitude at 636 Hz [%] (a) (b) (C)

Figure 9.8 Pdf and cpf of the voltage magnitude, at bus 49:RECT, [%] of its maximum value during a working day by means of the third procedure: a) 5th harmonic, b) 11th harmonic and c) interharmonic at 636Hz.

0.18 1 0.18 1 0.18 1 MAX MAX MAX MAX IMAX=11.3 A (VMAX=0.92 V) 0.16 I =260.20 A (V =12.75 V) 0.16 I =81.74 A (V =7.11 V) 0.16 T T _ 1 _ 1 T1 0.14 0.14 0.14 _T5 T2 T1 T1 0.12 0.12 0.12 T6 T2 T2

f f T f _ 1 T4 f f p p p

T6 T6 d d d c c c 0.1 0.1 0.1 p T3 p p T4 T4 T5 T3 T3 0.08 0.08 0.08 T5 T5 _T3 _T3 0.06 _T2 0.06 _T2 0.06 _T3 _T6 _T6 _T4 _T5 _T4 _T5 _T6 _T2 0.04 0.04 0.04 _T4

0.02 0.02 0.02

0 0 0 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 Current (voltage) magnitude at 300 Hz [%] Current (voltage) magnitude at 660 Hz [%] Current (voltage) magnitude at 636 Hz [%] (a) (b) (C)

Figure 9.9 Pdf and cpf of the current magnitude (and of voltage magnitude for constant impedances of rated values reported in Tab. III), at bus 49:RECT, [%] of its maximum value during a working day: a) 5th harmonic, b) 11th harmonic and c) interharmonic at 636Hz.

The differences among the histograms of ?? and ??.9 are evident. It is clear how taking into account impedance variability gives rise to an increase in the dispersion of the results; the values of the main statistic parameters result heavily influenced.

A comparison between the values of the statistic parameters, with reference to a whole week, obtained by means of the three methods, can be done with reference to ??. It suggests the following considerations:

 mean values are close enough for the harmonics (no more than 20% of difference), very different for the interharmonic (reaching 60% of differences);

th  the maximum value for the 11 harmonic (and for the interharmonic at 636 Hz) is overestimated by the second and underestimated by the third procedure;

 95% percentile values behave similarly to the maximum values.

Figure 9.10 Comparison of the statistic parameters of the voltage at bus 49:RECT during a whole week ( max 95% mean ), obtained “Zh constant” by the first method, “Zh&Ih directly correlated“ by the second method and “actual” by the third method: a) 5th harmonic, b) 11th harmonic and c) interharmonic at 636Hz. 5 Conclusions

The main outcomes of the chapter are: 11  it is necessary to take into account system impedance variability;

 procedures taking into account impedance time variability but neglecting the actual correlation among impedances and currents, that are particularly relevant in an industrial system, give unreliable results;

 assuming direct correlations among impedances and currents, in particular the same behaviour in time - as done with the second method -, gives precautionary (sometimes too pessimistic) results;

 interharmonics require the most accurate analyzes also due to their ability in producing immediately perceptible damages (light flicker) once a certain level has been reached.

6 References

[1] P. Marino, F. Ruggiero and A. Testa, “On the vectorial summation of independent random harmonic components”, 7th ICHQP, Las Vegas (USA), 1996. [2] R. Langella, P. Marino, F. Ruggiero and A. Testa, “Summation of random harmonic vectors in presence of statistic dependences”, 5th PMAPS, Vancouver (CA), 1997. [3] A. Cavallini, R. Langella, F. Ruggiero, A. Testa, “Gaussian Modeling of Harmonic Vectors in Power Systems”, 8th ICHQP, Athens (GR), 1998. [4] P. Caramia, G. Carpinelli, F. Rossi, P. Verde.: "Probabilistic iterative harmonic analysis of power systems", IEE Proc. on general Transmission Distribution., Vol. 141, N°4, July 1994. [5] R. Carbone, D. Castaldo, R. Langella, P. Marino, A. Testa, "Network Impedance Uncertainty in Harmonic and Interharmonic Distortion Studies", IEEE Budapest Power Tech '99, Budapest (UG), 1999. [6] R. Carbone, D. Castaldo, R. Langella, A. Testa, “Probabilistic Modeling of Industrial Systems for Voltage Distortion Analysis” 9th ICHQP, Orlando (USA), 2000. [7] IEC 1000-3-6, Electromagnetic compatibility, assessment of emission limits for distorting loads in MV and HV power systems. [8] IEEE Recommended Practices and Requirements for Harmonic Control in Electrical Power Systems, IEEE Standard 519-1992,1993. [9] D. Castaldo, R. Langella, A. Testa, “Probabilistic Aspects of Harmonic Impedances”, invited paper at IEEE Winter Power Meeting 2002, New York (USA), 27-31 January 2002. [10] Task Force on Harmonics Modelling and Simulations: “Test Systems for Harmonics Modelling and Simulations”, IEEE Trans. on Power Delivery, Vol. 14 2, pp 579–587, April 1999. [11] G. Carpinelli, V. Mangoni, S. Oliva, M. Russo, A. Testa: “Scelta dello schema elettrico e funzionamento di un sistema elettrico industriale con cogenerazione”, Autoproduzione e Cogenerazione nell’Industria, Febbraio 1995, Cassino (I). [12] A. Papoulis, “Probability, Random Variables and Stochastic Processes”, 3rd edition, 1991, McGraw Hill.