# Online Static Voltage Stability Monitoring for Power Systems Using PMU Data

Authors: Jianhong Pan; Aidi Dong; Jiashu Fan; Yang Li
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

A new online static voltage stability monitoring method for power systems is proposed by using phasor measurement unit (PMU) data in this paper. This approach uses the real-time power, voltage, and phase angle data collected by the PMU to estimate the power flow Jacobian matrix of the system, and then the static voltage stability is monitored via the minimum singular values (MSVs) of the power flow Jacobian matrix. The novelty of the approach lies in the fact that it only utilizes PMU data for implementing online monitoring of the power system static voltage stability, independent of the physical model and its parameters. The application results on the IEEE 57-bus test system verify the effectiveness of the proposed approach.


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## Body

## 1. Introduction

Voltage stability assessment (VSA) has been recognized as an important task to ensure the secure and economical operation of power systems [1, 2]. Problems arising from the growing integration of intermittent renewable power generation in a variety of forms such as active distribution networks [3] and microgrids [4, 5] and integrated energy systems [6] are nudging power systems toward potential dynamic instability scenarios due to the inherent uncertainties of renewable generation [7]. As a new generation of DC transmission technique, voltage source converter-based high voltage direct current (VSC-HVDC) has become a popular opinion of power transmission due to its significant advantages such as independent adjustments of active and reactive powers [8], asynchronous interconnection between islands [9], and black-start capability [10]. In addition, the growing integration of energy storage [11], electric vehicles [12], cyber attacks [13–15], and increasingly diversified demands [16, 17] affect the stable operation of the system to a certain extent. All these shifts pose new challenges in maintaining the system working reliably and securely. In recent years, the wide-area measurement system (WAMS) using time-stamped phasor measurement units (PMUs) has been receiving ever increasing attention from both academia and industry, which makes it possible to explore wide-area protection and control (WAPaC) schemes to avoid the system collapse [18–21]. Therefore, the voltage stability monitoring and assessment of power systems based on PMU data are of great significance in the new context [22–25].Since the first dQ/dU criterion proposed by the Soviet scholar N.M in the 1940s, a large number of voltage stability analysis approaches of power systems have been developed, such as the sensitivity method [26], continuous power flow method [27], the singular value decomposition method [28], and so on. The sensitivity method is only suitable for simple power systems, and sometimes there are discriminant errors when it is used in multimachine systems. In theory, the feasible solution domain method can calculate the voltage stability margin of a given operation mode, but the calculation of critical injection vectors involves complex nonlinear problems, and the computation is heavy. The multivalue method of power flow needs to track and calculate the multisolution value of the injection quantity repeatedly, and the static stability margin of the system voltage is approximate. References [29–31] illustrate the feasibility of the singular value decomposition theory of the trend Jacobian matrix, but the above methods need to know the topology and parameters of the system and cannot monitor the static voltage stability of the grid online. The widespread deployment and successful application of PMUs break through this bottleneck and shed new light on online monitoring of the static voltage stability of power systems [32–34].In recent years, PMU data have been successfully used for real-time power system stability assessment due to its powerful data acquisition capabilities. In [35], an extreme learning machine- (ELM-) based transient stability assessment (TSA) model optimized by improved particle swarm optimization is proposed by using PMU data; to address the issue of lacking online learning ability in current methods, an online sequential ELM- (OS-ELM-) based TSA method is presented for power systems in [36]; a feature selection approach is presented for TSA based on improved maximal relevance and minimal redundancy criterion and PMU measurements in [37]; a new online learning mechanism based on an ensemble of OS-ELM (EOS-ELM) is developed for TSA in [38]; in [39], a rule extraction method is developed by using ELM and improved ant-miner algorithm for TSA; based on PMU data, reference [40] proposes a TSA method by using kernelized fuzzy rough sets and memetic algorithm for feature selection; in [41], a real-time TSA approach based on EOS-ELM coupled with binary-based feature selection is proposed. In [42], a short-term voltage stability assessment method is proposed by calculating the Lyapunov exponent using PMU data. However, research reports on online monitoring of power system static voltage stability using PMU data have still been relatively rare for a long time. Lim and DeMarco put forward a VSA approach with the use of singular value analysis of PMU data in [43, 44]. Inspired by these investigations and data-driven decision making [45, 46], this paper presents an online static voltage stability analysis method based on PMU data. This method is not limited by the network model and parameters. It only needs to use the active power, reactive power, and voltage amplitude and phase angle measurement collected from PMUs to monitor the static voltage stability of a power system. Finally, the effectiveness of the proposed method is verified by using the IEEE 57-bus system.


## 2. Establishment of the Jacobian Matrix Model for Power Flow

The power flow equation of the power system in polar form can be expressed as follows:(1)Pi=Vi∑j∈iVjGijcosθij+Bijsinθij,Qi=Vi∑j∈iVjGijsinθij−Bijcosθij,where j∈i denotes that buses i and j are connected; Ui and Uj represent the voltage amplitude of buses i and j, respectively; Gij and Bij represent the values of the real and imaginary parts of the admittance matrix between buses i and j, respectively; θij represents the phase angle difference between buses i and j; and Pi and Qi represent the active and reactive power of bus i, respectively.Excluding the equilibrium bus of the system, according to the Newton–Raphson method, formula (1) can establish the following linearization modified equations:(2)ΔPΔQ=HijNijJijLijΔθΔV,where ΔP and ΔQ represent the microincrement of active power and reactive power of the injection bus; Δθ and ΔV represent the microincrement of the voltage phase angle θ and the voltage amplitude V, respectively; Hij and Nij represent the partial derivative of the active power P to the voltage phase angle θ and the voltage amplitude V, respectively; and Jij and Lij represent the partial derivative of the reactive power Q to the voltage phase angle θ and the voltage amplitude V, respectively. In equation (2),(3)J=HijNijJijLij,where J is the Jacobian matrix of the linearized power flow equation.


## 3. Evaluation of Voltage Stability by Singular Value Decomposition

Suppose the system hasn buses in addition to the slack bus, of which there are m PV buses, and the singular value decomposition (SVD) [47, 48] is performed on equation (3). Then, the following formula can be obtained:(4)J=EδUT=∑i=12n−meiδiuiT,where ei and ui represent the i-th column elements of E and U, respectively, and δi is the diagonal element of the diagonal matrix. If the Jacobian matrix is nonsingular, the effect of the increment ΔP and ΔQ of the injection powers P and Q on θ and V can be obtained by equations (2) and (4):(5)ΔθΔV=J−1ΔPΔQ=∑i=12n−mδi−1uieiTΔPΔQ.It can be seen from equation (5) that when a singular value δi is very small (close to zero), the small change of injection power P and Q will cause great fluctuation of θ and V. The response of the system is completely determined by the minimum singular value (MSV) δmin and its corresponding left and right singular vectors emin and umin, that is:(6)ΔθΔV=∑i=12n−mδi−1umineminTΔPΔQ,where umin=θ1,θ2,…,θn;v1,v2,…,vn−mT and emin=P1,P2,…,Pn;Q1,Q2,…,Qn−mT.Here,emin and umin are normalized to(7)∑i=1nθi2+∑i=1n−mvi2=1,∑i=1nPi2+∑i=1n−mQi2=1,and then(8)ΔPΔQ=emin,and thus(9)ΔθΔV=uminδmin.From equations (8) and (9), it can be concluded that since the MSV is sufficiently small, a small change in the injection power can cause a large change in voltage. The MSV of the Jacobian matrix can be used as a good indicator of static voltage stability [49, 50].


## 4. Evaluation of Static Voltage Stability by PMU Data

The aforementioned static voltage stability method by using the MSV of Jacobian matrix can give the static voltage stability quantitatively, but it depends on the fixed physical model and cannot judge the static voltage stability online.


### 4.1. Estimation of Power Flow Jacobian Matrix by PMU Data

In polar coordinates, the terms of the Jacobian matrix of power flow can be represented by the partial derivative ofθj and Vj by Pi and Qi, respectively, and then formula (3) can be expressed as(10)Hij=∂Pi∂θj,Nij=∂Pi∂VjVj,Jij=∂Qi∂θj,Lij=∂Qi∂VjVj.Suppose bothθj and Vj have a very small change, represented by Δθj and ΔVj, respectively. Then, Pi and Qi will also change with the slight changes above, denoted by ΔPiΔθj, ΔPiΔvj and ΔQiΔθj, ΔQiΔvj, respectively; then, equation (10) can be expressed as [51](11)Hij=∂Pi∂θj≈ΔPiΔθjΔθj,Nij=∂Pi∂VjVj≈ΔPiΔVjΔVjVj,Jij=∂Qi∂θj≈ΔQiΔθjΔθj,Lij=∂Qi∂VjVj≈ΔQiΔVjΔVjVj.At timet:(12)ΔPit≈∑j∈Ng∪NlΔPiΔθjt+∑j∈NlΔPiΔVjt,ΔQit≈∑j∈Ng∪NlΔQiΔθjt+∑j∈NlΔQiΔVjt,where Ng represents the number of PV buses and Nl represents the number of PQ buses, and substituting equation (11) into (12) yields(13)ΔPit≈∑j∈Ng∪NlΔθjtHij+∑j∈NlΔVjtNij,ΔQit≈∑j∈Ng∪NlΔθjtJij+∑j∈NlΔVjtLij.Further consolidation of equation (13) is available:(14)ΔPi≈Δθjj∈Ng∪NlΔVjj∈NlHiNi,(15)ΔQi≈Δθjj∈Ng∪NlΔVjj∈NlJiLi,where Hi=Hij,j∈Ng∪Nl, Ni=Nij,j∈Nl, Ji=Jij,j∈Ng∪Nl, Li=Lij,j∈Nl.


### 4.2. Model Solution

In the assumption of equations (14) and (15), both the regression matrix and the measurement vector have measurement errors. For the convenience of expression, here we make A=Δθjj∈Ng∪NlΔθjj∈Nl, bi=Pi; then, equation (14) can be expressed as(16)bi≈AHiTNiTT.Since equation (16) is an overdetermined equation, in this paper, HiNiT is calculated by using the total least square (TLS) [52, 53]. This method is also suitable for equation (15).In the ordinary least squares estimate (LSE), since the regression matrix is assumed to be error-free, the principle of this method is to correctbi as little as possible under the Euclidean norm [51], which forms the following optimization problem [54]:(17)minbi^∈RMΔbi2,bi^=AHiTNiTT,where M represents sets of synchronized measurements, Δbi=bi−bi^ (bi^ is a very small value), and any Hi^Ni^T that satisfies bi^=AHi^Ni^T is the solution of LSE of equation (16).Assuming thatA is full rank, the special solution of the closed form of equation (17) is(18)Hi^Ni^T=ATA−1ATbi.Unlike LSE, TLS also takes into account the measurement error inA; similar to the Euclidean norm, the problem is to find a minimized F-norm, as follows:(19)minA^b^i∈RM∗N+1ΔAΔbiF,b^i=A^HiTNiTT,where N−=Ng+2Nl and ΔA=A−A^,Δbi=bi−bi^.The TLS solution of equation (19) depends on the SVD. We describe this process and write equation (16) as follows:(20)AbiHiTNiT−1T≈0.By using the SVD, the above formula can be rewritten as(21)Abi=EδUT,where E=e1,e2,…,em, U=u1,u2,…,uN+1 is a symmetric matrix whose diagonal element δi is the singular value of Abi. If δn+1≠0; then, the rank of Abi is N+1, and the only solution of equation (20) is zero vector. In order to obtain the nonzero solution of equation (20), Abi must be reduced to the rank N. The matrix approximation theorem shows that(22)A^bi^=Eδ^UT.Since the rank of the approximate matrixA^bi^ is equal to N, equation (20) has a nonzero solution. According to the nature of the SVD, uN+1 is the unique vector belonging to the zero vector space A^bi^, and the TLS solution is obtained by scaling the vector uN+1 until the last element is −1:(23)H^iTN^iT−1T=−1uN+1N+1uN+1,where uN+1N+1 represents the N+1th element of uN+1; then, the unique TLS solution of equation (16) is(24)H^iTH^iTT=−1uN¯+1N¯+1uN¯+11,uN¯+12,…,uN¯+1N¯+1.


## 4.1. Estimation of Power Flow Jacobian Matrix by PMU Data

In polar coordinates, the terms of the Jacobian matrix of power flow can be represented by the partial derivative ofθj and Vj by Pi and Qi, respectively, and then formula (3) can be expressed as(10)Hij=∂Pi∂θj,Nij=∂Pi∂VjVj,Jij=∂Qi∂θj,Lij=∂Qi∂VjVj.Suppose bothθj and Vj have a very small change, represented by Δθj and ΔVj, respectively. Then, Pi and Qi will also change with the slight changes above, denoted by ΔPiΔθj, ΔPiΔvj and ΔQiΔθj, ΔQiΔvj, respectively; then, equation (10) can be expressed as [51](11)Hij=∂Pi∂θj≈ΔPiΔθjΔθj,Nij=∂Pi∂VjVj≈ΔPiΔVjΔVjVj,Jij=∂Qi∂θj≈ΔQiΔθjΔθj,Lij=∂Qi∂VjVj≈ΔQiΔVjΔVjVj.At timet:(12)ΔPit≈∑j∈Ng∪NlΔPiΔθjt+∑j∈NlΔPiΔVjt,ΔQit≈∑j∈Ng∪NlΔQiΔθjt+∑j∈NlΔQiΔVjt,where Ng represents the number of PV buses and Nl represents the number of PQ buses, and substituting equation (11) into (12) yields(13)ΔPit≈∑j∈Ng∪NlΔθjtHij+∑j∈NlΔVjtNij,ΔQit≈∑j∈Ng∪NlΔθjtJij+∑j∈NlΔVjtLij.Further consolidation of equation (13) is available:(14)ΔPi≈Δθjj∈Ng∪NlΔVjj∈NlHiNi,(15)ΔQi≈Δθjj∈Ng∪NlΔVjj∈NlJiLi,where Hi=Hij,j∈Ng∪Nl, Ni=Nij,j∈Nl, Ji=Jij,j∈Ng∪Nl, Li=Lij,j∈Nl.


## 4.2. Model Solution

In the assumption of equations (14) and (15), both the regression matrix and the measurement vector have measurement errors. For the convenience of expression, here we make A=Δθjj∈Ng∪NlΔθjj∈Nl, bi=Pi; then, equation (14) can be expressed as(16)bi≈AHiTNiTT.Since equation (16) is an overdetermined equation, in this paper, HiNiT is calculated by using the total least square (TLS) [52, 53]. This method is also suitable for equation (15).In the ordinary least squares estimate (LSE), since the regression matrix is assumed to be error-free, the principle of this method is to correctbi as little as possible under the Euclidean norm [51], which forms the following optimization problem [54]:(17)minbi^∈RMΔbi2,bi^=AHiTNiTT,where M represents sets of synchronized measurements, Δbi=bi−bi^ (bi^ is a very small value), and any Hi^Ni^T that satisfies bi^=AHi^Ni^T is the solution of LSE of equation (16).Assuming thatA is full rank, the special solution of the closed form of equation (17) is(18)Hi^Ni^T=ATA−1ATbi.Unlike LSE, TLS also takes into account the measurement error inA; similar to the Euclidean norm, the problem is to find a minimized F-norm, as follows:(19)minA^b^i∈RM∗N+1ΔAΔbiF,b^i=A^HiTNiTT,where N−=Ng+2Nl and ΔA=A−A^,Δbi=bi−bi^.The TLS solution of equation (19) depends on the SVD. We describe this process and write equation (16) as follows:(20)AbiHiTNiT−1T≈0.By using the SVD, the above formula can be rewritten as(21)Abi=EδUT,where E=e1,e2,…,em, U=u1,u2,…,uN+1 is a symmetric matrix whose diagonal element δi is the singular value of Abi. If δn+1≠0; then, the rank of Abi is N+1, and the only solution of equation (20) is zero vector. In order to obtain the nonzero solution of equation (20), Abi must be reduced to the rank N. The matrix approximation theorem shows that(22)A^bi^=Eδ^UT.Since the rank of the approximate matrixA^bi^ is equal to N, equation (20) has a nonzero solution. According to the nature of the SVD, uN+1 is the unique vector belonging to the zero vector space A^bi^, and the TLS solution is obtained by scaling the vector uN+1 until the last element is −1:(23)H^iTN^iT−1T=−1uN+1N+1uN+1,where uN+1N+1 represents the N+1th element of uN+1; then, the unique TLS solution of equation (16) is(24)H^iTH^iTT=−1uN¯+1N¯+1uN¯+11,uN¯+12,…,uN¯+1N¯+1.


## 5. Algorithm Flow and Basic Steps

The flowchart of the proposed method is shown in Figure1. The specific steps are as follows:Figure 1
Flowchart of the proposed method.(1)
Collect multiple sets of PMU data from different buses at thet+Δt time, and the number of collected groups is greater than the order of the Jacobian matrix.(2)
ObtainΔPi,ΔQi,Δθi,ΔVi, by difference.(3)
Taking the data obtained in step 2 into equation (14), the solution of the overdetermined equations is obtained by the TLS method, and the Jacobian matrix of power flow is obtained.(4)
Calculate the MSV of the Jacobian matrixJobtained in step 3 and compare it with the thresholdτ, if the difference exceeds the margin ξ.(5)
Repeat steps 1 to 4 to realize real-time monitoring of the power grid.By collecting the electrical quantities of different buses at timet+Δt, including Pi, Qi, θi, and Vi, the power flow Jacobian matrix can be obtained according to equations (14) and (15). This process is shown in Figure 2.Figure 2
Data collection of PMU.Note that the number of samples is greater than the order of the Jacobian matrix. The results of the MSV of the power flow Jacobian matrix estimated by PMU data and calculated by the Newton–Raphson method are shown in Table1.Table 1
MSV based on Newton–Raphson method and PMU data.
Bus systemsNewton–Raphson methodPMU dataIEEE 9-bus system0.89420.8928IEEE 14-bus system0.54640.5441IEEE 30-bus system0.21650.2068IEEE 57-bus system0.17450.1725IEEE 118-bus system0.18480.1836IEEE 300-bus system0.03970.0302Comparing the data in Table1, it can be seen that the difference is small in terms of the MSV obtained by the Newton–Raphson method and the PMU data. This shows that the proposed method can be used to evaluate the static voltage stability of the system.


## 6. Case Study

In this study, it is assumed that all buses of the system are equipped with PMUs. In order to simulate the process of voltage collapse in the power system, the active powerP is increased by 0.01 steps in the fifth node of the IEEE 57-bus system, and the MSV of the Jacobian matrix calculated by the proposed method is tracked in real time. The results are shown in Figure 3:Figure 3
P-V/δmin curves of the IEEE 57-bus system.P/V8−δmin, P/V13−δmin, P/V31−δmin, and P/V33−δmin above represent the process of the voltage V changing and the MSV δmin of the buses 8, 13, 31, and 33 changing with the change of power P, respectively. P/δmin represents the process in which the MSV of the system varies with power P.It can be seen from theP/δmin curve that as the power increases for the different buses, the MSV δmin of the system decreases continuously. When δmin is close to zero, the system collapses. Comparing the P/δmin and P/Vpucurves, it can be seen that as δmin decreases (P increases continuously), the voltage value also decreases and the rate of decrease is different. When δmin is close to zero, a small change of P causes a large fluctuation in voltage, which is consistent with the context in equations (8) and (9).During normal operation of the two systems, the sampling frequency of PMU is taken as 50 Hz/s, and the MSV values of the power flow Jacobian matrix are calculated every 4 seconds according to equations (14) and (15). In this way, the evolution curves of the MSV values are demonstrated in Figure 4.Figure 4
Evolution curves of the MSV values in the IEEE 57-bus system.δ1,min and δ2,min represent the smallest singular value obtained by the Newton–Raphson method and PMU data, respectively. The mean square errors of the two methods are 0.0023 and 0.0002, respectively. By comparing the matching degree of the curves of Figure 4, it can be seen that the curves obtained by the two methods are basically consistent, indicating that the method proposed in this paper can monitor the static voltage stability of the system online.The active powerP is increased by the step of 0.005 at bus 6 of the IEEE 57-bus system, and the static voltage stability of the power system is estimated from every 200 PMU measurements, as shown in Figure 5.Figure 5
Curves ofP-δmin of IEEE 57-bus system.By comparing the matching degree of the curves in Figure5, we can see that the curves obtained by the two methods are basically consistent, which shows that the method proposed in this paper can realize online monitoring in the process of system instability.


## 7. Conclusion

A method for real-time evaluation of static voltage stability of a power system based on PUM data is proposed. The effectiveness of the proposed method is verified by the IEEE 57-bus system. The conclusions are as follows:(1)
The proposed method is capable of realizing online monitoring of static voltage stability of power systems.(2)
Compared with the traditional method, the proposed method does not need any model and parameter information of the power grid and only needs PMU data for the implementation.(3)
The proposed method can be used in an online stability detection system of power systems for situational awareness improvement.Our future work will focus on extending the proposed approach to estimate an accurate voltage stability boundary and address stability constraints in voltage stability-constrained optimal power flow [55–58]. It is interesting to investigate the voltage stability of power systems with new elements such as distributed generations [59–61] and combined heat and power plants [62]. Besides, the optimal placement of PMU will be further studied to realize the monitoring of the whole network with fewer PMUs. Another potential topic in future research is to develop voltage stability assessment model using new machine learning techniques such as the lasso algorithm [63], least squares support vector machines [64], and deep learning [65].



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