TUNING STRUCTURALLY CONSTRAINED ...

TUNING STRUCTURALLY CONSTRAINED STABILIZERS FOR LARGE POWER SYSTEMS VIA NONSMOOTH MULTI-DIRECTIONAL SEARCH Paulo C. Pellanda∗, Pierre Apkarian†, Nelson Martins‡ ∗

†

IME - Instituto Militar de Engenharia - Electrical Engineering Dept. Praca General Tib´ urcio, 80, Praia Vermelha 22290-270 - Rio de Janeiro, RJ, Brazil

ONERA - Control Dept. and Paul Sabatier University - Maths. Dept. 2, Avenue Edouard Belin 31055 - Toulouse, France ‡

CEPEL - Centro de Pesquisas de Energia El´etrica Avenida Hum s/n - Cidade Universit´ aria, PO 68007 20001-970 - Rio de Janeiro, RJ, Brazil Emails: [email protected], [email protected], [email protected] Abstract— This paper proposes the use of a recently available algorithm combining multi-directional search techniques with nonsmooth optimization methods to design structurally constrained controllers for large power systems. Our aim is to achieve stability as well as a guarantee on a feasible user-defined minimum damping ratio for the closed-loop dynamics. The multivariable Brazilian North-South interconnected power system model, which has more than 16 hundred open-loop states, is used to test the proposed methodology. Three decentralized lead-lag power system stabilizers are simultaneously tuned according to time-domain performance specifications. Note that controller structure constraints and the system dimension make the problem difficult to handle through available conventional control design techniques. Resumo— Este artigo prop˜ oe o uso de um algoritmo recentemente dispon´ıvel que combina t´ ecnicas de busca multi-direcional com m´ etodos de otimizac˜ ao n˜ ao-suave no projeto de controladores com restric˜ oes estruturais para sistemas de potˆ encia de grande porte. O objetivo ´ e alcancar a estabilidade com a garantia de um fator de amortecimento m´ınimo vi´ avel, predefinido pelo projetista, para a dinˆ amica de malha fechada. O modelo do sistema de potˆ encia interconectado norte-sul brasileiro, que possui mais de 1600 estados em malha aberta, ´ e utilizado para testar a metodologia proposta. Trˆ es controladores descentralizados do tipo atraso-avanco s˜ ao simultaneamente ajustados levando em conta especificac˜ oes de desempenho no dom´ınio do tempo. As restric˜ oes na estrutura do controlador e a dimens˜ ao do sistema tornam dif´ıcil a soluc˜ ao do problema pelo uso de t´ ecnicas de controle convencionais existentes. Key Words— Nonsmooth Optimization, N P -Hard Design Problems, Pattern Search Algorithm, Moving Polytope, Power System Stabilizers, Multivariable Systems, Small-Signal Stability, Large Scale Systems, Structurally Constrained Controllers, Fixed-Order Synthesis, Simultaneous Stabilization.

1

Introduction

In (Apkarian and Noll, 2006), the authors describe an algorithm combining Multi-Directional Search (MDS) (Torczon, 1991; Torczon, 1997) with nonsmooth techniques to (locally) solve several difficult synthesis problems in automatic control. More specifically, they show how to combine Direct Search (DS) techniques with nonsmooth descent steps in order to ensure convergence in the presence of nonsmoothness. Typical nonsmooth criteria appearing in control problems include the spectral abscissa, the maximum eigenvalue function and the H∞ -norm. The proposed algorithm, here named MDSN, is intended to solve several nonconvex and even N P -hard problems, for which LMI techniques or algebraic Riccati equations are impractical. Hence, the algorithm can be applied to constrained and unconstrained optimal control problems, including static and fixed-order output feedback controller design, simultaneous stabilization and mixed H2 /H∞ synthesis. As the approach avoids using Lyapunov variables, it is suit-

ably applied in the synthesis of small and medium size controllers for plants with large state dimension, constituting an alternative to LMI- or BMIbased nonlinear programming algorithms. This paper presents the first large-scale application results of the MDSN algorithm, exploring the design of multiple fixed-parameter stabilizers in interconnected power system models. The problem of obtaining decentralized lead-lag controllers that provide good damping enhancement to inter-area oscillation modes of large power systems is a typical nonconvex control problem solved in practice through conventional (smallsignal) analysis/design techniques (Yang and Feliach, 1994; Martins et al., 1999). However, the efficiency and effectiveness of this procedure is strongly dependent on the designer experience and can lead to a great amount of trial-and-error before obtaining satisfactory specifications in terms of time-domain properties. The MDSN algorithm provides a systematic framework to solve this problem and is used here to compute structurally constrained Power Systems Stabilizers (PSS) by

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maximizing the minimum closed-loop damping ratio. The MDSN algorithm is briefly described in Section 2. Section 3 presents the test system used to verify the effectiveness of the method in solving large-scale problems. In section 4, numerical results are discussed. Section 5 concludes the paper. 2

The MDSN algorithm

This section contains a brief description of the MDSN algorithm. For an in-depth discussion of MDS in the smooth case the interested reader is referred to (Torczon, 1991). The MDS algorithm requires a ‘seed’ or base point v0 and an initial simplex S in Rn with vertices v0 , v1 , . . . , vn . The vertices are then relabeled so that v0 becomes the best vertex, that is, f (v0 ) ≤ f (vi ) for i = 1, . . . , n, where f (·) : Rn → R is a C 1 function to be minimized. The initial S is chosen from one of the three different shapes shown in Figure 1. The scaled simplex is used when prior knowledge on the problem scaling is available, but right-angled and regular simplices are generally preferred in the absence of information. The algorithm updates the current simplex S into a new simplex S + by performing two types of linear transformations and driving the search towards a point having a lower function value (better point): reflection and expansion/contraction (Figure 2). First vertices v1 , . . . , vn are reflected through the current best vertex v0 to give r1 , . . . , rn . If a reflected vertex ri gives a better function value than v0 , the algorithm tries an expansion step. This is done by increasing the distance between v0 and ri for i = 1, . . . , n and yields new expansion vertices ei for i = 1, . . . , n. The current simplex S is then replaced by either S + = {v0 , r1 , . . . , rn } or S + = {v0 , e1 , . . . , en }, depending on whether the best point was among the reflection or expansion vertices. If neither reflection nor expansion provide a point better than v0 , a contraction step is performed. This is done by decreasing the distances from v0 to v1 , . . . , vn . If a point better than v0 is found among the contraction vertices c1 , . . . , cn , the simplex S is replaced by S + = {v0 , c1 , . . . , cn }. To complete one iteration (or sweep) of the algorithm, v0+ is taken to be the best vertex of S + .

Figure 1: Selection of initial simplex

Note that the MDS algorithm is guaranteed to converge to a local minimum for smooth functions but may fail at a point of nonsmoothness for general non-differentiable functions. In our tests we have observed that it is beneficial in such a situation to switch between the geometries (regular, right-angled) in order to give MDS some additional help to move on. But all these considerations are clearly heuristic, depend on the context and will need further testing. We sum up the above discussion in the following pseudo-code. MDS with nonsmooth steps (MDSN)

1.

2. 3. 4.

5. 6.

Select initial simplex S = {v0 , . . . , vn }, where v0 is the best vertex. Fix an expansion factor µ ∈ (1, ∞) and a contraction factor θ ∈ (0, 1). Stop if the relative size of S is below threshold ε Perform a reflection step ri = v0 − (vi − v0 ). Compute f (ri ). If improvement f (ri ) < f (v0 ) perform expansion step ei = (1 − µ)v0 + ri . Compute f (ei ). If improvement f (ei ) < f (v0 ) put S + = {v0 , e1 . . . , en }. Goto step 5. else put S + = {v0 , r1 . . . , rn }. Goto step 5. else perform contraction step ci = (1 + θ)v0 − θri . Compute f (ci ). Put S + = {c0 , . . . , cn }. Perform a nonsmooth step w. Compare best vertex in S + to f (w). If w is better, replace S + by new simplex containing w as a vertex. Otherwise accept S + . Go back to step 2 to loop on.

The adopted stopping criterion is based on the relative size of the current simplex: 1 max kvi − v0 k < ε , max(1, kv0 k) 1≤i≤n

(1)

where v0 is the current best vertex of S = {v0 , . . . , vn } and ε > 0 is a specified tolerance. The choice of the initial simplex S is a relatively unexplored topic. The convergence proof in (Torczon, 1991) only requires that S be nondegenerate, which means that the n + 1 points {v0 , v1 , . . . , vn } defining the simplex must span Rn . Otherwise MDS would only search over the subspace spanned by the degenerate simplex. Note that in this preliminary study we shall only use a simpler version of the MDSN algorithm

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practical controller parameters and operating conditions. The CPU time for the QR eigensolution of the 1,676-state Brazilian system matrix is about 50 seconds on a Pentium personal computer (CPU 1.86 GHz - Intel Centrino).

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Figure 2: Reflection, expansion and contraction of current simplex

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where nonsmooth steps are not systematically invoked. The use of extra nonsmooth steps to accelerate and secure convergence is currently under investigation.

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Test system

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Figure 3: Sparse structure of BIPS matrix

The large system model utilized to test the MDSN algorithm is the Brazilian Interconnected Power System (BIPS) (Martins et al., 1999; Gomes Jr. et al., 2003). The North-Northeast and SouthSoutheast subsystems were interconnected in 1999 through a 1,000 km long, series-compensated 500 kV transmission line (Figure 5). Thyristor Controlled Series Compensators (TCSC) were placed at the two ends of this line and equipped with Power Oscillation Damping (POD) controllers in order to damp the North-South (NS) mode, a low-frequency, poorly damped inter-area oscillation mode associated with this interconnection. The BIPS model for the year 1999 has about 60 GW of generating capacity, 2,370 buses, 3,401 lines/transformers, 2,519 voltage dependent loads, 123 synchronous machines, 122 excitation systems, 46 PSS, 99 speed-governors, 4 static Var compensators, 2 TCSCs equipped with POD controllers, 1 HVDC link with two bipoles. Each synchronous machine and associated controls is the aggregate model of a whole power plant. All system equipment relevant to the study were modeled in detail, yielding a system Jacobian matrix with 13,165 rows/columns and 48,532 nonzero elements (Gomes Jr. et al., 2003). The system has 1,676 state variables and more than 11,500 algebraic variables. The pattern of nonzero elements of this large descriptor system matrix is pictured in Figure 3. QR routines from standard mathematical libraries (Patel et al., 1994; Mathworks, 1984-2005), have performed quite reliably for power system state matrices of about 2,000 states, considering

Figure 4 pictures the multivariable feedback control system focusing on the decentralized PSS loops of three major plants located at the Northeast Brazilian region. The system outputs are the machine rotor speeds (ωi (t), i = 1, 2, 3) at the same power plants. The output of the PSS controllers are supplementary control signals to be applied to the machine voltage regulator terminals (viref (t)). For the application of MDSN algorithm, these three PSSs are considered disconnected together with the POD controllers of the BIPS model, yielding an open-loop system model with 1,637 states.

Figure 4: Multivariable feedback control system including PSSs at three Northeast power plants Figure 6 shows the full eigensolution of the open-loop BIPS model. It is seen that the NS mode becomes unstable (0.1089 ± 1.2052j) in the

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Figure 5: Brazilian North-South interconnected power system - geographical location

15

ζ = 0.0538

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NS mode

Imaginary Axis

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(one or a set of eigenvalues and associated eigenvectors at a time). In spite of the fact that some of these methods allow dealing with multivariable systems in the analysis step, the design is normally performed by tuning one stabilizer loop at a time, which may require a number of redesigns. In the next section, we show that the design of multivariable structurally constrained controllers for large systems can be performed in a more direct way while considering time-domain specifications.

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ζ = 0.0538 −15 −5

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−2 Real Axis

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Figure 6: QR eigensolution for the open-loop BIPS model (1,637 states)

absence of the three PSSs depicted in Figure 4 and the two POD controllers. The objective is to stabilize the NS mode only through improved designs of these three PSSs, so that the POD controllers would not be needed in the small signal stability context. Adequate inter-area oscillation damping is achieved in practice by combining classical frequency response techniques with efficient methods for small signal stability analysis. Most of the available methods are based on the use of a descriptor state-space form, exploiting the Jacobian matrix sparsity (See Figure 3), and rely on iterative steps to obtain the dominant eigenstructure

Results

This section presents numerical results from the application of MDSN algorithm (Section 2) to design the three decentralized PSSs in Figure 4. The control synthesis objective is to maximize the minimum closed-loop damping ratio (ζ) by tuning the controller parameters, i.e., the gains Ki and the time constants T1i , T2i , T3i , T4i , i = 1, 2, 3. This can be cast as minimizing the function f (K) = max {cos (arg(λi )) : λi ∈ spec(A(K))} 1≤i≤ne

(2) where A ∈ Rne ×ne is the closed-loop dynamic matrix and K is the vector of controller parameters. The results in Table 1 were obtained by setting the following parameters and minimizing f (K) through the MDSN algorithm:

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• expansion factor: µ = 2; • contraction factor: θ = 0.5; • stop criterion: ε = 1e − 5;

• initial simplex shape: right-angled; 8

• initial simplex edge length: 5;

ζ = 0.0538

• initial controller parameters (seed 1): randomly generated in the interval [0, 30].

Open loop Iterations Closed loop

uncontrollable mode

7 6

Imaginary Axis

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Table 1: MDSN algorithm iterations (seed 1) Simplex transformations − 1 expansion 2 contractions 1 expansion 1 reflection 3 contractions 1 expansion 1 expansion 2 contractions 1 expansion 1 expansion 1 reflection 1 expansion 2 contractions 1 contraction 1 expansion 1 expansion 1 reflection 1 expansion 1 reflection 1 reflection 1 contraction 1 expansion

ζ -0.105 -0.079 -0.076 -0.073 -0.052 -0.052 -0.044 -0.027 -0.022 -0.018 -0.010 -0.003 0.005 0.010 0.010 0.012 0.016 0.019 0.035 0.042 0.049 0.049 0.054

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NS mode

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Figure 7: Effectiveness of the MDSN algorithm in increasing the NS mode damping ratio

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ζ = 0.054

10 NS mode

5 Imaginary Axis

Iteration 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

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−5

−10 ζ = 0.054 −15 −5

Figure 7 shows the MDSN algorithm is effective in increasing the NS mode damping ratio. The MDSN code was stopped after 23 iterations, when ζ = 0.054 was obtained (see Table 1). This barrier (lower bound) is explained by the existence of an uncontrollable mode (λ = −0.372±6.90j) for which f (K) = 0.054. Figure 8 shows that the full set of closed-loop eigenvalues are kept in a welldamped region (ζ > 0.054). The CPU time for this solution was about 23 hours and 42 minutes on a Pentium personal computer (CPU 1.86 GHz - Intel Centrino). The large number of full QR eigensolutions required in each iteration responds for the largest part of the computational cost of the MDSN algorithm. However, the number of iterations can be kept at acceptable levels if the initial controller parameters are suitably chosen. For this test system, when the parameters were randomly generated in the interval [0, 1] (seed 2) instead of the interval [0, 30] (seed 1), only three iterations were necessary to achieve the minimum damping ratio solution (Table 2), requiring only about 190 minutes of CPU time. Final controller parameter obtained by running the algorithm with seed 2 are given in Table 3.

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Figure 8: QR eigensolution for the closed-loop BIPS model (1,646 states)

Table 2: MDSN algorithm iterations (seed 2) Iteration 1 2 3

Simplex transformations − 1 contraction, 1 reflection 1 expansion

ζ -0.032 -0.029 0.054

Table 3: Final controller parameters P SS1 K1 = 7.9186 T11 = 0.8462 T21 = 5.5252 T31 = 0.2026 T41 = 0.6721

P SS2 K2 = 0.8381 T12 = 0.0196 T22 = 0.6813 T32 = 0.3795 T42 = 0.8318

P SS3 K3 = 0.5028 T13 = 0.7095 T23 = 0.4289 T33 = 0.3046 T43 = 0.1897

Note that the search was kept free over the entire controller parameter space, but it can easily be constrained to a desired domain by imposing

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an infinite barrier on the objective function: ½ max {cos (arg(λi ))} if K is feasible f (K) = +∞ otherwise 5

Conclusions

This paper shows the MDSN algorithm is a valuable method for tuning structurally constrained stabilizers for large-scale power systems. The existence of a proof of convergence (Torczon, 1997; Apkarian and Noll, 2006) represents an advantage of MDSN method over genetic algorithms. A comparison between the two methods will be presented in a later publication. The number of free PSS parameters in the power system application of this paper is excessive from a practical view point. Constraints such as specified center frequency for phase lead blocks or equal parameters for the two phase lead blocks would drastically reduce the search process, while turning the results more engineering oriented. These are left for future investigations. The use of partial eigensolution methods focused on a specific multivariable system dominant dynamics (Rommes and Martins, 2006) will considerably reduce the computation time of the MDSN method.

Torczon, V. (1991). On the Convergence of the Multidirectional Search Algorithm, SIAM J. on Control and Optimization 1(1): 123–145. Torczon, V. (1997). On the Convergence of Pattern Search Algorithms, SIAM J. on Control and Optimization 7(1): 1–25. Yang, X. and Feliach, A. (1994). Stabilization of Inter-Area Oscillation Modes Through Excitation Systems, IEEE Trans. Power Syst. 9(1): 494–500.

References Apkarian, P. and Noll, D. (2006). Controller Design via Nonsmooth Multi-Directional Search, SIAM J. on Control and Optimization 44(6): 1923–1949. Gomes Jr., S., Martins, N. and Portela, C. M. J. (2003). Computing Small-Signal Stability Boundaries for Large-Scale Power Systems, IEEE Trans. Power Syst. 18: 747–752. Martins, N., Barbosa, A. A., Ferraz, J. C. R., Santos, M. G., Bergamo, A. L. B., Yung, C. S., Oliveira, V. R. and Macedo, N. J. P. (1999). Retuning Stabilizers for the NorthSouth Brazilian Interconection, IEEE Power Engineering Society Summer Meeting, Alberta, pp. 58–67. Mathworks (1984-2005). MATLAB Mathematics - Version 7, Mathworks, Inc. Patel, R. V., Laub, A. J. and Van Dooren, P. M. (1994). Numerical Linear Algebra Techniques for Systems and Control, IEEE Press, New York, NY. Rommes, J. and Martins, N. (2006). Efficient Computation of Multivariable Transfer Function Dominant Poles Using Subspace Acceleration, University of Utrecht, UU Preprint 1344, accepted for publication in IEEE Trans. PWRS p. 16. 430 of 430