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Wide‐area damping control for inter‐area oscillations using inverse filtering technique

2015; Institution of Engineering and Technology; Volume: 9; Issue: 13 Linguagem: Inglês

10.1049/iet-gtd.2015.0027

1751-8695

R. Goldoost-Soloot, Yateendra Mishra, Gerard Ledwich,

Power Systems Fault Detection

IET Generation, Transmission & DistributionVolume 9, Issue 13 p. 1534-1543 Research ArticleFree Access Wide-area damping control for inter-area oscillations using inverse filtering technique Reza Goldoost-Soloot, Reza Goldoost-Soloot School of Electrical, Electronic and Computer Engineering, Queensland University of Technology, Brisbane, QLD, AustraliaSearch for more papers by this authorYateendra Mishra, Corresponding Author Yateendra Mishra Yateendra.mishra@qut.edu.au School of Electrical, Electronic and Computer Engineering, Queensland University of Technology, Brisbane, QLD, AustraliaSearch for more papers by this authorGerard Ledwich, Gerard Ledwich School of Electrical, Electronic and Computer Engineering, Queensland University of Technology, Brisbane, QLD, AustraliaSearch for more papers by this author Reza Goldoost-Soloot, Reza Goldoost-Soloot School of Electrical, Electronic and Computer Engineering, Queensland University of Technology, Brisbane, QLD, AustraliaSearch for more papers by this authorYateendra Mishra, Corresponding Author Yateendra Mishra Yateendra.mishra@qut.edu.au School of Electrical, Electronic and Computer Engineering, Queensland University of Technology, Brisbane, QLD, AustraliaSearch for more papers by this authorGerard Ledwich, Gerard Ledwich School of Electrical, Electronic and Computer Engineering, Queensland University of Technology, Brisbane, QLD, AustraliaSearch for more papers by this author First published: 01 October 2015 https://doi.org/10.1049/iet-gtd.2015.0027Citations: 5AboutSectionsPDF 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 this study, a non-linear excitation controller using inverse filtering is proposed to damp inter-area oscillations. The proposed controller is based on determining generator flux value for the next sampling time which is obtained by maximising reduction rate of kinetic energy of the system after the fault. The desired flux for the next time interval is obtained using wide-area measurements and the equivalent area rotor angles and velocities are predicted using a non-linear Kalman filter. A supplementary control input for the excitation system, using inverse filtering approach, to track the desired flux is implemented. The inverse filtering approach ensures that the non-linearity introduced because of saturation is well compensated. The efficacy of the proposed controller with and without communication time delay is evaluated on different IEEE benchmark systems including Kundur's two area, Western System Coordinating Council three-area and 16-machine, 68-bus test systems. Nomenclature E′qi quadrature component of internal voltage of generator δi rotor angle ωi rotor angular velocity Efdi field voltage VRi exciter input Xij tie-line reactance Xd direct-axis synchronous reactances X′d direct-axis transient reactances Xq quadrature-axis reactance T′do open-circuit d-axis transient time constant ws synchronous speed δ rotor angle J rotor inertia constant D damping coefficient Pm mechanical power input of generator Id armature current of d-axis Iq armature current of q-axis Us supplementary control input KT excitation controller gain fKE kinetic energy function KA automatic voltage regulator gain Ta exciter time constant 1 Introduction Inter-area oscillations are key issues for interconnected power systems that if not addressed properly can deteriorate the power system stability and decrease available transfer capability of transmission lines between interconnected power areas [1, 2]. The deregulated electricity markets have encouraged power systems to operate closer to their stability limits, and this has made power systems more vulnerable to low-damped low-frequency inter-area oscillatory modes [3]. To tackle this issue, various controllers have been designed using a variety of control approaches; utilising flexible ac transmission systems such as static var compensators [4–6], controllable series devices (unified power flow controller, controllable series capacitor) [7–11], high-voltage dc link [12, 13] and power system stabiliser (PSS) [14, 15]. These provide some of the options for damping low-frequency electromechanical modes. Considering the installation cost and the simplification of implementing the mentioned devices for the purpose of enhancing the damping of inter-area modes, PSS for excitation system is used. However, most of the early methods applied for improving dynamic performance rely on system linearisation at an operating point, limiting their robustness [14]. Moreover, the effectiveness of conventional PSSs diminishes when the inter-area modes are not observable and controllable for a local signal [16, 17]. The enhancement of dynamic stability and voltage regulation simultaneously using conventional PSS and automatic voltage regulator (AVR) can be challenging given the conflicting objective of these controllers. Recently, advanced methods such as multi-band PSS [18], non-linear PSS [19], wide-area PSS [20] and coordinated PSS [21] have been introduced to improve damping while maintaining sufficient voltage regulation. The rapid deployment of wide-area measurement system (WAMS) using phasor measurement units (PMUs) [22] has been effective in designing wide-area coordinated PSS. Maintaining system stability and avoiding voltage collapse after large disturbance, a high-gain AVR with a small time constant is chosen. However, assigning high gains for exciters tends to degrade the damping, which can contribute to instability [23]. To solve this problem, several techniques for controlling excitation system have been investigated in the literature. In [24], a fuzzy logic excitation controller is proposed to ameliorate excitation control by an online trade-off between PSS and AVR output values. On the basis of online measurements and fuzzy logic rules, an automatic weighted supplementary control input is injected to the exciter. In [25], an iterative linear matrix inequality is used to calculate an optimal static gain via an H∞ static output feedback control is proposed. The resultant optimal static gain vector is applied in parallel with the conventional control devices. The deviation of generator terminal voltage, active power and generator velocity are used as the inputs of optimal gain vector and require linearisation about an operating point. Although applied in a single machine system, H∞ control is hard to realise for large power systems. Lyapunov energy-based controller is another category of control approaches that has been applied for controlling excitation system. The Lyapunov theorem is also used to design decentralised excitation controller for the multi-machine power systems using local measurements [26]. Lyapunov function for this method is built as a quadratic expression of variations of rotor speed, active electrical power and generator terminal voltage from their reference values. The main advantage of this methodology with respect to previous Lyapunov-based methods [27, 28] is the ability to maintain voltage regulation while securing power system stability. Direct feedback linearisation has also been proposed by using either switching strategies [29] or membership function to weight local controllers [30]. The prior knowledge of the fault period and the efficiency of the membership function are the common drawback for actual implementation. Excitation system control using model predictive control is proposed in [31, 32]. Despite its robustness to enhance dynamic stability, the necessity to compute optimal control input by solving a dynamic optimisation problem online for each sampling period, leading to heavy computational burden, makes it harder to deploy in real-world. This issue has been addressed in [33] by proposing non-linear optimal predictive control for excitation system. This technique does not require online optimisation and hence reduces computational effort significantly. In this paper, the proposed method adds a supplementary control input on the exciter and the rest of the excitation system remains unchanged. This controller first determines the desirable generator flux for wide-area control (WAC) and then applies a special controller to track that flux. The contributions of this paper lie mainly in the extension of an initial work on inverse filtering for power systems [34] to multi-machine test system and employing (WAMS) to obtain a general expression for the desired flux by using equivalent angle and frequency of each area. Whereas in [34], future values for required variables are obtained by running the time-domain simulation in forward time, in this paper the WAC uses a non-linear Kalman filter [35]. From this the equivalent rotor angle and frequency for each coherent area is estimated. Having equivalent data for each area, future system states are calculated and from this required future fluxes are computed. Moreover, communication time delay which exists in WAMS is also considered to show the performance of the controller in presence of latency for remote signals. Power data concentrators are used to collect data obtained by dispersed PMUs throughout the power system. The PMU data are used for identifying area equivalent parameters and estimating angle and frequency of each area. This part of the proposed controller is carried out in a centralised WAC unit which gives desired flux for each area that would maximise the reduction rate of the kinetic energy of any disturbance in the power system. Then, by using inverse filtering method which is explained in detail in Section 2, a supplementary control input is generated for generators in each area to ensure the required changes in the flux of the area. Therefore this part of the proposed controller is related to flux tracking and is implemented on the excitation system of each generator. The rest of this paper is organised as follows: detailed steps of the proposed controller are discussed in Section 2. Section 3 describes the case study for three-machine nine bus, two-area test benchmarks, 16-machine, 68-bus test system and results followed by conclusions in Section 4. 2 Inverse filtering approach for excitation systems The inverse filtering approach is used to obtain the input of the system to set the desired value for the output. The concept can be seen in a simple form of deconvolution [36] and this paper shows how to use it in power systems. The desired flux for the generator is calculated using the kinetic energy of the system and then the supplementary control input to the exciter is designed to track this flux. The desired flux is obtained without using any linearisation technique and thus the result is independent of operating conditions. The overall concept of inverse filtering can be summarised using Fig. 1a. The transfer function relating generator flux to the field voltage is the first stage of inverse filtering. The effect of flux saturation and other algebraic variables are accounted in the next stage. The last stage of inverse filtering is to obtain the supplementary control input by using field voltage from previous stage. Fig. 1Open in figure viewerPowerPoint Overall concept of inverse filteringa Inverse filtering for excitation systemsb Dynamics of synchronous generator and excitation system The dynamic model of an n-machine power system using a flux-decay model for generators and first-order exciter can be described by the following differential and given algebraic equations in [37] (1) (2) (3) (4) (5)The flux saturation, which is the non-linearity of the output is also considered in a saturation function (SE) proposed by [38] as SE(E′q) = 1 + b/a(E′q)p−1 and the suggested values for parameters a and b used by [39] are a = 0.95; b = 0.051; and p = 8.727. The dynamics of a generator and a simple exciter can be shown using block diagram as in Fig. 1b. The Us is the supplementary control input which is added to the excitation system. The following inverse filtering procedure, (6)–(8), leads to the calculation of the supplementary control input (6) (7) (8)A larger time constant, TR, is considered to avoid the interference of terminal voltage deviations to the performance of stabiliser just after fault occurrence. As a result, variation of VR is neglected for a short period after disturbance. To obtain the control signal, (6)–(8) is discretised using sampling frequency significantly larger than two times of the inter-area frequency [40] and hence 20 Hz sampling signal is used in this paper. An autoregressive (AR) model is obtained using 'invfreqz' function in MATLAB. The control signal is calculated using the inverse filtering in (9)–(11). The effect of saturation is considered in (7) and (10). Therefore Us for the next sampling time is calculated using inverse filtering as in (11) after performing discretisation and obtaining AR model. The acquired AR model is invertible as a moving average process and can provide control signal using inverse filtering method and predicted values. This sampling frequency ensures maintaining similar frequency response with continuous transfer function in the range of interest. On the basis of the order of AR model, the number of required future values for flux can be specified. The bode diagram of discretised and continuous transfer function needs to be identical in the vicinity of the inter-area mode (9) (10) (11)where a0, …, an and b0, …, bn, are coefficients which are obtained after carrying out discretisation and inverse filtering and subscript k represents the time step. Using the non-linear Kalman filter, estimated parameters for each coherent area such as rotor angle, generator speed, reactance of tie-lines between coherent areas are obtained, and by updating Kaman filter the required values for next time steps will be achieved. The obtained actual flux after applying Us in (11) on the exciter tracks the desired flux. When a power system is in steady operating condition and the generator velocity is measured by using centre of inertia (COI) reference, the kinetic energy is zero. After the system is subjected to a fault, generator velocities start oscillating, and therefore the overall envelope of kinetic energy of the system in COI reference increases. The objective for this WAC is maximising the reduction rate of kinetic energy of the power system. The basis of constructing this function for a multi-machine power system is explained in [41]. The advantage of using kinetic energy for our purpose is that it can be expressed as a function of aggregated generator flux which is treated as a control variable. Decreasing kinetic energy of a system after fault occurrence enhances the transient stability of a power system. In doing so, the derivative of kinetic energy should be made as negative as possible (subject to saturation constraints). Our wide-area controller operates on an aggregate classical generator model for an area. The tracking problem of each generator uses the higher-order model as in (1)–(5). After substituting (3) and (14) in (13), derivative of kinetic energy is (15) and by differentiating it with respect to E′qi, the desired function of flux with respect to system angles can be achieved. The COI reference frame is used for computing the variations in the desired flux, it can be concluded that in steady operating conditions, this value is zero using (16) (12) (13) (14) (15) (16)where m is the number of coherent areas, and are angle and velocity differences between areas i and j, respectively, and KT is a user designed gain. The value of KT should not be too high creating bang–bang effect leading to undesirable transients after the first/second swing. Moreover, the smaller value of KT may not be effective in damping the inter-area modes. It is found that the acceptable range of value of KT is between 0.1 and 1 and depends on the time-domain simulations under severe-most credible contingency. The sensitivity analysis using different values of KT is performed and its effect on the damping of the inter-area oscillations is observed. The next section is devoted to a brief explanation of the method used in [35] to obtain estimated rotor angle and frequency of each aggregated area. The controller aims to improve the damping of inter-area modes by using a reduced system that includes these inter-area modes and exclude local modes. A non-linear Kalman filter is used to estimate the equivalent states for each area by combining the PMU measurements from each area and significantly decreases the presence of local modes in the estimated angles and frequencies. The parameters of Kalman estimator are obtained based on the identification of the reduced model as presented in [35] (17)where is the estimation of equivalent area states, is the non-linear function representing the reduced model, L is the gain of the non-linear Kalman filter, YPMU are the measurements observed from PMUs across the system and CA represents the relationship between reduced states and measurements. Therefore the steps required to achieve the wide-area damping control using inverse filtering can be summarised as follows: Step 1: Determine the coherent generators and obtain a reduced model by representing each area by an equivalent machine and then estimating equivalent area angles and frequencies using non-linear Kalman filter. Step 2: Calculate kinetic energy of the whole system which is expressible as a function of rotor angle, generator speed and flux. Step 3: Obtain the desired flux by differentiating the derivative of kinetic with respect to generator flux to minimise ḟKE. Step 4: Choose an appropriate system gain, KT, for the desired flux. Step 5: On the basis of model and parameters of generator and exciter, an AR model is derived for each generator. This yields an MA model for the inverse filtering which incorporate the inverse of the saturation function. Step 6: Update Kalman filter to calculate desired flux values for next sampling steps and the number of steps depends on the order of the AR model. 3 Simulations results The performance of the proposed excitation controller is tested on IEEE benchmark systems for power system stability studies. Western System Coordinating Council (WSCC), three-machine, nine bus test system (Fig. 2a), the two-area four-machine Kundur system (Fig. 4a) and 16-machine 68-bus test system (Fig. 6a) are the test systems used. The first test system shows the performance of the proposed controller while having more than one inter-area frequency and the second and third test systems show its effectiveness when the test system is expressed as coherent areas. Full order time-domain simulation is employed to validate the effectiveness of the proposed excitation controller. Fig. 2Open in figure viewerPowerPoint WSCC, three-machine and nine bus test systema WSCC test system with wide-area excitation controllerb Improvement in damping because of the proposed excitation controller for WSCC test system for G1, G2 and G3 3.1 WSCC test system A simple first-order exciter model is assumed for the WSCC test system [37]. The state perturbation method is used to obtain two low-damped inter-area modes (Table 1). The equivalent area parameters of each area are obtained using PMU data and a non-linear Kalman filter is used to estimate rotor angle and velocity. The next step is expressing the kinetic energy of the whole system and computing the desired flux for the generators in each area. The desired fluxes for generators are calculated using (10) (18) (19) (20)where KT is chosen as 0.3. After obtaining AR model and inverse filtering, the supplementary control input for a generator, say G1, is expressed as given in (23) (21) (22) (23) Table 1. Inter-area modes of three-area test system Mode index F, Hz ξ 1 1.874 0.0347 2 1.354 0.0354 The order of AR model for the last step is two, but to obtain , the desired fluxes need to be calculated for three steps ahead. It is notable that Id is a function of rotor angles, velocities and tie-line reactance [42], which are available through PMU data and Kalman filter. In practice, a limiter should be imposed on the controller to avoid excessive values for control signals as given in (24) (24)where US_min and US_max are lower and upper limits which are assumed as −0.15 and +0.15, respectively. Similarly, the supplementary control inputs for G2 and G3 are computed. The system is subjected to a three-phase short-circuit self-clearing fault for 50 ms at bus 5 at 1 s and the effect of the controller on damping inter-area modes is plotted in Fig. 2b. The performance of WAC inverse filtering is compared with conventional PSS [43]. Although, the PSS settings offer good performance, robustness and generality in the range of low-frequency modes [18], the proposed controller performs better. Generator flux tracking is well achieved, as shown in Fig. 3a, through WAC inverse filtering. The supplementary control input variations are limited to 0.15 pu as shown in Fig. 3b. It can also be observed from Fig. 3c that the generator bus voltages maintain their reference voltage. Fig. 3Open in figure viewerPowerPoint Generator flux trackinga Flux trackingb Supplementary control inputsc Generator bus voltagesd Velocity of generator 3 for several values of gain KT The performance of the proposed controller for different values of gain, KT is evaluated for the given system. The gain of KT = 0.05 is too low as its effect on the performance of the controller is negligible, whereas KT = 0.5, 0.8 results in satisfactory performance. The gain of 2 leads to poor performance. The variations of the velocity of generator 3 for different values of KT are illustrated in Fig. 3d. It was found that the performance of the controller is satisfactory for any gains between 0.1 and 1, which is used for the next two test power systems. 3.2 Two-area, four-machine Kundur's test system The state perturbation method is used to calculate system eigenvalues. Two local modes and one inter-area mode are shown in Table 2 [44]. Using PMU data and Kalman filter, the estimated equivalent angle and frequency of each coherent group is achieved as shown in Fig. 4a. The desired flux for each area is calculated using (25) and a supplementary control input for each generator is obtained using inverse filtering as (26)–(28). The value of the constant KT is chosen as 0.1 (25) (26) (27) (28) Table 2. Inter-area mode of two-area test system Mode index F, Hz ξ 1 0.5219 0.042 The system is subjected to a ten cycles three-phase short-circuit self-clearing fault at bus 7. The performance of the controller to track the desired flux is illustrated in Fig. 4b and shows that the generators are able to track the flux. The effect of the proposed excitation controller on the velocity (COI) is compared with conventional PSS and AVR and is shown in Fig. 4c. It is evident that the controller outperforms the conventional PSS. In addition, generator bus voltages are maintained in acceptable range after a severe fault occurs in the test system. Variations of generator bus voltages are illustrated in Fig. 5a. To avoid chattering of the controller, values <0.005 for control signal are neglected. Fig. 5b shows the injected control signals to G1 and G3, which produced desired flux for area one and two. The control efforts of excitation controller and PSS are illustrated in Fig. 5c. Fig. 4Open in figure viewerPowerPoint Assessing the proposed controller for two area test systema Single line diagram of the two-area test system with the wide-area excitation controllerb Flux tracking of G1 and G3c Effect of the proposed excitation controller for G1 and G3 on velocity Fig. 5Open in figure viewerPowerPoint Variations of generator bus voltagesa Generator bus voltages because of a ten cycle three-phase short-circuit fault with excitation controllerb Supplementary control inputc PSS output injected to the excitation system of G1 and G3 3.3 16-machine 68-bus test system New York–New England New York–New England (NY–NE) test system consists of five coherent areas which are separated by dashed lines in Fig. 6a [45]. This test system includes four inter-area modes which their frequency and damping are presented in Table 3 while PSSs are in service. To carry out the identification task and to use the outcome in the Kalman filter, several PMUs should be installed in this test system [35]. The PMUs are installed at generator buses G1, G3, G5, G7, G9 from area 1, G10 and G12 from area 2 and G14, G15 and G16 from areas 3–5. Installed PMUs at selected generators provide sufficient observation of the whole test system. The PMUs installed on generator buses record angle of each generator which is used to identify the tie-line reactances, damping and inertia of each area and finally lead into having COI velocities and angles of the coherent areas by using the proposed Kalman filter in [35]. These are the variables that are required to obtain desired flux for each area. Having desired fluxes, supplementary control input for each generator can be calculated. The 16-generator NY–NE system has four lumped generators which are the equivalent for a set of generators, and as a result these generators are not equipped with exciters, since it is not practical to have exciters for such big size generators. The rest of generators which are G1–G12 are using static exciters which have KA = 100 and Ta = 0.01 as used in [45]. Therefore just generators in areas 1 and 2 are benefiting from excitation controller. This test system has also 12 PSSs on generators G1–G12 which without them has unstable eigenvalues. For demonstrating the impact of non-linear excitation controller, all PSSs are out of service and replaced with our proposed excitation controller. Desired fluxes for generators in area 1 and area 2 are calculated using (29) and (30). Since area 1 is just connected to area 2, it just has one term. However, area 2 is connected to two other areas (area 3 and area 5). The gain KT is assumed equal to 1 to provide required supplementary control inputs for generators and also avoid excessive saturation for control action. The procedure of calculating desired flux and supplementary control input is similar to the two cases described in sections A and B (29) (30) Table 3. Inter-area mode of two-area test system Mode index F, Hz ξ 1 0.38 0.0898 2 0.49 0.1 3 0.67 0.0294 4 0.79 0.0869 The test system is subjected to 50 ms three-phase short-circuit self-clearing fault at bus 8 in area 1 which is connected by a tie-line to bus 9 in area 2. This fault excites the inter-area modes, especially the one which its most participated states are velocity and rotor angles of areas 1 and 2. The COI velocities of areas 1 and 2 show that at this inter-area mode frequency, these two areas are oscillating against each other. All of excitation controllers are installed on generators in these two areas and the effect of them could be more obvious on these two COI velocities. The proposed controller has damped inter-area oscillations as shown in Fig. 6b which illustrates the variations of generator velocities with and without controllers for G1–G16. To show the effectivity of the proposed controller more clearly, the COI velocities for all coherent areas for both cases including with and without proposed excitation controllers and also while equipping the system with PSS are illustrated in Fig. 7. To ensure that inverse filtering has performed its duty perfectly, flux tracking and control input for generator 1 are shown in Figs. 8a and b, respectively. Fig. 6Open in figure viewerPowerPoint Effect of the inverse filtering controller on NY–NE test systema Single line diagram 16-generator 68-bus system (NY–NE)b Generator velocity variations without controller and with controller Fig. 7Open in figure viewerPowerPoint Area velocity variationsa Without controllerb With proposed controllerc With PSS Fig. 8Open in figure viewerPowerPoint Flux tracking and control input for G1 and effect of time delaya Flux trackingb Supplementary control inputc COI velocity variations for various time delays for area 1 and area 2 3.4 Impact of time delay To make sure that the performance of the controller remains effective in the range of time delay for the data received from PMUs, it is assumed that the excitation controller receives data with a latency of 100 ms which is a normal delay for WAMS [46]. This amount of delayed input signal for excitation controller does not deteriorate the satisfactory performance of the proposed excitation controller and it still damps low-frequency inter-area oscillations well up to 200 ms. However, increasing latency to 300 ms causes under-damped low-frequency oscillations and time delays more than 320 ms destabilise the power system. The variations of velocities of areas 1 and 2 are illustrated in Fig. 8c for several values of time delay. It can be concluded that the proposed excitation controller is robust to the standard range of time delay associated to WAMS. 4 Conclusions In this paper, a new method for controlling excitation system is proposed to damp inter-area oscillations and regulate bus voltages. Wide-area measurements using PMUs installed in each area are used and a non-linear Kalman filter is used to determine the equivalent area parameters. The desired generator flux for each area is calculated using the kinetic energy function approach. Having the desired generator fluxes for the next sampling times, the supplementary control input for each generator is obtained using inverse filtering. The efficacy of the proposed method is verified on three well-known test systems. The proposed excitation controller provides significant improvement in damping inter-area modes while maintaining the generator bus voltages within limits. The communication time delay of remote signals is also incorporated and it is found that the proposed controller is effective in damping the inter-area oscillations for up to 200 ms of time delay. 5 References 1Klein, M., Rogers, G., Kundur, G.: 'A fundamental study of inter-area oscillations in power systems', IEEE Trans. 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