Modeling and Testing of a Phasor Measurement Unit Under Normal and Abnormal Conditions Using Real-Time Simulator ^†

Abstract

Abnormal operations, such as faults occurring in an electrical power system (EPS), disrupt its balanced operation, posing potential hazards to human lives and the system’s equipment. Effective monitoring, control, protection, and coordination are essential to mitigate these risks. The complexity of these processes is further compounded by the presence of intermittent distributed energy resources (DERs) in active distribution networks (ADNs) with bidirectional power flow, which introduces a fast-changing dynamic aspect to the system. The deployment of phasor measurement units (PMUs) within the EPS as highly responsive equipment can play a pivotal role in addressing these challenges, enhancing the system’s resilience and reliability. However, synchrophasor measurement-based studies and analyses of power system phenomena may be hindered by the absence of PMU blocks in certain simulation tools, such as PSCAD, or by the existing PMU block in Matlab/Simulink R2021b, which exhibit technical limitations. These limitations include providing only the positive sequence component of the measurements and lacking information about individual phases, rendering them unsuitable for certain measurements, including unbalanced and non-symmetrical fault operations. This study proposes a new reliable PMU model in Matlab and tests it under normal and abnormal conditions, applying real-time simulation and controller-hardware-in-the-loop (CHIL) techniques.

1. Introduction

Phasor measurement units (PMUs) originate from the research publications of an American mathematician and electrical engineer named Charles Proteus Steinmetz in 1893 [1], in which a simplified mathematical description of the alternating current waveform was presented and named a phasor. Almost a century later, in 1988, Dr. Arun G. Phadke and Dr. James S. Thorp from Virginia Polytechnic Institute invented the first PMU [1]. They developed the calculation of synchronized phasor measurements known as synchrophasors, from Steinmetz’s phasor calculation techniques and the symmetrical component distance relay (SCDR) algorithm developed earlier to protect high-voltage transmission lines [2]. The first PMU prototypes were built later in the same institute [3].

The deployment of PMUs in electrical power systems (EPS) has progressively followed their invention, especially for monitoring transmission lines, and has played a substantial role in preventing catastrophic blackouts [3]. However, their massive use in the power networks was hindered by their cost, which was relatively high, and the required communication links [4]. Now, affordable micro-phasor measurement units (µPMUs) are being developed and can be used in distribution networks, not only at the sub-station level but also in several network nodes, including tap-changing transformers, complex loads, distributed energy resource (DER) buses, etc. [5]. These µPMUs consider specific particularities of distribution networks and their vulnerabilities, including smaller power angles, shorter lengths of the lines, and high levels of harmonic distortion [6]. PMUs provide a high degree of precision monitoring and grant the proper system observability, identifying the system’s faults and failures before they become fatal, enabling smart and preventive control decisions [7,8,9].

The occurrence of a fault in the EPS disrupts normal operations, can endanger lives, and may cause significant damage to equipment [10]. Therefore, the fault current must be quickly interrupted with minimal disturbance to upstream devices. This necessitates proper protective device coordination and precise trip command settings. However, the evolving dynamic nature of today’s distribution systems—becoming more active with bidirectional power flow due to high levels of renewable and distributed energy resources—makes coordinating, controlling, and monitoring power systems very difficult [11]. To tackle these issues, the synchrophasor measurement approach using PMUs, which are highly responsive tools [12], has been applied successfully. They allow for more accurate system state estimation, enabling wide-area monitoring of the grid’s operational status and comprehensive power network observability [12,13]. They are emerging as powerful and essential devices for electric power grid measurement and monitoring, considered the most important tools for the future of power systems [14,15]. Due to their high measurement reporting rate—typically up to 60 and 240 samples per second for newer technologies [12]—they play a vital role in real-time system monitoring. They are suitable for handling dynamic phenomena, making power systems fully observable [15].

Different techniques for phasor estimation mainly rely on the discrete Fourier transform (DFT), the same method outlined by the IEEE standard. It offers advantages such as being easy to implement and demonstrating proven robustness against harmonics [16]. However, when the signal frequency deviates from the nominal, errors can occur due to spectral leakage [17]. Reference [18] used the recursive DFT algorithm with a digital signal processor (DSP) and GPS applications, considering CTs and VTs signals to design a low-cost PMU with higher accuracy and faster information. In reference [19], the authors introduced a fast iterative-interpolated discrete Fourier transform (FiIpDFT) technique to estimate the phasor for signals contaminated with out-of-band interference (OOBI), using a three-point interpolated DFT approach. It first applies a non-iterative amplitude, phase, and frequency estimator to remove the interference from negative frequencies, then provides a simple formula along with a two-stage criterion to limit the computation involved in the OOBI removal process. Besides DFT-based methods, other techniques have also been employed in PMU design and modeling. These include the weight least square (WLS) of a Taylor approximation [20], the Taylor-Kalma filters [21], adaptive cascaded filters [22], the space vector approach [23], Taylor–Fourier transform (TFT) [24], and phase-locked loop algorithm [25,26,27,28,29]. These include the enhanced PLL (EPLL)-based algorithm [25], space vector and PLL-based technologies [26,27], the proportional-integral-derivative, digital synchronous reference frame (PID DSRF)-PLL-based PMU algorithm [28], and harmonic–interharmonic and DC offset (HIHDO)-PLL-based algorithm [29].

The present study, which focuses on modeling and testing phasor measurement units under both normal and abnormal conditions, was motivated by the observation that different simulation software used by various researchers and engineers to analyze power network behavior through synchrophasor measurements have significant technical limitations that impede obtaining relevant results. For instance, PSCAD is a powerful simulation tool adept at handling power system phenomena, but its library lacks a PMU block; it only includes a fast Fourier transform (FFT) block, which poorly processes phasors when analyzing the dynamic operations of the power system and frequency deviation [30]. Similarly, the Matlab/Simulink library does contain a PMU block, but it only provides the positive sequence component of the measurements, failing to offer specific information about the individual phases. Unable to measure either negative or zero sequence components, this PMU block is unsuitable for applications under unbalanced or unsymmetrical fault conditions.

This work proposes a new reliable PMU model that measures positive, negative, and zero sequence components, offering measurements taken from each system phase and the neutral current. This makes it suitable for operations under normal and symmetrical fault conditions, as well as during system imbalances and unsymmetrical faults involving ground current flows. The proposed PMU model is tested in the electrical power network operating in normal and faulty conditions by applying a real-time simulator and the controller-hardware-in-the-loop (CHIL) whose principles are described in [31].

The rest of this work is structured as follows: Section 2 presents an overview of Fourier transform, the phase locked loop and the phasor measurement unit; Section 3 describes the methodological approach adopted to conduct the study; Section 4 discusses the proposed PMU model and different procedures applied in testing processes; Section 5 presents the test simulation results and their discussion; Section 6 concludes the study.

2. Overview of Fourier Transforms, Phase Locked Loop, and Phasor Measurement Unit

2.1. Fourier Transform

The Discrete Fourier Transform (DFT) is analogous to the continuous Fourier Transform for signals known only at intervals separated by sample times. Let f(t) represent the continuous signal and the source of the data [32]; N is the number of samples, T the corresponding time, K a number less than N, while n, k, and r are integers, and ω is the angular frequency in radians per second.

Let the samples be denoted $f\left[0\right],f\left[1\right],f\left[2\right],\dots ,f\left[k\right],\dots ,f[N-1]$, the Fourier Transform of the original signal, $f\left(t\right)$, is given by (1)

$$F\left(jw\right)={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f\left(t\right)e^{-jwt}dt$$

(1)

We could regard each sample as an impulse having an area $f\left[\mathrm{k}\right]$ [33]. Then, since the integrand exists only at the sample point:

$$F\left(jw\right)={\int }_0^{\left(\mathrm{N}-1\right)\mathrm{T}}f\left(t\right)e^{-jwt}dt$$

(2)

$$F\left(jw\right)={\sum }_{k=0}^{N-1}f\left[\mathrm{k}\right]e^{-jwkT}$$

(3)

Since there are only a finite number of input data points, the DFT treats the data as if it were periodic.

Fast Fourier Transform (FFT) algorithms were developed as highly efficient tools to handle DFTs since the standard DFT involved considerable redundant calculation [33]. Equation (3) can be rewritten as given in (4) and (5):

$$F\left[n\right]={\sum }_{k=0}^{N-1}f\left[\mathrm{k}\right]e^{-j{\displaystyle \frac{2\pi }{N}}nk}$$

(4)

$$F\left[n\right]={\sum }_{k=0}^{N-1}f\left[\mathrm{k}\right]W_N^{nk}$$

(5)

In (5), $W_N^{nk}$ is redundant during the computation process, due to not only the integer product $nk$ which is repeated in various combinations of $k$ and $n$, but also to the fact that $W_N^{nk}$ is an N-value periodic function.

For phasor estimation, there are two types of algorithms: non-recursive and recursive algorithms [34]. Let $f_0$ be the nominal frequency of a signal with constant input sampled at N times; the corresponding phasor estimate, which represents the non-recursive algorithm, is shown in (6)

$$X^{N-1}={\displaystyle \frac{\sqrt{2}}{N}}{\sum }_{n=0}^{N-1}{x}_{n}cos n\theta -j{\displaystyle \frac{\sqrt{2}}{N}}{\sum }_{n=0}^{N-1}{x}_{n}sin n\theta$$

(6)

where xn is the sampled value at the time $n∆T$, $∆T$ is the time difference between two consecutive samples and $\theta$ is equal to ${\displaystyle \frac{2\pi }{N}}$.

Denoting $X_k$ the phasor estimate in (6), $X_{kr,}$ and $X_{ki}$ its real and imaginary parts, we have then:

$$X_k=X_{kr}-j{X}_{ki}$$

(7)

$$\left|X_k\right|=\sqrt{{X_{kr}}^2+{X_{ki}}^2}$$

(8)

$$Phase angle={tan}^{-1}{\displaystyle \frac{X_{ki}}{X_{kr}}}$$

(9)

The recursive algorithm which is modified from the previous one, is presented in (10). If the last sample in the window is the N + rth, the phasor estimate is given as follows.

$${\overline{\mathrm{X}}}^{N+r}={\overline{X}}^{N+r-1}+{\displaystyle \frac{\sqrt{2}}{N}}{\sum }_{n=0}^{N-1}\left(x_{N+r}-x_r\right)e^{-jr\theta }$$

(10)

2.2. Phase-Locked Loop

2.2.1. Phase-Locked Loop Principle

A phase-locked loop (PLL) is a closed-loop feedback-based control system that generates a signal of the same frequency as its input signal. This is performed by locking and continuously adjusting its phase difference to the reference signal [35]. The PLL circuit is composed of three parts: (1) a phase detector that measures the phase difference between two periodic signals, (2) a low-pass filter that translates the phase difference into a feedback signal, and (3) a voltage-controlled oscillator (VCO) whose frequency is adjustable by the feedback signal and subsequently generates the output signal. The negative feedback loop keeps the output at the same frequency as the input signal by monitoring the phase difference between the input signal and the signal generated by the VCO. Based on this monitoring, the output frequency is modified to align with the phase difference, thereby ensuring the stability and consistency of the output [36]. The block diagram for a PLL configuration is illustrated in Figure 1.

2.2.2. Phase-Locked Loop Modeling in Continuous-Time Domain (S-Time Domain)

The modeling method discussed in this section is based on linear control theory. From Figure 1, the linear model of the PLL in the continuous-time domain (S-domain) is given in Figure 2. Due to the negligible phase error, i.e., mathematically sin(θ) $\approx$ θ, a PLL can be accurately represented by a linear model. In this model, θ_in(t) and θ_fd(t) represent, respectively, the phases of the input and feedback signals [37], as illustrated in Figure 2.

The Laplace transform helps to express the transfer functions of the loop filter (H1(S)), the VCO (H2(S)), and the closed-loop (H_cl(S)) of the PLL as illustrated, respectively, in Equations (11)–(13)

$$H1\left(S\right)={\displaystyle \frac{G_{lp}}{G_{lp}+S}}$$

(11)

$$H2\left(S\right)={\displaystyle \frac{G_{VCO}}{S}}$$

(12)

$$H_{cl}\left(S\right)={\displaystyle \frac{G_{lp}{G}_{VCO}}{{S^2+G}_{lp}S+G_{lp}{G}_{VCO}}}$$

(13)

Equation (13) illustrates the second-order system [37,38], and its transfer function is given in Equation (14)

$$H_S\left(S\right)={\displaystyle \frac{{\omega }_n^2}{S^2+2\zeta {\omega }_{n}S+{\omega }_n^2}}$$

(14)

where ω_n denotes the natural undamped frequency and $\zeta$ a damping ratio. Such a system is known as a standard prototype second-order system [38], whose characteristic equation is given by (15)

$${∆\left(S\right)=S}^2+2\zeta {\omega }_{n}S+{\omega }_n^2$$

(15)

From the roots of Equation (15), two poles of the system are defined as shown in (16) and (17).

$$S_0={\zeta \omega }_n+j{\omega }_n\sqrt{1-{\zeta }^2}=-\alpha +j\omega$$

(16)

$$S_0={\zeta \omega }_n-j{\omega }_n\sqrt{1-{\zeta }^2}=-\alpha -j\omega$$

(17)

where α denotes the damping factor and ω the damped frequency. When $\zeta$ and ω_n are given, the system poles are obtained.

2.2.3. Phase-Locked Loop Modeling in Discrete-Time Domain (Z-Time Domain)

Modeling and design of a PLL in the discrete-time domain is required for digital PLL applications. The output response of its control system is also a function of the time variable t. The purpose of this process is to provide a system that satisfies the time-response exigencies defined by $\zeta$ and ω_n to a corresponding second-order model in discrete-domain [37,39]. The block diagram of a linear model of a PLL in Z-domain is given in Figure 3.

The transfer function of the loop filter, the digital controller oscillator (DCO), and the closed loop of the digital PLL are given, respectively, by Equations (18)–(20).

$$H1\left(Z\right)={\displaystyle \frac{aZ-1}{Z-1}}$$

(18)

$$H2\left(Z\right)={\displaystyle \frac{cZ}{Z-1}}$$

(19)

$$H2\left(Z\right)={\displaystyle \frac{acZ-c}{Z^2+\left(ac-2\right)Z+(1-c)}}$$

(20)

The second-order system’s poles can be mapped from the S-domain to the Z-domain. The general format of the corresponding transfer function of the PLL is given by (21)

$$H\left(Z\right)={\displaystyle \frac{N\left(Z\right)}{(Z-Z_1)(Z-Z_0)}}$$

(21)

where N(Z) can be a constant scaling factor, Z₀ and Z₁ represent the two poles in the Z-domain. The characteristic equation is given in (22).

$$∆\left(Z\right)=\left(Z-Z_1\right)\left(Z-Z_0\right)=Z^2-\left(Z_1+Z_0\right)Z+Z_1{Z}_0$$

(22)

Let C₁ and C₀ be the coefficient of the characteristic equation as shown in (23)

$$C_1=-(Z_1+Z_0) \mathrm{and} {C_0=Z}_1{Z}_0$$

(23)

The characteristic equation given in (22) can then be simplified as shown in (24)

$$∆\left(Z\right)=Z^2+C_{1}Z+C_0$$

(24)

The mapping of the two system poles in the discrete-time domain (Z-domain) is derived from those provided in the continuous-time domain, as shown in (25).

$$Z_0=e^{S_0{T}_s}=e^{\left({-\zeta \omega }_n{T}_s+j{\omega }_n{T}_s\sqrt{1-{\zeta }^2}\right)} \mathrm{a}\mathrm{n}\mathrm{d} Z_1=e^{S_1{T}_s}=e^{\left({-\zeta \omega }_n{T}_s-j{\omega }_n{T}_s\sqrt{1-{\zeta }^2}\right)}$$

(25)

where T_s denotes the discrete system sampling period. Considering the mapped poles in the discrete domain, the coefficients of the characteristic equations in (24) are given by Equation (26)

$$C_0=e^{{-2\zeta \omega }_n{T}_s} \mathrm{a}\mathrm{n}\mathrm{d} C_1={-2e}^{{-\zeta \omega }_n{T}_s}\mathit{cos}\left({\omega }_n{T}_s\sqrt{1-{\zeta }^2}\right)$$

(26)

Equations (21) and (24) will determine the transfer function of the digital PLL, as the characteristic equation impacts responses of the system’s transient [38].

2.2.4. The Second-Order Digital PLL Implementation

The architecture diagram describing the implementation of the digital PLL based on the mapping results of the model is given in Figure 4.

The loop filter and the digital controlled oscillator (DCO), the two main blocks are described below:

• Loop filter: this is an infinite impulse response (IIR) filter [37,39]. Its transfer function H1(Z) is given in (27)

$$H1\left(Z\right)={\displaystyle \frac{G_1+G_2-G_1{Z}^{-1}}{1-Z^{-1}}}$$

(27)

where G1 and G2 represent the gains of the IIR filter

• Digital controller oscillator (DCO): this is a digitally controlled VCO also known as discrete-time oscillator (DTO). Its transfer function H2(Z) is given in (28)

$$H2\left(Z\right)={\displaystyle \frac{G_{VCO}}{1-Z^{-1}}}$$

(28)

where G_VCO stands for gain of the discrete VCO. The transfer function of the closed-loop is expressed as given in (29)

$$H\left(Z\right)={\displaystyle \frac{{\theta }_{VCO}\left(z\right)}{{\theta }_{in}\left(z\right)}}={\displaystyle \frac{H1\left(z\right)H2\left(z\right)Z^{-1}{G}_{pd}}{1+H1\left(z\right)H2\left(z\right)Z^{-1}{G}_{pd}}}$$

(29)

where Gpd denotes the gain of the phase detector. The transfer function in (29) can be extended to the format provided in (30)

$$H\left(Z\right)={\displaystyle \frac{{\theta }_{VCO}\left(z\right)}{{\theta }_{in}\left(z\right)}}={\displaystyle \frac{{(g}_1+g_2)Z-g_1}{Z^2+{(g}_1+g_2-2)Z+(1-g_1)}}$$

(30)

where $g_1=G_{pd}{G}_{VCO}{G}_1$ and $g_2=G_{pd}{G}_{VCO}{G}_2$. From comparison with Equation (20), the coefficients C₀ and C₁ can be expressed as shown in (31).

$$C_0=1-g_1 \mathrm{a}\mathrm{n}\mathrm{d} C_1=g_1+g_2-2$$

(31)

Considering Equations (22) and (27), $g_1$ and $g_2$ can be given as in (32)

$$g_1=1-e^{{-2\zeta \omega }_n{T}_s} \mathrm{a}\mathrm{n}\mathrm{d} g_2=1+e^{{-2\zeta \omega }_n{T}_s}{-2e}^{{-\zeta \omega }_n{T}_s}\mathit{cos}\left({\omega }_n{T}_s\sqrt{1-{\zeta }^2}\right)$$

(32)

2.3. PMU Overview

2.3.1. Phasors and Synchrophasors

A phasor is defined as a rotating vector that represents a sinusoidal quantity, such as an AC signal. The concept may also be delineated as a vector comprising magnitude and angle, corresponding to a sinusoidal waveform, as illustrated in Figure 5. The subject is frequently posited as a complex number, accompanied by phase and amplitude [14,16].

As with any complex number, a phasor quantity $\overline{X}$ can be expressed in rectangular, exponential, and polar forms as in (33)

$$\overline{X}=X_r+j{X}_i={\displaystyle \frac{X_m}{\sqrt{2}}}\left[\mathit{cos}\left(\mathsf{\Phi }\right)+jsin\left(\mathsf{\Phi }\right)\right]={\displaystyle \frac{X_m}{\sqrt{2}}}{e}^{j\mathsf{\Phi }}={\displaystyle \frac{X_m}{\sqrt{2}}}\mathrm{\angle }\mathsf{\Phi }$$

(33)

where ${\displaystyle \frac{X_m}{\sqrt{2}}}$ and Φ represent the magnitude and phase of the phasor while X_r and X_i are the phasor’s real and imaginary parts, respectively. The sinusoidal waveform representing the voltage and the current of the electrical system is shown in (34).

$$x\left(t\right)=X_m\cos \left(\omega t+\mathsf{\Phi }\right)=\mathrm{R}\mathrm{e}\left(\sqrt{2}\overline{X}{e}^{j\omega t}\right)$$

(34)

where Re stands for the real part of the sinusoid.

With the further transformation of the phasor, considering the time-dependence aspect of the frequency and amplitude [16,30], Equation (34) can be represented as shown in Equation (35):

$$x\left(t\right)=X_m\left(t\right)cos\left(2\pi f_{0}t+2\pi \int g\left(t\right)dt+\mathsf{\Phi }\right)$$

(35)

$$g\left(t\right)=f\left(t\right)-f_0$$

(36)

where f(t) and f₀ respectively denote the actual and nominal frequency, while g(t) represents a time-varying function that defines the actual frequency deviation from the nominal. In terms of synchrophasors, Equation (35) becomes as shown in (37)

$$\overline{X}\left(t\right)={\displaystyle \frac{X_m\left(t\right)}{\sqrt{2}}}{e}^{j\left(2\pi \int g\left(t\right)dt+\mathsf{\Phi }\right)}$$

(37)

From Equations (35) and (37), the frequency f(t) and the rate of change in frequency ROCOF(t) can be given, respectively, by (38) and (39)

$$f\left(t\right)=f_0+{\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{d\mathsf{\Phi }}{dt}}$$

(38)

$$ROCOF\left(t\right)={\displaystyle \frac{df\left(t\right)}{dt}}={\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{d^2\mathsf{\Phi }}{d{t}^2}}$$

(39)

The phasor of a sinusoid is derived using the Fourier transform and data samples of the signal in a specified or selected time window. Concerning the signal’s steady state, its magnitude is constant, while its angle value is a relative quantity contingent on the initial point of the samples. Consequently, reference selection emerges as a pivotal component in this process [14,16].

2.3.2. PMU Description

The IEEE standard C37.118 in [40], and its amendment in [41] define a PMU as “a device that produces synchronized phasor, frequency, and rate of change in frequency (ROCOF) estimates from voltage and/or current signals and a time synchronizing signal.” The total vector error (TVE) is defined as the discrepancy between the theoretical synchrophasor, which has been properly sampled, and the estimate derived from the PMU under evaluation at a particular moment in time. Since difference and error could originate from magnitude, phase inaccuracy, or both, TVE combines both magnitude and phase errors, as illustrated in (40).

$$TVE=\sqrt{{\displaystyle \frac{{\left({\widehat{X}}_r-X_r\right)}^2+{\left({\widehat{X}}_i-X_i\right)}^2}{X_r^2+X_i^2}}}$$

(40)

In (40), the real and imaginary parts of the measured synchrophasor are represented by ${\widehat{X}}_r$ and ${\widehat{X}}_i$, while the input signal’s real and imaginary parts are denoted by $X_r$ and $X_i$ respectively [42].

PMUs collect analog measurements from the secondary windings of instrument transformers (CTs and PTs). The data are transmitted to the local phasor data concentrators (PDCs) via an anti-aliasing filter, which serves to eliminate the signal with a frequency that exceeds or equals one-half of the sample rate, also referred to as the Nyquist frequency. This process effectively restricts the analog signal bandwidth to the necessary level [13,14]. To obtain the desired digital signal, data are conveyed through the analog-to-digital (A/D) converter to be discretized. The resulting digital signal is then transmitted to the central processing unit (CPU), where mathematical computation of its magnitude and phase takes place. At this level, data time synchronization by a sampling clock occurs with the signal received from the GPS receiver [43,44]. The PMU constituent modules are illustrated in a block diagram of Figure 6.

3. Methodological Approach

The methodological approach adopted in this work, which models and tests the phasor measurement unit under normal and abnormal working conditions, is described herein. The abnormal operations considered consist of three scenarios: (1) the presence of 3rd, 5th, and 7th harmonics, (2) operation under symmetrical fault, and (3) unsymmetrical fault working conditions. The electrical power system (EPS) in which the tests and measurement processes are conducted is illustrated in Figure 7. This system is directly grounded and made of a 13.8 kV, 50 Hz generator and a 13.8 kV, 50 Hz, 8 MW, 2.63 MVar connected RLC load. From the given parameters, some required values were calculated; these include the phase voltage amplitude of 11.268 kV, the current amplitude of 498.2 A, the apparent power of 8.42 MVA, and the power factor of 0.95, corresponding to the phase angle of 18.2°. The proposed PMU model and the existing Matlab PMU block are both used for synchrophasor measurements in different operation coditions.

Referring to the performance requirements of the IEEE standard C37.118.1-2011 [40] related to the use and functionalities of the PMUs, considering Fourier transforms and phase-locked loop (PLL) techniques, a P-class PMU is modeled, implemented and simulated in MATLAB/Simulink R2021b and the real-time simulator OPAL-RT OP5707XG. For validation, the offline and real-time simulation measurement results from the proposed PMU model are compared to the measurement results of the real PMU laboratory tests and the PMU performance requirements outlined in the IEEE standard C37.118.1-2011 and C37.118.1a-2014 [40,41]

The laboratory experiment was conducted on testing and measuring a PMU incorporated into the protective relay SEL-351A. The Omicron CMC 256, whose inputs were derived from the Comtrade of the simulation signal inputs, was utilized to conduct the measurements of the real PMU. The synchronization to Coordinated Universal Time (UTC), a widely used common time source, was accomplished through the implementation of the interface box CMIRIG-B. The measured voltage and current synchrophasors are obtained through the utilization of the software program WIRESHARK 4.4.6, and the communication between different devices was facilitated through the Ethernet network. The methodological approach adopted for the PMU modeling and testing under normal and abnormal conditions is illustrated in Figure 8.

The OPAL RT-LAB 2022.1 software adapts the EPS model, which is then built, loaded, and executed in the OPAL-RT OP5707XG simulator as C code. The controller hardware-in-the-loop (CHIL) technique enables the interaction between the system model and the real controller in Raspberry Pi 4, using the Modbus TCP/IP protocol for communication with the real-time simulator. This technique was used to control the fault occurrence by determining the adding and removal time of faults during the PMU-monitored EPS test process. For faulty operations, a symmetrical fault with a three-phase line-to-line (3LL) short-circuit and an unsymmetrical fault with a single line-to-ground (1LG) short-circuit in phase A are considered.

4. Proposed Phasor Measurement Unit Model and Test Procedures

4.1. Proposed Phasor Measurement Unit Model

The initiative of proposing an improved PMU model using Matlab/Simulink was motivated by the awareness of challenges faced by researchers while analyzing the power system phenomena using synchrophasor measurement techniques with different simulating software. For example, PSCAD, whose library does not contain any PMU block, has only the FFT block, which poorly handles phasors in the power system’s dynamic operations. The other shortage problem is found in Matlab/Simulink, whose existing PMU block in its library provides only a positive sequence component of the measurements. This constitutes technical limitations since individual phases’ situations and neutral flows cannot be fully analyzed, and this PMU block becomes non-suitable for measurements in case of imbalances and unsymmetrical faults when the system experiences ground current flows.

A more reliable PMU model is proposed to mitigate the above-listed challenges. The model is capable of providing current and voltage synchrophasor measurements for each of the phases and the neutral. It is developed in Matlab/Simulink R2021b and abided with the IEEE C37.118.1-2011 standard [40] performance requirements. The Fourier transform and the phase-locked loop (PLL) have been applied to estimate the phasor amplitude, phase, and frequency. The Nyquist filters were used as anti-aliasing filters to eliminate signals whose frequency is greater or equal to the Nyquist frequency (Nyquist frequency) and maximize the reconstruction of signals from sampled signals, as illustrated in Figure 9. Some of the preliminary results related to this research were published in the conference papers in [14,42,43].

4.2. Offline Simulation

In this process, the proposed PMU model, together with the PMU block existing in the Matlab library, is used to conduct the voltage and current synchrophasor measurements in a given electrical system. The system is modeled and simulated in Matlab/Simulink R2021b under different working conditions; the simulation model is illustrated in Figure 10.

4.3. Real-Time Simulation

It is a computer operating mode that verifies algorithmic design behaviors when running simulation models at a specific speed to meet exact time expectations. Different hardware, such as actuators and sensors, execute the models. Digital signal processing (DSP), fast control prototyping, and vision system prototyping constitute an integral part of real-time simulation [43]. This process can be conducted in three main stages:

Stage 1: Host PC—This consists of the workspace responsible for executing functions including modeling, graphical interface, and simulation management.

Stage 2: Real-Time Simulator (RTS)—it consists of a multi-core Central Processing Unit (CPU) that compiles and computes the model as C code. The RTS’s main functions include model execution and computation, data logging, and I/O management.

Stage 3: FPGA and I/O Boards—the field-programmable gate array (FPGA) and the input/output (I/O) board function as the interface to connect with physical devices. This is primarily used in a power hardware-in-the-loop (PHIL), a practice in which one or more external hardware components or a physical subsystem are involved in conjunction with the system’s model being simulated.

The Ethernet connection is used to ensure the necessary communication between the workspace utilizing the RT-LAB software and the real-time simulator OPAL-RT OP5707XG. The real-time simulator engine computes and executes the signals and responses of the modeled system, aligning with actual time as in a real electrical network [45]. The tests and simulations for this research were conducted in Benguerir, Morocco, at the Green Energy Park, in the Smart Grids Test Lab. The entire work process is summarized in the flowchart in Figure 11.

4.4. Controller-Hardware-in-the-Loop (CHIL) Technique

During the PMU model tests in the given power network under fault conditions, the controller hardware-in-the-loop (CHIL) technique is used to validate the control algorithm for the fault controller, creating a virtual real-time environment that represents the physical system to control. The process was conducted in four main steps:

Step 1, choosing the communication method: Different approaches are typically used for this purpose, such as analog input and output (I/O), Modbus RTU, and Modbus TCP/IP protocols. Modbus TCP/IP is used in this work.

Step 2, information about input and output data: the fault control data from the Raspberry Pi 4 controller is considered as input data, and synchrophasor signals provided by the PMU model and the Matlab Simulink PMU block from the EPS model in RT-LAB are taken as output.

Step 3, assigning registers: at this step, single Read/Write Single Holding registers are chosen, and then data addresses are assigned as follows: the three-phase line-to-line fault is assigned to address 0, the single-phase line-to-ground fault is assigned to address 1, and the required synchrophasor outputs were assigned to addresses 3 to 11.

Step 4, configuration of communication in OPAL RT-LAB. This is performed by assigning the addresses to the model inputs and outputs using Opin/Opout blocks.

The fault control was implemented using a Raspberry Pi 4. Utilizing Modbus TCP/IP, signals were transmitted from the Raspberry Pi serving as a master to the real-time simulator OP5707XG functioning as the slave, enabling precise fault addition or removal. The control implementation was realized through the Raspberry Pi support package and master write blocks within Matlab/Simulink.

In the used setup, Modbus TCP/IP was selected over other common communication protocols such as CAN (Controller Area Network) or serial-based methods (e.g., RS-232/RS-485) due to its high fidelity, reliability, and superior data throughput. Modbus TCP/IP benefits from being built upon the Ethernet infrastructure, allowing faster data transmission, larger payloads, and broader system scalability, which are crucial in real-time monitoring and control scenarios. In terms of communication latency and packet loss, it is important to highlight that

• Modbus TCP/IP inherently supports higher communication rates with reduced latency compared to serial protocols.
• The TCP/IP layer ensures reliable data delivery through acknowledgment and retransmission mechanisms, which minimizes the risk of packet loss affecting system performance.

During the CHIL experiments, the communication latency remained within acceptable bounds, and no significant packet losses were observed, thus preserving the accuracy and timing requirements of PMU data transmission.

4.5. Measurements with the Real PMU

The laboratory measurement with a real PMU has been conducted. The PMU considered in this exercise is incorporated in a protective relay SEL-351A (see Figure 12a), which is designed to ensure complete distribution system protection. It protects lines and equipment using phase, negative sequence, residual ground, and neutral-ground overcurrent elements with directional control. The SEL-351A relay has many other built-in features, including synchrophasor measurement functions provided by a built-in PMU that provides real-time system state measurement with time-synchronized voltages and currents in the format and characteristics that comply with the IEEE standard C37.118.1-2011 [40]. These features significantly improve the system performance by offering complete synchrophasor solutions, including hardware, communications, data collection, viewing and analysis software, and data archiving.

Additional equipment was utilized during the laboratory measurements with the real PMU of the relay SEL-351A. These include the Omicron 256 (see Figure 12b), regarded as a high-precision relay test set and universal calibrator that can be used with a variety of measuring devices, including energy meters, measuring transducers, power quality measurement devices, and phasor measurement units. In this study, the Omicron was assigned the Comtrade data obtained from the conversion of the input signals of the electrical system utilized in the offline and real-time simulation. The interface box CMIRIG-B (see Figure 12c) is also employed to ensure synchronization to the Coordinated Universal Time (UTC), which serves as the standard time reference. The Ethernet switch depicted in Figure 12d was utilized to establish the network infrastructure and facilitate communication between disparate devices. The visualization of the measured synchrophasors was obtained using the WIRESHARK software 4.4.6 and a personal computer (PC). The experimental tests were conducted for a power system under normal operating conditions, encompassing third, fifth, and seventh harmonics, as well as defective operating conditions. The measurement results are presented in

Table 1. Real PMU measurements.

Phases	Ph A	Ph B	Ph C	Ph A	Ph B	Ph C
	Normal (balanced, with no harmonics) Operation			Operation with 3rd, 5th, and 7th Harmonics
V [V]	10,823	10,820	10,830	8227	8238	8241
Vangle [°]	−94.2	145.8	25.8	−99.7	140.2	20.2
I [A]	479	479	478	364	364	364
I angle [°]	−112.6	127.4	7.3	−118.2	121.8	1.8
f [Hz]	50	50	50	50	50	50
	Symmetrical Fault			Unsymmetrical Fault
V [V]	28.6	28	29	31	10,830	10,830
Vangle [°]	−82.1	157.9	39	−80.6	−1247	11.3
I [A]	5871	5852	5843	5843	479	479
I angle [°]	−82.3	157.9	39	−80.8	−142.9	97.1
f [Hz]	501	50.1	52.2	50.2	50	50

.

5. Tests and Result Comparison

5.1. Test of the PMU Model Under Normal Conditions

In this case study, the electrical system is simulated in normal balanced operation with no harmonics or faults. The proposed PMU model and the Matlab PMU block are used to provide the power system synchrophasor measurements, and their results can be compared when applicable, i.e., when the working conditions involve only positive sequence components. The voltage and current waveforms for all phases of all treated scenarios cannot be shown in this paper, but they are represented by the graphs of phase A and shown in Figure 13, Figure 14, Figure 15 and Figure 16, respectively, for both PMUs.

In normal and balanced working conditions, the offline and real-time simulations for both PMUs yield almost the same results. Since the system experiences neither negative nor zero sequences, the positive sequence measurements exhibited by the PMU block existing in Matlab (see Figure 13 and Figure 14) are comparable to the phase A measurements provided by the proposed PMU model (see Figure 15 and Figure 16). It has been demonstrated that both PMUs are suitable for these working conditions. However, the proposed model has more advantages, as it provides synchrophasors measured in each phase, facilitating smoother system analysis. As demonstrated in the preceding scenarios treated above, the transient period at the commencement of the measuring process for offline simulation is longer than that of real-time simulation. Conversely, the steady-state situations are nearly equivalent for both simulations. This shows that the real-time simulation runs faster and more efficiently than the offline one.

5.2. Test of the PMU Model Under Operation with Harmonics

The input signals of voltage and current are considered with the 3rd, 5th, and 7th harmonics and are consequently distorted as illustrated in Figure 17. The voltage and current synchrophasor measurements from the simulation of both PMUs are presented in this sub-section. Since the graphical representations for all the phases and cases cannot be represented in the paper, only the waveforms of phase A from the proposed PMU model and the positive sequence measurement from the Matlab PMU block are shown in Figure 18, Figure 19, Figure 20 and Figure 21.

Figure 18 and Figure 19 show the voltage synchrophasors, measured by the proposed PMU model and the existing PMU block in the Matlab library, respectively. These figures display the results of offline and real-time simulations for a power system operating under third, fifth, and seventh harmonics. In both the transient phase at the start of the measurement and the steady-state phase, the offline and real-time simulators behave similarly, producing almost identical results. In this case, the measured voltage is relatively low compared to normal operating conditions due to significant voltage drops caused by harmonics.

The graphical representations of the current synchrophasors in Figure 20 and Figure 21, measured by both PMUs indicate that real-time and offline simulations produce approximately the same waveforms during the transient period at the beginning of the process as well as within the steady-state operation. Both PMUs (the Matlab PMU bloc and the proposed model) were found to be suitable for such a measurement. The low current measurements were observed due to power losses resulting from the harmonic presence.

5.3. PMU Tests Under Fault Conditions

This case study examines the system under symmetrical fault involving a three-phase line-to-line fault (3LL) and a non-symmetrical single-phase-to-ground fault (1LG) in phase A. The controller-hardware-in-the-loop technique was employed, utilizing a Raspberry Pi 4 to control the fault occurrence by introducing and eliminating faults within a specified period for symmetrical and non-symmetrical fault scenarios.

5.3.1. Synchrophasor Measurements Provided by the Proposed PMU Model

Figure 22, Figure 23 and Figure 24 illustrate the current and voltage synchrophasors measured simultaneously by the proposed PMU model under symmetrical (3LL) and non-symmetrical (1LG) fault conditions. Due to limitations inherent in the scope of the study, the voltage and current waveforms are represented by the graphs of phase A for the proposed PMU model. As depicted in Figure 22, the measured current exhibits an increase reaching nearly ten times the normal load current during both symmetrical and non-symmetrical fault occurrences. This is known as fault current. As shown in Figure 23, the current flowing through the neutral conductor is negligible under the symmetrical fault because there is no system imbalance. However, during a single-phase-to-ground fault, the neutral current increases significantly due to the current imbalances caused by the asymmetrical fault. In Figure 24, the voltage variations are illustrated, comparing the pre-fault measured voltage with the voltage during the fault. The latter is found to be negligible for both fault scenarios. An increase in phase angle and fluctuations in frequency have been observed during both fault periods, across both current and voltage measurements.

5.3.2. Synchrophasor Measured by the Matlab PMU Block

The proposed PMU model accurately measures the system synchrophasors under symmetrical and non-symmetrical faults (3LL and 1LG), where their occurrences result in the voltage passing to the lowest values (approximately zero) and the current reaching very high values around ten times the normal load current, corresponding to the fault currents as discussed above. However, it should be noted that the Matlab PMU block suitably measures the system’s synchrophasors exclusively under symmetrical faults, in which case neither negative nor zero sequences are available. This PMU block is unable to effectively process the measurements under unsymmetrical faults where all symmetrical components are present. This is illustrated in Figure 25, where the PMU block fully measures the symmetrical fault as the voltage drops to its lowest and negligible level. However, the unsymmetrical fault is not accurately measured due to the presence of negative and zero sequence components that the PMU block in question does not detect.

5.4. Result Discussion

A novel PMU model was proposed and implemented in Matlab/Simulink 2022b, employing a combination of Fourier transform and phase-locked loop techniques. The proposed PMU model was assessed by measuring an electrical power system’s voltage and current synchrophasors. The system’s parameters are as follows: a nominal voltage of 13.8 kV, an active power of 8 MW, a reactive power of 2.63 Mvar, and a frequency of 50 Hz. The power factor is 0.95, corresponding to a phase angle of 18.2°. Four scenarios were considered, consisting of normal working conditions, operation under the 3rd, 5th, and 7th harmonics, symmetrical fault, and unsymmetrical fault conditions. To validate the PMU model’s test results, a comparison was conducted between its measured synchrophasors and those provided by the real PMU measurements. The real PMU measurements were regarded as references (or true measurements), while the synchrophasor measurements provided by the PMU model were considered the observed values, and the measuring errors were determined. Subsequently, a comparison was made between the measuring errors and the IEEE standard admissible tolerance, which is ±10% in the case of amplitudes. As illustrated in

Table 2. Comparison between synchrophasors.

Operations		PMU Model	Real PMU	Measument Error [%]	IEEE Std Tolerence
Normal Operation	V [V]	10,828	10,823	0.046	±10%
	I [A]	478.8	479	−0.042	±10%
3rd, 5th, 7th Harmonics	V [V]	8057	8227	−2.110	±10%
	I [A]	356	364	−2.247	±10%
Symmetrical fault (3LL)	V [V]	28.4	28	1.408	±10%
	I [A]	5855	5871	−0.273	±10%
Unsymmetrical fault (1LG)	V [V]	30.6	30	1.961	±10%
	I [A]	5851	5843	0.137	±10%

, a comparison of the results is presented.

In normal operation, the measured phase angle, determined by the difference between the voltage and current angles, is 18.2° across all phases for both offline and real-time simulations. This corresponds to a measurement error of 0°, while the standard tolerance is ±10°. Therefore, the phase differences observed between the measurements of the PMU model, the real PMU, and the existing Matlab PMU block arise from different references, since the phase angle designated as φ remains nearly the same in all cases. The maximum frequency error (FE) recorded was 0.002 Hz, whereas the IEEE standard stipulates a permissible frequency error tolerance of 0.005 Hz. The rate of change in frequency error (RFE) of the proposed PMU model is 0.152 Hz/s. The RFE tolerance for the IEEE standard, as amended in 2014, is 0.4 Hz/s. The total vector error (TVE) ranges from 0.0397% to 0.041% across different phases for offline and real-time simulations, whereas the standard acceptable tolerance for TVE is 1%. As illustrated in

Table 3. Measurement errors comparison.

Type of PMU Model	TVE [%]	FE [Hz]	RFE [Hz/s]
Proposed PMU Model	0.041	0.002	0.152
PID DSRF-PLL-based PMU model [28]	0.09	0.00295	0.11
EPLL-based PMU algorithm [25]	0.8	0.006	0.5
IEEE Std requirements	1	0.005	0.4

, a comparative analysis of the measuring errors is presented among the proposed PMU, the EPLL-based PMU, and the PID DSRF-PLL-based PMU models, as delineated in [25,28].

In the context of the normal operation test, the absence of zero and negative sequence components ensures that the positive sequence component of the measurements corresponds to the phase values. Consequently, the measured current and voltage phasors by the PMU model in phase A, the positive sequence component measured by the Matlab PMU block, and the measurements of the real PMU, phase A, exhibited close similarity, with a marginal discrepancy of less than 0.1%. This finding aligns with the established normal load current parameters for both PMUs and the voltage under normal operating conditions. The current flowing in the neutral is negligible, amounting to nearly zero, due to the absence of imbalance in the system.

In operation under the 3rd, 5th, and 7th harmonics, both PMUs were found to be equally applicable for this measurement, with results that were approximately equivalent to those of the real PMU. However, low voltage and decayed current measurements were observed due to voltage drops and power losses caused by harmonics. The proposed PMU model is more advantageous in both operations because it directly provides information and measurements from all the phases and the neutral conductor.

When testing the EPS under faulty conditions, the controller-hardware-in-the-loop approach was applied. The management of the fault occurrence was facilitated by a Raspberry Pi 4 controller. In the context of a symmetrical fault, the system experiences the flow of substantial fault currents, which precipitates a decline in voltage to remarkably low and negligible levels, approaching zero. A comparison of both PMU measurements reveals that they are nearly equal, with minimal disparities (typically less than 0.1%), as illustrated in Figure 24 and Figure 25. The fault current is balanced across all phases, and the current flowing in the neutral is found to be nearly zero and negligible. Zero and negative sequence components are absent, and the positive sequence component is found to be equivalent to the phase current measured by the PMUs. This corresponds to the fault current, which is approximately 10 times the nominal load current.

During the non-symmetrical fault (phase A to ground), the system’s heavy fault current flows through phase A, causing the voltage to drop to zero only in phase A. The system is unbalanced, with all the sequence components (zero, negative, and positive) present, and the fault current is accurately measured by the proposed PMU model, as depicted in Figure 22. The model measures the heavy imbalance current flowing through the neutral. This current is nearly zero during the pre-fault and symmetrical fault operation, as illustrated in Figure 23. The Matlab PMU block is incapable of measuring negative and zero sequences; consequently, its measurements are not an accurate reflection of the actual situation for both current and voltage, as illustrated in Figure 25.

Compared with some existing PMU models, the proposed model presents various advantages including the fact that it measures all the symmetrical components (positive, negative, and zero sequence components). It provides the measurements of individual phases and neutral current. These features make the proposed PMU model suitable for application under all tests in normal and abnormal working conditions such as operation under harmonics, system imbalances, symmetrical and unsymmetrical faults involving ground current flows, etc.

Table 4. Comparison of PMU models.

Model Type	Technology	Dynamic Response	Harmonic Rejection	Compliance with IEEE Std	Observation
Proposed PMU	DFT and PLL	Fast	Good	Yes	Suitable for applications in EPS under steady-state and dynamic conditions
Matlab PMU Block	DFT and PLL	Fast	Good	Yes	Suitable for applications with positive sequence components only
PID DSRF-PLL [28]	PLL	Fast	Good	Yes	Low computational burden
EPLL [25]	PLL	Fast	Good	Yes	Medium computational burden and minor deviation from IEEE Std

shows additional property comparisons between the proposed PMU model and some existing models

6. Conclusions

In this study, a PMU model was proposed and tested under normal and abnormal conditions through both offline and real-time simulations. The abnormal operations consist of the presence of 3rd, 5th, and 7th harmonics, as well as two case studies of faulty operations: a case of three-phase line-to-line (3LL, symmetrical short-circuit) and a single line-to-ground fault (1LG, unsymmetrical short-circuit in phase A).

The analysis revealed that the maximum measuring errors were −2.11% and −2.24% for voltage and current amplitudes, respectively, and 0° (0%) for the phase angle. The IEEE standard C37.118.1-2011 [40] tolerance is ±10% for amplitudes and ±10° for phase angle. The frequency error (FE) registered at 0.002 Hz, whereas the admissible frequency error tolerance as delineated by the IEEE standard is 0.005 Hz. The total vector error (TVE) ranges from 0.0397% to 0.041% across various phases in both offline and real-time simulations. It is important to note that the TVE standard admissible tolerance is set at 1%. Consequently, the proposed PMU model is in accordance with the requirements stipulated in the IEEE C37.118.1-2011 standard [40] and its amended version C37.118.1a-2014 [41].

For the power system under normal and symmetrical fault conditions, the proposed PMU model and the existing PMU block in Matlab/Simulink provide appropriate measurements that meet the aforementioned standard requirements, rendering both models equally suitable. Nevertheless, in the event of an asymmetrical fault, the utilization of the Matlab PMU block is not recommended, as evidenced by Figure 25. The PMU block in question provides solely the positive sequence components of the synchrophasors. Consequently, the measurements it produces do not accurately reflect the actual situation of the system. Conversely, the measurements derived from the proposed PMU model in this case are found to be precise. Therefore, the proposed PMU model was found technically superior, as it provides essential measurements for all phases during any operation, along with the current in the neutral conductor for unbalanced scenarios. This enhances the analysis of power system phasors and facilitates a more comprehensive and nuanced assessment of system performance.

The controller hardware-in-the-loop technique facilitates the interaction of real physical devices with system models for real-time control and testing, constituting a powerful practice for designers and researchers. This affords them a broad array of control and testing possibilities that could be difficult to handle, costly, time-consuming, and sometimes hazardous when conducted in actual physical systems.

The main contribution of the proposed model is to deliver accurately measured synchrophasors during the system’s steady state and dynamic operations. Its capacity to provide individual phase measurements and neutral current, providing consequently all the symmetrical components, enables a proper fault analysis of the system and makes it a robust and powerful analytical tool for power system phenomena.