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A robust damping controller for power systems using linear matrix inequalities

A robust damping controller for power systems using linear matrix inequalities
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  A Robust Damping Controller for Power Systems using Linear Matrix Inequalities Aaron F. Snyder’, Student Member Mohammed A.E. A1Ali2 Nouredine HadjSaYd’, Member Didier Georges’, Member Thibault Margotin3 Lamine Mili4, Senior Member ‘Laboratoire d’Electrotechnique de INPG I ENSlEG Saint Martin d’H&res, France Laboratoire d’ Automatique de INPG I ESISAR Saint Martin d‘Hkres, France Grenoble Grenoble Abstract: This paper presents the application of a linear matrix inequalitiy tuning approach for the design of remote feedback controllers for power systems. In particular, the determined controllers are both robust and optimal for a wide range of operating conditions, This duality is assured through the application of the proposed tuning process. The controllers presented in this paper were conceived for and tested on a 4-machine test power system with regard to their ability to damp the inter-area oscillation present in the test system and to allow an increase in tie-line power flow without a loss of stability. Keywords: Inter-Area Oscillations, Power System Stabilizers, Synchronized Phasor Measurements, Robust Control, Linear Matrix Inequalities. I. INTRODUCTION Low-frequency oscillations, related to the small-signal stability of a power system, are detrimental to the goals of maximum power transfer and power system security. Automatic Voltage Regulators (AVRs) help to improve the steady-state stability of power systems, but are not as useful for maintaining stability during transient conditions. The addition of Power System Stabilizers (PSSs) in the AVR control loop provides the means to damp these oscillations [1,2]. Before its inclusion into the control scheme, the PSS should be tuned in a manner that allows it to be robust over the expected range of operating points of the power system 131. If possible, the PSS should also be optimized for these operating points. These two conflicting design goals of robustness coinciding with optimality are often difficult to achieve simultaneuously [4]. The added AVRs and PSSs are designed to act upon local measurements such as bus voltage, generator shaft speed, or the rotor angle of the associated machine. This type of feedback control is useful for local and control mode oscillations, but may be limited for inter-area oscillations. Synchronized phasor measurements (SPMs) have been proven useful for many control applications in power systems, including state 0-7803-4403-0/98/$10.00 998 IEEE 519 3ElectricitC de France Direction des Etudes et Recherches ’?he Bradley Department of Electrical and Computer Engineering Clamart, France Blacksburg, Virginia, U.S.A. Departement Etudes desRCseaux Virginia Tech estimation, transient stability, and FACTS device control [5,6]. A Remote Feedback Controller (RFC) that includes SPMs (both local and remote) as inputs should be able to realize a more efficient alperation based on the information received in real- time from geographically separated points in the power system. However, one remaining difficulty is in determining a tuning methodology that is capable of treating these types of measurements. A new tuning methodology, called Linear Matrix Inequalities (LMIs), h(as been recently determined to provide a solution to the multi-objective controller design problem. By formulating the desired goals in the form of matrix inequalities, the numerical solution of the multi-objective problem may be achieved, even where the analytical solution is not possible. This type of solution could be considered ideal for large dynamic :systems, such as power systems, where analytical solutions are entirely out of the question. The LMI approach is also well-adapted to the power system controller problem since a posteriori determination of the optimality or robustness requires many long, time-consuming simulations. By definition, the LMI solution, if it exists, is robust and optimal. A caveat of the LMI approach is that a solution may not be possible given the operator-defined constraints. In this case, no information about the solution is available [7,8]. This paper addresses the placement of a PSS and a RFC tuned using LMIs for a two-area, four-machine test system [9] which contains a poorly damped inter-area oscillation. The test system Wiis analyzed using conventional small-signal analysis tools through the use of Eurostag [ 101 and Matlab [ 111. The “RFC-LM I” developed here is compared to a conventional AVWSS (CAPSS) to demonstrate its effectiveness. Section 2 briefly discusses the small-signal analysis tools used in this study. Section 3 provides an introduction to the LMI theory while Section 4 describes the design choices used in its application for the controller conceived demonstrated in this paper. Section 5 describes the continuation power flow method and its results as applied to the test power system while Section 6 provides a time-domain comparison of the different controller structures. 11. POWER SYSTEM MODEL ND SMALL-SIGNAL ANALYSIS A four-machine test power system (Fig. 1, following page) was used in this study. With a resemblance to the system in [SI, this system contains four generators in symmetry around a transmission network. The two loads, on Buses 3 and 13 respectively, are modeled as constant impedances. The generators were modeled following the two-axis method from  [8,9], and the governor, turbine and constant gain excitation systems were modeled in detail [8,12]. I I GEN 2 GEN 12 I Fig. 1: Test System 2.1 Power System Representation Once the models were constructed, a non-reduced Jacobian formulation is finalized in order to finish the system representation. The linearized form of the differential equations is then given in matrix form [ 131: AX = MAX NAz 0 = PAX + QAz BNRAU (1) Ay = CNRAZ where Ax is the state vector representing the deviation from the operating point, Au the input vector, Ay the output vector, z the algebraic variable vector, BNR he non-reduced input matrix, CNR he non-reduced output matrix, and (2) the non-reduced Jacobian matrix. Once the algebraic variables have been eliminated, the classical state-space representation is obtained: AX = AAx BAu Ay = CAX + DAu (3) where A is the state matrix, B the input matrix, C the output matrix and D is the feedforward matrix. 2.2 Small-Signal Analysis Once the system has been linearized about a certain operating point, traditional indices derived from small-signal theory [SI, such as participation factors for machine speed and angle, are used in conjunction with the right eigenvectors (mode shape) to identify and classify the oscillatory modes in the test system. Table 1  provides the “oscillation profile” of the three most weakly damped modes for the nominal power flow solution of the test system (recall Fig. 1). TABLE TEST SYSTEM OSCILLATION PROFILE From the right eigenvector entries (not presented here), it can be seen that Mode 3 is clearly an inter-area mode, with generators 1 and 2 swinging against generators 11 and 12. This mode is also considered to be critical due to its instability at the nominal operating point. The system can now be defined as having two areas: the sendidg end machines (1 2) and the receiving end machines (11 12). In determining where to place a supplementary controller PSS or RFC), the controllability, observability and residue magnitudes [S, 141 for the critical inter-area mode are shown in Table 2. TABLE : CRITICAL ODE ANALYSIS RESULTS The observability factors and residues are given for controller inputs of speed and electrical power, respectively. From these indices, it should be obvious that a controller located on generator 12 (highlighted values) will provide the greatest benefit with respect to the critical oscillatory mode. 111. LINEAR MATRIX NEQUALITIES The linear matrix inequality (LMI) approach to control systems is a synthesis of several robust control approaches. For certain types of well-posed control problems, the LMI formulation allows the exact numerical solution of control problems which have no analytical solution. Another advantage of the use of LMIs is the possibility of solving so-called multi- objective control problems [7]. LMI techniques are applicable to three general types of control problems: feasibility 4), inear objective minimization 5) and generalized eigenvalue minimization (6) problems. These problems are respectively expressed as follows: 1. a solution x to A(x) < 0 (4) 2. minimize c’x subject to A(x) < 0 5) 3. minimize ubject to B(x) > 0 6) A(x) < w 4 C X) < 0 7) In general, LMI constraints are of the form A(x) = A0 + x~A] ... xJ, < 0 where x = (xI, x2, ... , x,) is an unknown scalar vector (decision or optimization variables) and (A,,, AI, ... , A,) are the given symmetrical matrices. The inequality in (7) signifies that the largest real part of the eigenvalues of A(x) is negative. To formulate the LMI problem for a linear time-invariant system, the following state-space representation is used: dx dt E- = Ax t) Bu t) y(t) = Cx t) + Du t) where A, B, C, D and E are real matrices and E is invertible. This “descriptor” formulation is useful for specifying parameter-dependent systems. Recalling the traditional state- space system (3) used for power systems, it is convenient that the LMI method is developed in the same general form. The 520  complete details of the LMI approach may be found in [6] and U]. IV. LMI TUNING ETHODOLOGY This section deals with the applied design and analysis of a remote feedback controller using the LMI theory presented in Section 3. 4.1 Problem Formulation The general multi-objective H2/H, state-feedback control problem is depicted in Fig. 2. This multi-objective approach includes the benefits of robustness and tracking from H_ theory, noise insensitivity from linear quadratic Gaussian (LQG) theory and, through the use of LMIs, controllers with reasonable feedback gains and good transient response through robust pole placement [6]. I U erz2 Fig. 2: General State-Feedback Controller Problem [6] In power systems, a state-feedback approach would never be possible since access to every state in the system is rarely, if ever, assured. Therefore, the H_ ynthesis relies on the so- called standard form: X = Ax+ B,w+ B2u z = C,x + D,,w + D u (9) y = C,,x+ DYlw with where P s) and K s) are state-space realizations of the open- loop system and controller respectively, and z- and z2 denote the output associated with the H_ performance (perturbation attenuation) and the output associated with the H2 erformance (energy minimization) respectively. The multi-objective algorithm can be used to create a H_ controller, a H2 controller, a mixed-objective H2/H, controller or a combination of these with regional pole placement [6] For our study, the H, approach with regional pole placement was selected [ 15,161. In determining the actual controller via the LMI-H_ synthesis approach, the use of non-structured uncertainties was also included in the control loop. These uncertainties represent in some small manner the range of operating points over which the controller is expected to maintain functionality (closed-loop stability and performance). In this paper, the use of additive uncertainties A was considered, such that P s)=P, s)+A s), where P, s) denotes the nominal system. The inclusion of the uncertainty transfer function A is achieved via a perturbed model description similar to the one used in the linear fractional transformation [ 17,181 in the case of structured uncertainties, seen here in Fig. 3. I-- .... .._ i j j IU Yl I t ..................................... . -... .i Fig. 3: System Representation with Uncertainties [18] Under this formulation, the so-called interconnection matrix M s determined by where M = F'(P,K) = PII P12K(I - P22H)-'P21=KSy 1 1) and S, is the ouput-sensitivity ransfer matrix. Z ~~ Fig. 4: Actual System Model with Uncertainties [19] Fig. 4 indicates the actual system model with uncertainties used for this paper [19]. The system P s), the controller K s), the measured output y, the performance output z, the perturbations w he uncertainty model Wd, the control U and the control error e are all represented. Then, the problem is to minimize IIKS,wdll_ o obtain the optimal perturbation attenuation factor under a pole placement constraint. According to the well-known small-gain theorem, stability rolbustness is guaranteed when this norm is less than 1. A LMI problem of type (5) is then computed: Minimize f subject to: -I 02 <O X>O 13) clX + cl xc: B ; CdIX Dcll Y21 i nd the LMI in X, arising from the pole placement constraint, with cl denoting the matrices of the closed-loop system. It can be shown that the matrix inequality (13) may be transformed to a true LMI with an appropriate change of variables [6]. 521  4.2 Controller Determination In practice, computation of LMI problems appears to be limited to systems of about order 50 when using the classical interior-point algorithm. With a complete representation of each generator in the system as well as their respective regulation structures, the test system can include 40 to 52 states. To accelerate the LMI algorithm calculations, the test system model was reduced to a 10th order system using the optimal Hankel approximation without balancing, described in [20]. As applied to power systems, this reduction method requires the computation of the observability and controllability Grammians and uses the left and right eigenvectors in order to perform the model reduction. By working with candidate signals that make the system observable and controllable, the reduced-order model is assured to have the same global characteristics as the srcinal system [21]. A comparison of the critical oscillatory modes of the srcinal system and the modes retained by the reduced-order model are shown in Table 3. TABLE : SELECTED OSCILLATORY MODES FOR FULL AND REDUCED MODELS 5 5 4.5 Mode Full Model Reduced Model -0.9076 .9854 -0.9308 2 6.9483 - - - RFC-LMI 2 -0.5803 k 7.4152 -0.5874 + 7.4080 3 0.0073 -+ 4.0772 0.0073 r 4.0772 The Hankel reduction thus retains the three principal oscillatory modes as targeted by the input and output signals selected before the reduction. The outputs were selected to be the input signal of a supplementary controller such as a PSS or RFC, and the inputs were selected to be the injection of the controller signal into the AVRs of each machine. Through this approach, the limitations of developing a controller based on the one-machinehfinite-bus methodology are avoided since more dynamical information about the real system is retained in this reduced-order model of the full test system. Once the order reduction has been performed, the system matrices are expressed in the LMI formulation and the controller goals are determined. For our study, the H_ approach with regional pole placement was selected. The use of different pole-placement regions heavily influenced the a priori robustness of the determined controller. Indeed, a controller achieving this goal provided very weak damping for the target oscillation 5 < O.l), while the controller reported in this paper determined with a more relaxed robustness objective achieves a damping factor of < 0.5. It should be stated that this still may not be the “absolute best” controller achievable due to these constraints; howeGer, the research results have shown that the developed controller performs quite well over the investigated operating range of this small test system. An investigation of this tuning approach on a larger test system and a study of the tradeoff between robustness and performance using the LMI approach is forthcoming. Finally, being based on robust control techniques, the LMI approach yields controllers of the same order as the system investigated. For implementation issues, the 10th order controller was reduced using the same method as applied to the test system model to yield a 4th order controller. The lower- order controller was verified under closed-loop conditions using frequency domain and small-signal analysis to assure that the performance goal of closed-loop stability was maintained. V. CONTINUATION ETHOD ANALYSIS Once the modes of the system have been identified and classified, and a controller sited and tuned, the control robustness and performance may be analyzed using the continuation power flow method [22]. This method consists of increasing the power flow in the system until a conjugate pair of eigenvalues transversely crosses the imaginary axis, signifying the instability of the power system. This is called a Hopf Bifurcation Point [23]. ’‘ CAPS> ?t igration Direction i Line Impedance 100 j I O‘ -2.5 -2 -1 5 -1 -0.5 Real Axis I Fig. 5: Eigenvalues for the Studied Control Configurations Fig. 5 shows the trace of the inter-area mode for tie-line power flows from 0 to 558MW for three control structures: “AVR Only”, “CAPSS” and “RFC-LMI”. The PSS and RFC controller configurations are given in Appendix A. The arrow labeled “migration direction” shows how the critical mode moves in response to the increase in tie-line power flow. The test system loses stability with only AVRs installed at a tie-line power flow of 158MW and is clearly unstable at the nominal power flow of 381MW. The addition of the PSS or the RFC results in the ability of the system to attain maximum power transfer (tie-line power = 558MW) without instability. The CAPSS and the RFC-LMI are also robust over the same range of operating points previously examined with only AVRs in the system; therefore, the Hopf Bifurcation Point conditions could not be met due to the voltage collapse limit. Finally, the RFC- LMI input signals are assumed to be synchronized and delay- free, which is not too boastful if a system employing phasor measurement units [ 241 and a high-speed communications network [25] is considered. In an attempt to avoid the voltage collapse limit (singular Jacobian matrix, implying loss of a load flow solution), the impedance of the lines connecting the two halves of the test system was varied between 50 and 150 of its nominal value. However, while this attempt was unsuccessful under this range of impedance values, this test provided another view of the robustness of the controller for a variety of operating points. The plots of the controllers remain roughly the same for whatever impedance value and are therfore not presented in this 52 2  paper. A plot of the damping factors for the controllers is shown in Fig. 6. Here, the clear advantage of the RFC-LMI controller over the CAPSS in terms of damping is demonstrated. 2 0.4 c” 0.3 - - . - - _ . 0 . U _ 0 100 200 300 400 500 600 Tie-Line Power Plow (MW) Fig. 6: Damping Factor for the Studied Control Configurations VI. TIME-DOMAIN NALYSIS To validate the controllers under extreme operating conditions, time-domain simulations using Eurostag were performed. In order to excite the inter-area oscillation, a short circuit of lOOms is initiated and cleared on the line 3-102 (recall Fig. I). Fig. 7 shows, through the plot of the electrical power of generator 12, that the system is clearly oscillatory and unstable with only the AVRs installed. Fig. 8 provides a comparison between the CAPSS and the RFC-LMI controllers under the same conditions for the same generator. It is visible that the RFC tuned using the LMI approach provide a remarkable improvement over the srcinal system, as well as the CAPSS. The effectiveness of the proposed RFC-LMI is also clearly evident through a plot of the speed of generator 12 (Fig. 9). A view of the tie-line power oscillation (Fig. 10)  shows that the RFC-LMI achieves a higher level of damping while requiring a much smaller increase in the amount of energy transferred between the two zones of the test power system. VII. CONCLUSIONS In this paper, a Remote Feedback Controller RFC) employing synchronized phasor measurements has been designed and placed in a four-machine test power system. The exploitation of model reduction and Linear Matrix Inequality (LMI) techniques resulted in a robust, optimal control structure. The proposed controller has been shown to improve the damping of the low-frequency inter-area oscillation present in the test system in comparison with a conventional AVRJPSS controller for a wide range of operating points. VIII. ACKNOWLEDGEMENTS The authors would like to thank Electricite‘ de France and the Fede‘ration Elesa for their continued financial support of this research under CERD/R45/1 K8495BR350. The authors would also like to thank Dr. Mario Rios for his contributions towards the power system model reduction and controller parametrization methodologies used during this project. .Oo0 ]Power I Time setjd 4 1~~ 320 3iO 9 do 980 Fig. 7: Generator 12 Electrical Power, AVR Only Case Time sec) 400 800.0 302.6 305.0 507.5 310.0 312.5 Slb.0 ~~ Fig. 8: Generator 12 Electrical Power, All Controllers 1.00e II CAPSS I Time sec) 920 530 do Fig. 9: Generator 12 Speed, All Controllers Time sec) 308 ai2 310 3dO Fig. 10: Tie-Line Power, All Controllers IX. REFERENCES [l] IEEE Power Engineering Society System Oscillations Work Group. “Inter-Area Oscillations in Power Systems,” IEEE #95-TP-101, October 1994. 523
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