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Wide‐area measurement system‐based model‐free approach of post‐fault rotor angle trajectory prediction for on‐line transient instability detection

2018; Institution of Engineering and Technology; Volume: 12; Issue: 10 Linguagem: Inglês

10.1049/iet-gtd.2017.1523

1751-8695

Dan Dan Huang, Xiuyuan Yang, Shuyong Chen, Tao Meng,

High-Voltage Power Transmission Systems

IET Generation, Transmission & DistributionVolume 12, Issue 10 p. 2425-2435 Research ArticleFree Access Wide-area measurement system-based model-free approach of post-fault rotor angle trajectory prediction for on-line transient instability detection Dan Huang, Corresponding Author Dan Huang meetzl@163.com School of Electrical Engineering, Beijing Jiaotong University, Beijing, 100044 People's Republic of ChinaSearch for more papers by this authorXiuyuan Yang, Xiuyuan Yang School of Automation, Beijing Information Science & Technology University, Beijing, 100192 People's Republic of ChinaSearch for more papers by this authorShuyong Chen, Shuyong Chen School of Electrical Engineering, Beijing Jiaotong University, Beijing, 100044 People's Republic of China China Electric Power Research Institute, Beijing, 100192 People's Republic of ChinaSearch for more papers by this authorTao Meng, Tao Meng State Grid Shanxi Electric Power Company, Taiyuan, 030001 Shanxi, People's Republic of ChinaSearch for more papers by this author Dan Huang, Corresponding Author Dan Huang meetzl@163.com School of Electrical Engineering, Beijing Jiaotong University, Beijing, 100044 People's Republic of ChinaSearch for more papers by this authorXiuyuan Yang, Xiuyuan Yang School of Automation, Beijing Information Science & Technology University, Beijing, 100192 People's Republic of ChinaSearch for more papers by this authorShuyong Chen, Shuyong Chen School of Electrical Engineering, Beijing Jiaotong University, Beijing, 100044 People's Republic of China China Electric Power Research Institute, Beijing, 100192 People's Republic of ChinaSearch for more papers by this authorTao Meng, Tao Meng State Grid Shanxi Electric Power Company, Taiyuan, 030001 Shanxi, People's Republic of ChinaSearch for more papers by this author First published: 18 April 2018 https://doi.org/10.1049/iet-gtd.2017.1523Citations: 10AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In order to achieve adequate time for emergency control against transient instability, a scheme to predict the post-fault rotor angle trajectory based on the grey Verhulst self-memory model is provided. The proposed scheme is model free and only uses wide-area measurement system data to provide timely, reliable information about the transient rotor angle stability. Moreover, the scheme is model-parameter-adaptive and takes full consideration of the non-linearity as well as uncertainty of power systems. Firstly, the grey Verhulst self-memory prediction model is presented to cope with the strong non-linear and non-autonomous nature of the power system. Secondly, the equal dimension and new information data model as well as the rolling prediction method are adopted to improve adaptability and robustness of prediction. Investigations with the IEEE-39 bus system and NCE China power system indicate that the proposed prediction scheme gives better prediction performance compared with the other two prediction methods, i.e. grey Verhulst prediction and auto-regressive prediction. Furthermore, combined with equal area criterion based on the severely disturbed generator pair, the proposed prediction scheme is conducted to detect transient instability. Simulation results indicate that the proposed prediction scheme is effective for prediction of transient stability. 1 Introduction In recent years, power systems are confronted with great challenges due to their increasing scale and more complex structure. Since the advancement of wide-area measurement system (WAMS) based on synchronised phasor measurements has made it possible to gain real-time information of transient stability status in power systems [1], the power system transient stability assessment and emergency control have received renewed interests. WAMS has provided key foundation for the development of on-line transient stability analysis based on the real-time response data [2]. However, there are some challenges to employ WAMS for real-time monitoring and control. The time frame of interest in transient stability studies is usually 3–5 s following the disturbance [3]. The relatively short-time period and the large scale of the power system make it difficult to take the timely emergency control actions to prevent the transient instability. If the post-fault trajectory information provided by WAMS is utilised solely after transient instability is detected, it would probably fail to prevent the transient instability. Therefore, we need to not only measure the post-fault trajectory about characteristics of the power system transient stability timely but also predict its developing trend accurately in real time. The real-time post-fault trajectory prediction in transient stability is to predict the changing trend of the key electric parameters based on the mathematical prediction model and the obtained response data from WAMS. The aim of the real-time post-fault trajectory prediction is to assess the transient stability of a power system by combining the obtained prediction data with a certain transient stability criterion, and the detection result is used for initiating emergency control process against transient instability. Therefore, the real-time post-fault trajectory prediction plays an important role for on-line transient stability assessment and emergency control of the power system, which attracts many researchers' focus. Existing methods for predicting the post-fault trajectory can be classified into two categories: model-based method [4] and model-free method [5]. Model-based methods employing the power system model are difficult for real-time transient stability analysis due to computation cost which depends on the size of power system. Model-free methods mainly include the artificial intelligence approach [6, 7] and curve fitting and extrapolation method [8–15]. Although the artificial intelligence approach combines off-line learning and online monitoring reasonably and its real-time computation cost is lower, it is difficulty to acquire sampling spaces in off-line learning, due to the large amount of the samples required. The curve fitting extrapolation method is an appropriate method to overcome the above difficulties. In [8], the Taylor series expansion for the rotor angle and angular velocity prediction is first proposed by Haque. The authors in [9] propose a real-time curve-fitting algorithm based on the least-squares method to estimate rotor angle and the unstable equilibrium point. In [10], the implicit integration method with trapezoidal rule is used to approximate the unobservable generator angles and angular speeds. The rotor angle, generator rotor speed and acceleration are estimated by using discrete phasor measurement unit (PMU) data based on the piecewise linear assumption in [11]. In [12], the time series from PMUs are modelled by short memory auto-regressive (AR) models in order to predict PMU measurements from different sensors. Based on a preliminary analysis of large-scale power systems' linearity, a linear power system dynamic model in the autoregressive with exogenous input model structure is proposed in [13]. By employing an error correction scheme, an improved grey Verhulst model for perturbed trajectory prediction of post-fault rotor angle is proposed in [14]. The authors in [15] present a Prony algorithm to calculate and predict the rotor angle trajectories, which are used to determine whether and when a power oscillation will lead to power system instability. However, the methods above turned out low accuracy and poor robustness, due to lack of consideration on the effect of model parameters on prediction, as well as short accurate prediction time horizon. The rotor angle data can be collected as time-series, thanks to the development of the information and communication technologies in WAMS. Essentially, the curve fitting and extrapolation of the post-fault trajectory approach is to compute and analyse the time-series data of the post-fault trajectory. The grey system theory (GST), a system engineering theory based on uncertainty of small samples, was first introduced by Deng in 1982 [16]. It is an effective theoretical tool which can be employed to predict future values of a time-series data according to the most recent data set. The Verhulst model in the GST is utilised to model and predict time-series with saturated trend and it has been successfully applied in real electricity problems [14, 17]. Self-memory principle was proposed in early 1990s by Cao [18] based on the fact that natural and social phenomena are irreversible. The self-memory equation was first applied for atmosphere motion prediction by Cao [19]. The self-memory model, which can recall observation data in the past and focus on the evolution law of system itself, is an effective statistical-dynamical method to overcome random fluctuations in non-linear dynamic systems. The non-linearity, uncertainty and non-autonomy of power systems make it difficult to predict the post-fault rotor angle trajectory in transient stability accurately. To address these challenges, this paper proposes a model-free prediction scheme for post-fault rotor angle trajectory prediction, where the grey Verhulst self-memory model is built and rolling prediction method is adopted to improve adaptivity and robustness of the scheme. Ridge regression for parameter estimation of the proposed model is used to improve the prediction model. Furthermore, combined with the equal area criterion (EAC) based on the severely disturbed generator pair (SDGP), the proposed scheme is employed to assess the transient instability of power system. Case studies show that this scheme designed for transient instability detection is fast, efficient and realisable. This paper is organised as follows: Section 2 describes the transient rotor angle stability problem. Section 3 provides the mathematical model and the prediction scheme for post-fault trajectory prediction. In Section 4, simulations results are presented and prediction performance is discussed. In Section 5, the transient instability detection based on the proposed prediction scheme is conducted. Section 6 gives the conclusion. 2 Transient rotor angle stability problem Transient stability problem studies the stability of the rotor angle dynamics following a severe fault or disturbance. The model can be represented by a set of non-linear differential and algebraic equations. For the multi-machine power systems, the motion equations of generator i can be expressed as (1), where δi is the angle of generator i ; ωi is the angular velocity with respect to a synchronous frame of the generator i ; Mi is the inertia constant of generator i ; Pmi and Pei are the mechanical power input and electric power output of generator i ; Gii is the self-conductance of generator i ; Gij is the transfer conductance between generator i and generator j ; Bij is the transfer susceptance between generator i and generator j ; Ei is the constant voltage of generator i ; δij is the angle differences between generator i and generator j (1) The transient stability problem can be summarised as: after a fault or disturbance occurs, if δi (t) tends to be a certain value as t → ∞, then the system is deemed stable [20]. Actually, the geometric characteristics of the generator rotor angle difference trajectories reflect the system stability, so the trend of the generator rotor angle difference curves can be used for rapid transient stability prediction. When a fault occurs, the rotor angle cannot accelerate instantaneously because of the slow electromechanical response of the generator, which provides a prerequisite for the prediction. The typical curve of a grey Verhulst model and the response curves of synchronous generator to a disturbance are shown in Figs. 1a and b, respectively. The curves in Fig. 1b represent three different stability conditions of the generator: single-swing instability, multi-swing instability, and multi-swing stability. According to local observation, during each swing, all the rotor angle trajectories follow an S-shape curve. In the multi-swing case, the angle increases gradually at the beginning, and the angular velocity starts to decrease after a certain time so that the angle reaches its wave peak or trough at last. In the single-swing case, the angle keeps increasing, presenting the S-shape ascending trajectory. Therefore, it is illustrated that the grey Verhulst model can be employed to predict and fit the rotor angle trajectories. However, from these angle curves, it can be seen that the non-linearity for them differs apparently and the shapes of the curves are totally different under different degrees of disturbance, while for each curve, its non-linearity changes with time. Moreover, the response curve of rotor angle to the disturbance on a real power system is more complex and variable due to strong non-linear and non-autonomous nature of the power system, which makes it too hard to predict accurately the rotor angle trajectory simply by grey Verhulst model prediction method. The self-memory model, which involves the historical information and emphasises on the relationship between fore and after status of the system itself, can solve the problem efficiently. Fig. 1Open in figure viewerPowerPoint Curves (a) Curve of the grey Verhulst model, (b) Rotor angle response curve to a disturbance 3 Prediction of post-fault rotor angle trajectory 3.1 Grey Verhulst self-memory model 3.1.1 Grey Verhulst model The input original sequence can be represented as (2) where x(0) (k) is the time-series data at time k. Then a new sequence X(1) is generated by applying the accumulating generation operator to the input original sequence (X(0)) as follows: (3) The contiguous mean value sequence of X(1) is represented by Z(1) : (4) The foundation model of the grey Verhulst model is defined as (5) Then the whitenisation equation of the grey Verhulst model is expressed as (6) where a and b are model coefficients, and the coefficient vector (a b)T can be obtained by the ridge regression [21] estimation as follows: (7) where By solving (6) and discretising the solution, the grey Verhulst prediction model can be obtained as (8) 3.1.2 Deriving the self-memory model equation The differential equation given in (6) is considered as the system dynamics equation of the grey Verhulst self-memory model. The dynamics equation is shown as (9) where x is the system variable, λ is the system parameter, t is time-series, and F (x, λ, t) is the system dynamic kernel. The inner product operation in a Hilbert space is defined as (10) A time set is defined as T = {t−p, t−p+1, …, t−1, t0, t1}, where t−p, t−p+1, …, t−1, t0 is historical observation time, t0 is the predicted initial time, t is the coming prediction time. The backtracking order is represented as p, and the time interval is △t. The memory function β (t) is introduced, then β (t) and (9) are computed by (10) as follows: (11) The left-hand side of (11) is integrated at every time interval, after that integration by parts is conducted. Hence, the following equation is obtained as (12) Both the variable x and the memory function β (t) are supposed to be continuous, differentiable and integral. According to the mean value theorem, (12) can be written as (13) Let β ≡ β (t), βi ≡ β (ti), xi ≡ x (ti), and x1 ≡ x (t1), a difference-integral equation named a self-memory prediction equation with the backtracking order p can be obtain as follows: (14) The first two items on the side of (14) involved the historical data, which are defined as the self-memory items, are considered as the contribution to the prediction of x1. The last item is represented as the contribution of the function F (x, λ, t) over the time interval . So it can be seen that (14) emphasises the serial correlation of the system by itself, which is the self-memorisation of the system. Let Fi = F (x, λ, ti), and by conducting the summation instead of the integration, (14) can be written as (15), where , i = −p, …, 0,1, namely the memory coefficient. Equation (15) is defined as the discrete form of the self-memory prediction function (15) 3.1.3 Solving the self-memory model equation If F and x are the input and output of the system, respectively, (15) can be seen as the typical equation of the system identification theory, which can be solved by ridge regression method. Given a time-series data x = {x1, x2, …, xm}, the data is divided into several groups by the sliding mode in sequence, in which each group has p + 2 data, so there are n = m − p − 1 groups. Substitute F−p, F−p +1, …, F0 in each group obtained by (6) into (15), system linear equations with symmetric coefficient matrix can be obtain as follows: (16) where where (17) Then the solution of ridge regression is written as (18) The coefficients α−p, α−p +1, …, α0, α1 can be solved by (18), and the original data can be predicted and fitted by (15). 3.2 Model-parameter-adaptive prediction scheme To improve the adaptability, an adaptive prediction scheme employed the equal dimension and new information data model is proposed. Set the step size of the prediction to be L to obtain a desired length of the prediction. Then p + 2 pieces of the historical rotor angle data can be get from WAMS, noted as: δ−p, δ−p+1, …, δ0, δ1. Calculate the next prediction value which is marked as by the proposed prediction model, then remove the first observation δ−p and let serve as the (p + 2)th observation. In each step, the above calculation is conducted repeatedly. Furthermore, rolling prediction method is adopted to improve the accuracy of the prediction, which means a certain amount of data need to be collected for prediction, so that the memory coefficients of the prediction model can be updated continually. The overall flowchart of the proposed adaptive prediction scheme is shown as in Fig. 2. Fig. 2Open in figure viewerPowerPoint Flowchart of the grey Verhulst self-memory prediction scheme 4 Simulation results The proposed prediction algorithm is tested by using the IEEE-39 bus system and a real power system, i.e. the interconnected North China power grid with Central China (CC) power grid, and East China (EC) power grid (NCE China power system). In order to demonstrate the stability and robustness of the proposed prediction scheme, the rolling prediction with different sampling points is simulated. To validate the proposed prediction algorithm, its results are compared with the other two estimation approaches in [14, 22], i.e. the grey Verhulst model prediction approach and AR model prediction approach. Considering the speed of the telecommunication infrastructure in WAMS as well as the calculation amount, and to ensure both prediction precision and length, set the sampling interval Δt = 10 ms, the number of sampling points m = 10 and the backtracking order p = 3. The response results to a disturbance by Power System Department- Bonneville Power Administration (PSD-BPA) (a China-version BPA software developed by China Electric Power Research Institute) are used as the real-time measurement data of WAMS. All the tests are performed in MATLAB environment (computer specifications: Intel Core i7-5500U CPU at 2.39 GHz, 8 G RAM). 4.1 IEEE-39 bus system IEEE-39 bus system has ten generators, 19 loads and 46 branches. The parameters about this system can be found in [23]. The fourth-order transient model is used for generators. All loads are represented in ZIP model and the ratios of the constant current, constant resistance and constant power loads are 0.5, 0.3, 0.2, respectively. The three phase to ground fault occurs on the line connecting buses 4 and 14 at the time 0 s, and the fault is cleared at 0.23 and 0.24 s for cases 1 and 2, respectively. Cases 1 and 2, which are critical stable cases, are used to verify the effectiveness of the proposed prediction scheme. The generator at bus 39 is assumed as the reference machine, and the time-domain simulation results are shown in Fig. 3a. Generator 35 serves as an example for observation. To verify the prediction performance of the proposed scheme, the relative rotor angle value of 1 s obtained by single rolling prediction based on the three different methods is compared. The comparison of different prediction results with the 0.4–0.5 s sampling points for cases 1 and 2 are shown in Figs. 3b and c, respectively. Moreover, to test the stability of the proposed scheme, the rolling predictions with sampling points at different time periods of the proposed scheme are conducted for cases 1 and 2, and the simulation results are shown in Fig. 3d. Fig. 3Open in figure viewerPowerPoint Time-domain simulation results, and prediction results for bus 35 of IEEE-39 bus test system considering a three-phase to ground fault (a) Time-domain simulation results, (b) Comparison of different prediction methods for case 1, (c) Comparison of different prediction methods for case 2, (d) Rolling prediction results with sampling points at different time periods for Cases 1 and 2 4.2 Real power system To evaluate the applicability of the proposed scheme, NCE China power system known as one of a large-scale interconnected power system is selected as a realistic test system. The power system covers 12 provinces and four municipalities in China, and is a real-world example of an ultra-high-voltage AC/DC hybrid power system with complex dynamics. The power flow diagram of NCE China power system in a certain operation mode is shown in Fig. 4 (G: generating capacity, MW; L: load capacity, MW). The number of the generators is 1996 and the loads are described by different proportions constant resistance load model and induction motors. The prediction scheme is applied to NCE China power system to better verify the performance of the prediction. The three phase to ground fault occurs on the 500 kV line which is the tie-line connected Sichuan (SC) power grid and Chongqing (CQ) power grid at the time 0 s, and the fault is cleared at 0.27 and 0.28 s for cases 3 and 4, respectively. In CC power grid, SC power grid (G: 51136 MW; L: 24696.1 MW) is the sending areas, whereas CQ power gird (G: 12327 MW; L: 15391 MW) is the receiving area. If the three phase to ground fault cannot be cleared timely, the transient instability will be inevitable because of unbalanced power. We take the generator Sanxia in EC power grid as reference, and time-domain simulation results are shown in Fig. 5a. One generator in SC power grid is selected as observation. The comparison of different prediction results with the 0.4–0.5 s sampling points for cases 3 and 4 is shown in Figs. 5b and c, respectively. The rolling predictions with sampling points at different time periods of the proposed scheme are conducted for cases 3 and 4, and simulation results are shown in Fig. 5d. Fig. 4Open in figure viewerPowerPoint Structure of NCE China power system Fig. 5Open in figure viewerPowerPoint Time-domain simulation results, and prediction results for one generator in SC power grid of NCE China power system considering a three phase to ground fault (a) Time-domain simulation results for cases 3 and 4, (b) Comparison of different prediction methods for case 3, (c) Comparison of different prediction methods for case 4, (d) Rolling prediction results with sampling points at different time periods 4.3 Different stability conditions To verify the application of the proposed prediction method for different stability conditions, we apply this method on three typical stability conditions, i.e. first-swing instability, multi-swing instability and stable condition, for IEEE-39 bus system. The three phase to ground fault occurs on the line connecting buses 21 and 22 at the time 0 s, and the fault clearing time is 0.12, 0.11 and 0.06 s for first-swing instability, multi-swing instability and stable condition, respectively. The generator at bus 39 is assumed as the reference machine and generator 35 serves as an example for observation. The sampling data is updated every 0.4 s, and simulation results are shown in Fig. 6. Fig. 6Open in figure viewerPowerPoint Prediction results for different stability conditions 4.4 Performance of the prediction analysis The performance indices of the rotor angle prediction with three prediction models, including accurate prediction time, standard deviation and average relative error, are compared, listed in Table 1. The prediction performance indices of the grey Verhulst self-memory prediction method with sampling points at different time periods, including the accurate prediction time, maximum relative error, mean absolute error and standard deviation, are summarised in Table 2. Table 1. Comparison of different method for prediction Accurate prediction time, s/standard deviation, s/sverage relative error, s Grey Verhulst self-memory model Grey Verhulst model Auto-regressive model IEEE-39 bus test system case 1 0.55/11.40/0.023 0.40/39.70/0.087 0.29/40.10/0.69 case 2 0.56/12.11/0.024 0.37/32.29/0.056 0.38/17.98/1.08 real power system case 3 0.58/12.92/0.098 0.40/27.40/0.219 0.36/37.66/0.59 case 4 0.53/9.68/0.044 0.41/34.67/0.128 0.34/35.93/0.27 Note : The accurate prediction time is defined as the duration when the absolute value of prediction error is <5°. Table 2. Prediction performance by grey Verhulst self-memory method with different sampling points Sampling duration Accurate time of prediction, s Maximum absolute error, deg. Mean absolute error, deg. Standard deviation, (deg. IEEE-39 bus test system case 1 0.5–0.6 s 0.55 8.1041 1.2016 2.8967 0.9–1.0 s 0.53 7.2469 2.3834 3.2155 1.4–1.5 s 0.56 8.0527 1.6931 2.8496 1.9–2.0 s 0.56 7.5483 2.2615 3.3259 case 2 0.1–0.2 s 0.56 7.6819 1.4726 2.6915 0.5–0.6 s 0.54 7.0470 1.7582 2.9653 0.9–1.0 s 0.54 6.9907 1.7410 2.8976 1.3–1.4 s 0.56 7.4163 1.6417 2.8022 NCE China power system case 1 0.9–1.0 s 0.55 6.3027 1.4492 2.5670 1.3–1.4 s 0.57 5.0002 1.5275 2.4450 1.7–1.8 s 0.55 6.4715 1.9423 2.7373 2.1–2. 5 s 0.55 6.8231 1.4873 2.3692 case 2 0.5–0.6 s 0.56 6.0436 1.9814 2.7094 0.9–1.0 s 0.55 6.4910 1.4120 2.4857 1.4–1.5 s 0.54 6.4485 1.5219 2.5453 1.9–2.0 s 0.54 7.0294 2.2915 3.1146 By comparing the prediction results of three prediction methods, it is easily observed that the grey Verhulst self-memory prediction scheme, whether in stable case or unstable case, presents the best prediction effect for both IEEE-39 bus system and the practical power system. Especially from Fig. 3b, it can be found that the stability trend of the rotor curve prediction obtained by the grey Verhulst self-memory prediction scheme is correct, while the prediction is inaccurate through the other two prediction methods. The closer the predicted points are to the measurement points, the higher the accuracy of prediction is. Moreover, the curves in Fig. 6 show that the post-fault rotor angle trajectories can be predicted well by using the proposed prediction scheme for any stability condition. The comparison of the prediction results in Table 1 shows that the grey self-memory prediction not only reduces the prediction error and improves the accuracy but also increases the accurate prediction time. According to the results obtained by the rolling prediction with sampling points at different time periods, the grey Verhulst self-memory prediction method has good stability and strong robustness. Results presented in Table 2 indicate that the accurate prediction time is about 0.55 s which is enough for transient stability control. The largest computing time which occurs in case 1, is <10−4 s, so the proposed method can satisfy real-time computing demand. 4.5 Effect of sampling data window size and data density For the implementation of the proposed prediction scheme, the size of sampling data window (T0) as well as the number of sampling points (N), i.e. the data density, needs to be determined. The appropriate size of the sampling data window and data density is critical for accurate, reliable and timely prediction. Fig. 7 shows the prediction results of different T0 and N for cases 1 and 2. In Fig. 7, for a fixed value of Δt, the larger N leads to a more accurate prediction, and for a fixed size of T0, the larger N leads to a more accurate prediction. The corresponding prediction performance indices of Fig. 7 are illustrated in Table 3. From these indices, it can be seen that a larger T0 and N results in a longer accurate prediction time. Moreover, the effect of data density is greater than the effect of the size of sampling data window for the proposed prediction scheme. Therefore, the accuracy and time length of prediction can be improved by increasing T0 and N appropriately. Fig. 7Open in figure viewerPowerPoint Prediction results of different T0 and N for cases 1 and 2 (a) Case 1, (b) Case 2 Table 3. Prediction performance index with different T0 and N for cases