A Comparative Study on and Frequency Measurement Techniques in Power Systems

Aravinth Sridharan Vaskar Sarkar Department of Department of Electrical Engineering Indian Institute of Technology Hyderabad Indian Institute of Technology Hyderabad E-mail: [email protected] E-mail: [email protected]

Abstract—The objective of this paper is to make a comparative to slow function like expansion planning. Hence, the real- study on different phasor and frequency measurement algorithms time measurement of power system quantities is helpful in available for power systems. The dynamic nature of the power improving the reliability of the power system network. With system signals has made the estimation of frequency and difficult. Over the years, many digital algorithms have been an increase in the usage of power electronic devices and other developed to overcome these difficulties and to provide accurate nonlinear components, there is an increase in the harmonic real-time frequency and phasor measurements. In this paper, four contamination in power systems. It is, therefore, essential for different algorithms for frequency estimation and three different utilities to seek and depend on reliable models for accurate algorithms for phasor estimation are studied and compared. A estimation of phasor and frequency. Supervisory Control and case study is performed by considering a three- voltage signal with time-varying and frequency to verify the Data Acquisition (SCADA) is an existing real-time monitoring effectiveness of these estimation algorithms. and control system that is available for power systems. Due Index Terms—Amplitude, complex valued least squares, dis- to high latency in data flow and slower estimation techniques crete , frequency, phase angle, phase locked used, the real-time dynamic variations in power system quan- loop, phasor, polynomial fitting, prony, smart discrete fourier tities are not observed by SCADA. transform. The synchronized phasor measurement over a wide re- NOMENCLATURE gion has led to the development synchrophasor measurement and Wide Area Measurement System (WAMS). The integral 𝜔 Actual signal frequency (rad/s). components of WAMS are the Phasor Measurement Unit 𝜔 𝑠 Nominal signal frequency (rad/s). (PMU) and Phasor Data Concenterator (PDC). Synchronized 𝜔ˆ Estimated signal frequency (rad/s). phasor measurement made with PMU provides high speed and 𝐹 Sampling frequency (samples/second). coherent real-time information of the power system that are 𝐿 Sampling window length (in terms of the number of not available in the traditional SCADA systems. The PDC is samples) for the least square estimation. generally used to collect data from several PMUs, rejects bad 𝑁 Number of samples per nominal frequency cycle of data and alignment of time stamped data obtained to create a the input signal. coherent record. 𝑟 Index representing the sampling instant. With the advent of digital processors, the computational 𝑡 Time variable. 𝑡 speed has increased significantly. Now, there is a need for 𝑟𝑒𝑓 Reference value of the time variable for the phase digital algorithms which can estimate the phasor and frequency angle calculation. accurately in real-time. Some of the available methods in 𝑣𝑚 Peak value of a sinusoidal signal. 𝑉 literature include zero crossing [1], wavelet transform [2], Actual phasor. modified zero crossing, and signal demodulation [3]. In pursuit 𝑉ˆ Estimated phasor by ordinary DFT. of higher accuracy for frequency and phasor measurement, the Δ𝜔 Signal frequency deviation from the nominal value optimization methods such as least mean square methods [4], (rad/s). Kalman filter [5], and Newton’s method [6] are being used. Δ𝑇 Sampling time interval (s). These above optimization methods are not widely used in real 𝜙 Actual phasor angle. time application as convergence being the main problem. 𝜙ˆ Estimated phasor angle by ordinary DFT. In this paper, a comparative study on different phasor and (.) (.) 0 Initial value of . frequency estimation algorithm that are available in power I. INTRODUCTION system is performed. The algorithms are tested with signals having time varying amplitude and frequency to verify their Real-time data of power system quantities has many appli- effectiveness. The algorithms that are employed in this paper cations ranging from rapid control action like relay functioning for frequency estimation include Prony method [7], Polyno- 978-1-4799-5141-3/14/$31.00 ⃝c 2016 IEEE mial Fitting (PF) [3], Complex Valued Least Square (CLS) [8], and Phase Locked Loop (PLL). For phasor estimation, the are considered are expressed with respect to the same time algorithms employed in this paper are Discrete Fourier Trans- variable. This in turn, indicates that the time reference should form (DFT) [9], PLL, and Smart Discrete Fourier Transform be considered to obtain all the voltage and current phasors. The (SDFT) [10]. relative phase angle between the voltage and current phasors The rest of the paper is organized as follows, Section II gives can be directly obtained as they are measured with respect a brief overview about the concept of phasor and time synchro- to the same time reference.Time synchronization in PMU is nization. In Section III, different phasor estimation algorithms provided by the GPS pulse signal. are explained. Section IV is focussed on explaining different The phasor concept was initially developed for pure sinu- frequency estimation algorithms. In Section V, a case study is soidal signal. However, the concept can be extended in the presented with the comparison details of the various estimation form of dynamic phasor for signals whose and algorithms that are explained in the previous sections. Finally, phase angles are slowly varying with time. Since, the signal the paper is concluded in Section VI. parameters are varying with time, accurate time stamping of the estimated phasor quantities is required for reconstruction II. CONCEPT OF PHASOR AND TIME SYNCHRONIZATION of the signal. Time stamping is also helpful in comparing the A phasor can be defined as a complex equivalent of a measurements made by different PMUs located at different sinusoidal quantity whose amplitude, angular frequency and locations. The time stamping in PMU is provided by high phase angle are time invariant. The phasor mentioned above accuracy GPS clocks. The techniques that are available for can also be called as static phasor. Consider a sinusoidal input estimating static phasors may also be suitable for dynamic signal of frequency 𝜔𝑠 as follows, phasors.

𝑣(𝑡)=𝑣𝑚 cos(𝜃(𝑡)) (1) III. PHASOR ESTIMATION TECHNIQUES 𝜃(𝑡)=𝜔𝑠𝑡 + 𝜙. (2) A. Discrete Fourier Transform Its phasor representation is given by, DFT is a technique used in signal processing to calculate 𝑣𝑚 𝑉 = √ ∕𝜙. (3) the amplitude and phase angle of individual signal frequency 2 components. The sinusoidal signal given in (1) is sampled 𝑁 𝐹 = 𝑁( 𝜔𝑠 ) Phasor representation is possible only for single frequency times per cycle such that 2𝜋 . The discrete time component in a pure sinusoid. In real time systems, a sinusoid representation of the signal in (1) is given by, is often corrupted by harmonics and noise. So, it is necessary 𝑣[𝑟]=𝑣 cos(𝜔 𝑟Δ𝑇 + 𝜙). to extract the single frequency component of the signal (usu- 𝑚 𝑠 (7) ally the fundamental one) and then represent it by a phasor. The time reference can be set to be static or dynamic in The phasor representation of a signal is defined with respect the phasor calculation. The dynamic time reference is shifted to specific time variable. The time variable is specified with forward by one sampling time step with each new sample respect to a reference time instant. The time reference can be (i.e., 𝑡𝑟𝑒𝑓 = 𝑟Δ𝑇 ). On the other hand, in the static case, the defined as instant at which the specified time variable is set to time reference is always maintained at the time origin (i.e., zero. The phase angle given in (3) is measured with respect 𝑡𝑟𝑒𝑓 =0). The phasor representations of a signal with respect 𝑡 =0 to the time reference 𝑟𝑒𝑓 . Now, consider that the time to the dynamic and static time references are shown below. 𝑡 = 𝑇 reference is shifted to 𝑟𝑒𝑓 . When the time reference is 1) Dynamic time reference: The phasor representation of shifted, the time variable also changes. The time variable is the signal 𝑣[𝑟] (7) with respect to the dynamic time reference 𝜏 𝑡 𝜏 now changed to . The relation between and is given by is obtained as follows. 𝑡 = 𝑇 + 𝜏. Now, representing (1) with respect to the new time √ 𝑁−1+𝑀 variable 𝜏, we get, 2 ∑ 𝑉ˆ [𝑟]= 𝑣[𝑟 + 𝑛]𝑒−𝑗𝑛𝜑 𝑑𝑟 𝑁 𝑣(𝜏)=𝑣𝑚 cos(𝜔𝑠(𝜏 + 𝑇 )+𝜙) (4) 𝑛=𝑀 √ 𝑁−1+𝑀 𝑣(𝜏)=𝑣𝑚 cos(𝜔𝑠𝜏 + 𝜔𝑠𝑇 + 𝜙). (5) 2 ∑ = 𝑣˜[𝑟 + 𝑛]𝑒−𝑗𝑛𝜑. 𝑁 (8) The phasor representation of (1) with respect to reference time 𝑛=𝑀 instant 𝜏 =0(or 𝑡 = 𝑇 ) is given by, 2𝜋 Here, 𝜑 = 𝑁 . Subscript “dr” stands for dynamic reference. 𝑣𝑚 𝑉 𝑛𝑒𝑤 = √ ∕{𝜙 + 𝜔𝑠𝑇 }. (6) Signal 𝑣˜[𝑟] contains 𝑣[𝑟] as well as is polluted with super- 2 harmonics. The value of 𝑀 can be set to zero, if the phasors Time synchronization is required for providing a common are calculated based on the future samples. But, this introduces time reference for phasor measurement made at different a time delay of one cycle in phasor estimation. The value of 𝑀 locations. In power system, the active and reactive power is set to 1 − 𝑁, when the phasors are calculated based on the flow in the grid are calculated using the power flow equations past signal samples. Equation (8) shows the non-recursive form based on the phasor values of the corresponding voltage and of phasor calculation. A recursive form of phasor calculation current quantities. The voltage and current quantities that with respect to the dynamic time reference can be written as follows. After estimating the values of 𝑉 0 and Δ𝜔,thevalueof √ 2{ } 𝑉 𝑑𝑟[𝑟] can be obtained from (14). The frequency estimation 𝑉ˆ [𝑟 +1]= 𝑣˜[𝑟+𝑁+𝑀]−𝑣˜[𝑟+𝑀] 𝑒−𝑗(𝑀−1)𝜑 𝑑𝑟 𝑁 techniques are discussed in the next section. The procedure for obtaining 𝑉 0 is explained below. +𝑉ˆ [𝑟]𝑒𝑗𝜑. 𝑑𝑟 (9) The discrete time waveform (7) can be expressed as follows. 𝑗𝑟(𝜔 +Δ𝜔)Δ𝑇 ∗ −𝑗𝑟(𝜔 +Δ𝜔)Δ𝑇 The recusrive technique is helpful in increasing the compu- 𝑉 0𝑒 𝑠 + 𝑉 𝑒 𝑠 𝑣[𝑟]= √ 0 . (15) tational efficiency since the new phasor is obtained only by 2 making a marginal update on the old phasor. After replacing (15) into (8), the following relationship is 2) Static time reference: Unlike in the case of the dynamic obtained. time reference, the phase angle of a pure sinusoid remains 𝑉 𝑉 ∗ constant if the time reference is chosen to be static. The static ˆ 0 −1 0 𝑉 𝑑𝑟[𝑟]=𝛼[𝑟]𝛽1 + 𝛼[𝑟] 𝛽2 (16) time reference based phasor calculation formula can be written 𝑁 𝑁 as, where, √ 𝛼[𝑟]=𝑒𝑗𝑟(𝜔𝑠+Δ𝜔)Δ𝑇 𝑁−∑1+𝑀 (17) ˆ 2 −𝑗𝜑(𝑛+𝑟) 𝑉 𝑠𝑟[𝑟]= 𝑣[𝑟 + 𝑛]𝑒 𝑁−∑1+𝑀 𝑁 𝑗𝑛Δ𝜔Δ𝑇 𝑛=𝑀 𝛽1 = 𝑒 (18) √ 2 𝑁−∑1+𝑀 𝑛=𝑀 = 𝑣˜[𝑟 + 𝑛]𝑒−𝑗𝜑(𝑛+𝑟). 𝑁 (10) 𝑁−∑1+𝑀 −𝑗𝑛(2𝜔𝑠+Δ𝜔)Δ𝑇 𝑛=𝑀 𝛽2 = 𝑒 . (19) Subscript “sr” stands for dynamic reference. The recursive 𝑛=𝑀 form of the above equation is shown below. The values of 𝛼[𝑟], 𝛽1 and 𝛽2 can be obtained by determining √ ˆ 2{ } 𝜔 and the value of 𝑉 𝑑𝑟[𝑟] is directly obtained the from 𝑉ˆ [𝑟 +1]= 𝑣˜[𝑟+𝑁+𝑀]−𝑣˜[𝑟+𝑀] 𝑒−𝑗(𝑀+𝑟)𝜑 𝑠𝑟 𝑁 ordinary DFT. Subsequently, Equation (16) can be solved for 𝑉 0 by separating out the real and imaginary parts on both the +𝑉ˆ [𝑟]. 𝑠𝑟 (11) sides of (16). B. Smart Discrete Fourier Transform C. Phase Locked Loop The SDFT algorithm developed in [10] is similar to the The PLL technique has been used as a common way of conventional DFT method. The SDFT method was originally recovering the phase angle and frequency information in elec- proposed for a constant frequency signal. However, unlike trical systems. The schematic diagram of PLL based phasor the ideal case for ordinary DFT, the signal frequency can estimator is shown in Fig. 2. The input signals which are in abc be different from the nominal frequency. The SDFT approach involves the following basic steps. 1) Calculate the phasor by means of ordinary DFT with respect to the dynamic time reference. 2) From the ordinary DFT-based phasor, estimate the signal frequency. 3) Correct the DFT-based phasor according to the estimated signal frequency. It is, therefore, necessary to first make a raw phasor estimate by means of the ordinary DFT, which is subsequently refined by means of the SDFT. The raw phasor estimation can Fig. 1. PLL based phasor estimator. be carried out by means of the recursive or non-recursive calculation. domain are first converted to dq domain using a transformation 𝑇 Consider the same signal as in Equation (7). Let, matrix . The transformation matrix considered here is given below, 𝜙 =Δ𝜔𝑡 + 𝜙 √ [ ] 0 (12) 2 2𝜋 2𝜋 𝑐𝑜𝑠(𝜃) 𝑐𝑜𝑠(𝜃 − 3 ) 𝑐𝑜𝑠(𝜃 + 3 ) 𝑇 = 2𝜋 2𝜋 . (20) where, 𝜙0 is the phase angle at 𝑡 =0. The initial phasor (i.e., 3 −𝑠𝑖𝑛(𝜃) −𝑠𝑖𝑛(𝜃 − 3 ) −𝑠𝑖𝑛(𝜃 + 3 ) 𝑡 =0 at ), irrespective of static time reference or dynamic time A first order PI controller is used to obtain the frequency reference, can, therefore, be written as follows. information of the input signals. When the input signals are un- 𝑣𝑚 balanced, there is the presence of high frequency components 𝑉 0 = √ ∕𝜙0. (13) 2 in 𝑉𝑑 and 𝑉𝑞. A low pass filter (LPF) is used to filter out these high-frequency components. The moving average block (MA) The relationship between 𝑉 𝑑𝑟 and 𝑉 0 can be expressed as, is used to smoothen the magnitude and frequency waveform 𝑗𝑟(𝜔𝑠+Δ𝜔)Δ𝑇 𝑉 𝑑𝑟[𝑟]=𝑉 0𝑒 . (14) by reducing the ripples present in it. IV. FREQUENCY ESTIMATION TECHNIQUES Finally, the signal frequency is estimated through the following It is necessary to make a phasor estimation for determining equation. ( ) the signal frequency. The phasor samples obtained from the 1 𝜔ˆ[𝑟]= 𝑐𝑜𝑠−1 0.5ˆ𝑤[𝑟] . ordinary DFT or PLL should be used as inputs to the frequency Δ𝑇 (29) estimation algorithms. It is, however, to be noted that the B. Prony method phasor values that are used in the frequency estimation may subsequently be refined based upon the estimated frequency. The Prony method [7] is quite similar to the CLS method. However, instead of using the entire phasor, only the real part A. Complex Valued Least Square of a phasor is used for the frequency estimation. The formulae The complex valued least square method is based upon the that are used in the Prony method are shown below. SDFT relationships that were derived in the previous section. ( 𝑻 ){ ( ) ( )} Re 𝑽ˆ [𝑟 − 1] Re 𝑽ˆ [𝑟] +Re 𝑽ˆ [𝑟 − 2] According to (19), 𝒅𝒓 𝒅𝒓 𝒅𝒓 𝑥 =  ( ) ( )2 𝑗(𝜔𝑠+Δ𝜔)Δ𝑇 𝑗(𝑟−1)(𝜔𝑠+Δ𝜔)Δ𝑇  ˆ ˆ  𝛼[𝑟]=𝑒 𝑒 = 𝑐𝛼[𝑟 − 1]. (21) Re 𝑽 𝒅𝒓[𝑟] +Re 𝑽 𝒅𝒓[𝑟 − 2]  𝑐 (30) ( ) Therefore, 1 0.5 𝜔ˆ[𝑟 − 1] = 𝑐𝑜𝑠−1 . ∗ Δ𝑇 𝑥 (31) ˆ 𝑉 0 −1 −1 𝑉 0 𝑉 𝑑𝑟[𝑟]=𝑐𝛼[𝑟 − 1]𝛽1 + 𝑐 𝛼[𝑟 − 1] 𝛽2 (22) 𝑁 𝑁 C. Polynomial Fitting ∗ In the polynomial fitting approach [3], the phase angle 𝜙 is ˆ 𝑉 0 𝑉 𝑉 [𝑟 − 2] = 𝑐−1𝛼[𝑟 − 1]𝛽 + 𝑐𝛼[𝑟 − 1]−1𝛽 0 expressed as a polynomial function of 𝑡. That is, 𝑑𝑟 1 𝑁 2 𝑁 (23) 2 𝑚 ( ) 𝜙(𝑡)=𝑝0 + 𝑝1𝑡 + 𝑝2𝑡 + ⋅⋅⋅+ 𝑝𝑚𝑡 . (32) ˆ ˆ −1 ˆ 𝑉 𝑑𝑟[𝑟]+𝑉 𝑑𝑟[𝑟 − 2] = 𝑐 + 𝑐 𝑉 𝑑𝑟[𝑟 − 1]. (24)  Therefore, 𝑤 𝑑𝜙(𝑡) 𝜔(𝑡)=𝜔𝑠 + The expression of 𝑤 can be simplified as follows. 𝑑𝑡 ∑𝑚 𝑘−1 𝑤 =2cos(𝜔Δ𝑇 ). (25) = 𝜔𝑠 + 𝑘𝑝𝑘𝑡 . (33) 𝑘=1 In the case the signal frequency remains perfectly constant, 𝑚 the value of 𝑤 can be precisely determined from (24) by The polynomial degree is arbitrarily chosen. By using the 𝑀 using three successive ordinary DFT based phasor samples. phasor estimates over a data window of future samples 𝐿 − 𝑀 Subsequently, 𝜔 can be obtained by solving (25). and past samples, Equation (32) can be rewritten as The CLS method [8] was basically proposed for the case of follows. ˆ time varying signal frequency. In the case the signal frequency 𝝓 = 𝑹 × 𝒑 + 𝒆 (34) varies over time, Equation (24) does not perfectly hold true. where, By introducing an error vector 𝒆 over a data window of 𝑀 𝑇 𝒑 =[𝑝0 𝑝1 ⋅⋅⋅ 𝑝𝑚] (35) future samples and 𝐿 − 𝑀 past samples, Equation (24) can { } be rewritten as follows. 𝑙−1 𝑅𝑘,𝑙 = (𝑟 + 𝑀 − 𝑘 +1)Δ𝑇 (36) ˆ ˆ ˆ 𝑤[𝑟]𝑽 𝒅𝒓[𝑟 − 1] = 𝑽 𝒅𝒓[𝑟]+𝑽 𝒅𝒓[𝑟 − 2] + 𝒆 (26) ⎡ ⎤ 𝜙ˆ[𝑟 + 𝑀] where, ⎡ ⎤ ⎢ 𝜙ˆ[𝑟 + 𝑀 − 1] ⎥ ˆ ˆ ⎢ ⎥ 𝑉 𝑑𝑟[𝑟 + 𝑀] 𝝓 = ⎢ ⎥ . (37) ⎢ ⎥ ⎣ . ⎦ ⎢ 𝑉ˆ [𝑟 + 𝑀 − 1] ⎥ . ˆ ⎢ 𝑑𝑟 ⎥ ˆ 𝑽 𝒅𝒓[𝑟]=⎢ . ⎥ . (27) 𝜙[𝑟 + 𝑀 − 𝐿] ⎣ . ⎦ ˆ As before, 𝒆 indicates the error vector. The phase angle 𝑉 𝑑𝑟[𝑟 + 𝑀 − 𝐿 +2] ˆ ˆ estimate 𝜙 in effect represents the angle of 𝑉 𝑠𝑟. The least It is to be noted that, for the time-varying frequency, 𝑤 is also square estimate of 𝒑 is obtained as, 𝑤[𝑟] time varying. From (27), is determined by minimizing ( )−1 𝑇 𝑇 𝒆𝐻 𝒆. Here, 𝐻 in the superscript indicates the Hermitian 𝒑ˆ = 𝑹 𝑹 𝑹 𝝓ˆ. (38) operation. Thus, the value of 𝑤[𝑟] is obtained as follows. [ { }] Finally, the signal frequency is estimated as follows. ˆ 𝑯 ˆ ˆ Re 𝑽 𝒅𝒓[𝑟 − 1] 𝑽 𝒅𝒓[𝑟]+𝑽 𝒅𝒓[𝑟 − 2] ∑𝑚 𝑤ˆ[𝑟]=   . (28) 𝜔ˆ[𝑟]=𝜔 + 𝑘𝑝ˆ (𝑟Δ𝑇 )𝑘−1.  ˆ 2 𝑠 𝑘 (39) 𝑽 𝒅𝒓[𝑟 − 1] 𝑘=1 V. C ASE STUDY The frequency error (absolute value) for the test signal The case study is performed with a three-phase voltage sig- whose phasors are obtained from PLL is provided in Table nal that contains both the fundamental frequency and harmonic III. From Table III, it can be seen that frequency error is components. Only third and fifth harmonic components are TABLE III considered here. There are all sequence (i.e., positive, negative FREQUENCY ERROR (HZ) FOR PHASORS OBTAINED FROM PLL and zero) components at the fundamental frequency. However, no negative or zero sequence component is considered at Methods Window length QC HC OC TC THC harmonic frequency. The amplitude of fundamental frequency CLS 0.046 0.025 0.0268 0.029 0.033 components is taken to be time varying. Table I shows the Prony 0.21 0.026 0.027 0.030 0.033 amplitude and initial phase angles (corresponding to Phase PF 0.050 0.046 0.030 0.033 0.040 ‘A’) of different components of the test voltage signal. The PLL 0.033 HC-half cycle, OC-one cycle, QC-quarter cycle fundamental signal frequency is also taken as a slow varying TC-two cycles, THC-Three cycles sinusoid as below. 𝜔 =2𝜋{50 + sin(2𝜋𝑡)}. (40) least in the case of CLS method when half cycle window length is considered. When prony method is considered, half Therefore, the signal frequency oscillates between 49 Hz and cycle window length is suggested and for the polynomial 51 Hz with a time period of one second. The simulations are fitting method, one cycle window length is recommended carried out in MATLAB/Simulink. The sampling frequency is for obtaining minimum frequency error. Phasors obtained set at 12.8 kHz. The number of samples per cycle is 256 (i.e., from DFT provide more accurate system frequency estimation 𝑁=256). Harmonic components are added to these signals to compared to phasors obtained from PLL for the same fre- verify the effectiveness of the algorithms mentioned earlier. quency estimation algorithm and window length. The errors in frequency estimation by different algorithms with PLL-based TABLE I COMPONENTS OF PHASE ‘A’ VOLTAGE phasor input are plotted in Fig. 4. The amplitude errors (absolute value) for the different Components Amplitude (V) Initial phase angle (rad) phasor estimation techniques are tabulated in Table IV. The Positive sequence 100+10sin(2𝜋𝑡) 0 Negative sequence 50+5sin(2𝜋𝑡) -1.0491 amplitude error is minimum in the case of SDFT method. The Zero sequence 30+3sin(2𝜋𝑡) -2.0769 plots of amplitude error are produced in Fig. 5. Third harmonic 12 0 Fifth harmonic 9.5 0 TABLE IV AMPLITUDE ERROR

The number of phasor samples required for the frequency Methods DFT PLL SDFT estimation algorithms is varied from quarter cycle to three Amplitude error (V) 0.09 0.244 0.04 cycles window length. The outputs of these algorithms (i.e., amplitude and frequency waveforms) is passed through a The phase angle errors (absolute value) for different phasor moving average block to reduce the ripples present in it. The estimation techniques are provided in Table V. The phase frequency error (absolute value) for the different estimation angle error is minimum in the case of SDFT method. The algorithms whose phasors are obtained from DFT are tab- plots of phase angle error are produced in Fig. 6. The SDFT ulated in Table II. From Table II, it can be seen that the TABLE V TABLE II PHASE ANGLE ERROR FREQUENCY ERROR (𝑚HZ) FOR PHASORS OBTAINED FROM DFT Methods DFT PLL SDFT Methods Window length Phase angle error (rad) 7.3e-3 0.03 3.6e-3 QC HC OC TC THC CLS 16.5 1.95 2.4 4.4 7.57 technique provides accurate amplitude and phase angle mea- Prony 40 4.15 4.49 6 9.1 PF 20 15 3.1 5.9 13.2 surement since the phasors are calculated based on the system HC-half cycle, OC-one cycle, QC-quarter cycle frequency deviation from its nominal value. TC-two cycles, THC-Three cycles VI. CONCLUSION frequency error is least for CLS method compared to the A comparative study on different phasor and frequency other frequency estimation algorithms. The half cycle window estimation algorithms that are available for power systems length is suggested for CLS and prony method for obtaining is presented in this paper. The DFT-based algorithms for minimum frequency error. When the polynomial fitting method phasor estimation provide good accuracy even when the signal is used for frequency estimation, one cycle window length is parameters are varying. The dynamically varying system fre- used for obtaining minimum frequency error. The errors in quency can be accurately estimated from its positive sequence frequency estimation by different algorithms with DFT-based phasor using CLS technique. For frequency estimation in the phasor input are plotted in Fig. 3. case of PLL, the phasors obtained from PLL can be used −3 −3 −3 x 10 x 10 x 10 6 2 4 4 2 1.5 2 0 1 0

−2 −2 0.5 Frequency error (Hz) Frequency error (Hz) Frequency error (Hz) −4 −4 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 Time (seconds) Time (seconds) Time (seconds) (a) CLS (Half cycle) (b) PRONY (Half cycle) (c) PF (One cycle) Fig. 2. Frequency error for DFT based phasor inputs.

0.03 0.04 0.02 0.02

0.02 0.01 0.01 0 0 0 −0.01

−0.01 −0.02

−0.02 −0.03 −0.02 Frequency error (Hz) Frequency error (Hz) Frequency error (Hz) −0.04 −0.04 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Time (seconds) Time (seconds) Time (seconds) (a) CLS (Half cycle) (b) PRONY (Half cycle) (c) PF (One cycle) Fig. 3. Frequency error for PLL based phasor inputs.

0.02 0.06 0.15 0 0.1 0.04 −0.02 0.05 0.02 0 −0.04 0 −0.05 −0.06 −0.1 −0.02

−0.08 −0.15 −0.04 Amplitude error (V) Amplitude error (V) Amplitude error (V) −0.2 −0.1 −0.25 −0.06 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Time (seconds) Time (seconds) Time (seconds) (a) DFT (b) PLL (c) SDFT Fig. 4. Amplitude error.

−3 −3 x 10 x 10 6 6 0.03

4 4 0.02 2 2 0.01 0 0 −2 0 −4 −2 −0.01 −6 −4 Phase angle error (rad) Phase angle error (rad) −8 Phase angle error (rad) −0.02 −6 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Time (seconds) Time (seconds) Time (seconds) (a) DFT (b) PLL (c) SDFT Fig. 5. Phase angle error.

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