SYNTHESIS OF LOW ORDER MULTI-OBJECTIVE CONTROLLERS FOR A VSC HVDC TERMINAL USING LMIs

Transcription

1 SYNTHESIS OF LOW ORDER MULTI-OBJECTIVE CONTROLLERS FOR A VSC HVDC TERMINAL USING LMIs Martyn Durrant, Herbert Werner, Keith Abbott Control Institute, TUHH, Hamburg Germany; m.durrant@tu-harburg.de; Fax: AREVA T&D UK Ltd. Power Electronic Systems,Stafford England Keywords: Power System Control, LMIs, Robustness Abstract In this paper an approach to designing low order controllers for a VSC HVDC terminal which are robust over a range of operating points is presented. An uncertainty region is used to characterise the operating range as described by a non-linear model of the terminal. An LMI based formulation is then used to synthesize low order controllers which maximize the size of the structured uncertainty region within which closed loop stability is maintained. The method uses an iterative scheme with a feasible low order controller as the initialising controller. The improvement provided by the iterative scheme and the performance of the resulting controllers is compared for different initialising controllers using the non-linear model. Introduction VSC HVDC transmission (Voltage Source Converter High Voltage Direct Current transmission) is an electrical transmission technology that has received considerable attention in recent years due to the development of high power transistor technology [] [4]. A VSC HVDC transmission system connects two AC networks using two AC-DC terminals and a DC link. In such systems it is necessary to control power flow between the terminals under different AC network operating conditions, and while the network moves between these conditions, while at the same time controlling the terminal AC and DC voltages. This is a robust control problem which becomes increasingly challenging as the impedance of the network increases [3]. With some conservatism, LMI based multi-objective output controller synthesis [7] allows full-order controllers to be designed which maximise the range of uncertain parameters over which stability is maintained while maintaining performance requirements for a nominal model. These are achieved respectively by minimisation of a H norm and fulfillment of closed loop performance constraints such as H 2 performance or pole positions. A potentially attractive feature of the design method is that the controllers designed are quadratically stable across the stability region found, so the controller can withstand arbitrarily fast changes in operating conditions within this region. In [2] this method was adapted to allow the structure of the uncertainty to be exploited using a D-K type iterative scheme. This iterative scheme was then used to design a full order power and voltage controller for one terminal of a VSC HVDC transmission system. This paper describes an alternative adaption of this method that generates low order controllers in addition to exploiting the structure of the uncertainty, and its application to the same problem. The motivation for this is that (high) full order controllers may not be acceptable in practical applications. The paper is organised as follows. The analytical model of the terminal is described in Section 2. The control objectives are outlined in Section 3 and a controller and uncertainty structure to achieve these is described in Section 4. The controller synthesis is formally formulated and an iterative procedure to solve the resulting non-convex optimisation problem as a series of LMIs are described in Section 5. In the remaining sections methods for generating initialising controllers are discussed, and the performance and robustness achieved with different initialising controllers are described. 2 VSC HVDC Model As is common for 3 phase power circuits [5], physical quantities in this paper are represented in the dimensionless per unit (pu) form, and phasor quantities are represented in d and q (direct and quadrature) units relative to a rotating reference frame (RRF). A non-linear model of VSC HVDC terminal attached to an AC network has been developed and successfully validated against a rigorous model including realistic converter behaviour [3]. This model is represented schematically in Figure. The converter itself is represented by a voltage source, the associated transformer is represented by R c and L c and the filter used for suppression of converter switching harmonics is represented by C f. The AC network is represented by a fixed voltage source v n and an

2 impedance that may vary, consisting of L n and R n. The RRF used is the voltage phasor of the converter terminal voltage, which is measured using a phase locked loop. The control inputs used are v cld and v clq, the voltages across the converter impedance as measured in the RRF. The dynamic behaviour between these inputs and the power (P) and converter terminal voltage magnitude ( v l ) depends on the operating point defined by R c, L c, P and v l. 3 Control Objectives The controller objectives for the system are: i) To control power P and converter terminal voltage v l using inputs v cld and v clq with no steady state error between P and v l setpoints and measurements. ii) To maintain closed loop stability at 52 representative operating points as detailed in Table and while moving between them. The operating points are defined by power flow and AC system reactance. iii) To minimise the response time to power setpoint steps for a given controller order. iv) During power setpoint steps of 0.5pu, to achieve terminal voltage variation v l < 0.pu, input variation v cld < 0.5pu, v clq < 0.5pu and power overshoot of less than 25%. The first objective was achieved using the controller structure discussed in Section 4. The trade-off between the second and third objectives is formalised using LMIs in Section 5. 4 Uncertainty And Controller Structure To explicitly allow consideration of the operating range during controller synthesis, the non-linear model was linearized at each of the operating points. The members of the resulting set of linearised models each have 8 states, the ith member having state space matrices A mi, B mi and C mi ; each of these matrices varies as a function of i. To allow offset free tracking of control setpoints, integral v n R n L n AC Network i nd, i nq C f PLL i cd, i cq v l RRF R c L c VSC P v cd, v cq Figure : One terminal of VSC HVDC and AC Network action was added to each input channel of the model. This has the added advantage of also embedding the variation of B mi in a set of augmented A i matrices. To embed the uncertainty in the C mi matrices in augmented A i matrices, each output channel of the model was augmented by a first order filter with a fast time constant (0 3 s); these filters can also be seen as representing the filtering behaviour of the P and v l measuring devices. The resulting augmented system {Āi, B, C} is 2th order and only has variation in the operating point A matrices, denoted by Ā i. The nominal plant matrix Ā0 is defined as the matrix of the Āi matrix element averages across the range of operating points; the operating points are then represented by the deviations of the augmented plant matrices Ā i from Ā0, Ā δi. The method of [] was then used to determine matrices B w and C z of suitable size such that: Ā δi = B w i C z i <,i =...52 () where the constant, real matrix i takes on different values at the different operating points i. The matrices B w and C z therefore characterise the uncertainty. As detailed and explained in [2], a large fraction of the uncertainty can be described by i being a 2 2 matrix with structure 2d = { = δi 2 2,δ R}, (2) This structure is taken advantage of in the controller formulation below. 5 Controller Formulation The matrices B w and C z from Section 4 are first used to create a generalised plant model, with the uncertainty connected between z and w: ẋ = Ā0x + Bu + B w w y = Cx z = C z x w = z The closed loop transfer function T zw (K) from w to z is a function of the controller K connected between y and u. From the Small Gain Theorem, if < γ the system is closed loop quadratically stable if T zw (K) < γ. Hence, the smaller the value of T zw (K), the greater the fraction of plants covered by the uncertainty that is quadratically stabilised by the controller. Network Low High Operating v l Reactance Power Power Points Table : VSC HVDC Terminal Operating Points (3)

3 From a modification of the Bounded Real Lemma as described in [2], T zw (K) < γ for an uncertainty with a structure if the following inequality and constraints on the scaling matrices S, T and Q are satisfied: A T cl X + X A cl X B clw + C T T T C clz S Bclw T X + TC Q + TD + D T T T Dclzw T S < 0 SC clz T SD clzw γ 2 I S 2 Q > 0 : T = T T, S = S,Q = Q (4) Where A cl, B clw,c clz,d clzw are the state space matrices of T zw (K). Note that B clw and C clz contain B w and C z, and S is the square root of the standard scaling matrix S. For the diagonal structured uncertainty set = 2d the structural conditions on the scaling matrices are satisfied if S = S T, T = T T and Q = Q T. If the state space matrices of K,A k, B k, C k and D k for any order n c are directly substituted into the matrices A cl, B clw, C clz and D clzw in inequality (4), this constraint is an LMI in the variables X, S, Q, T and γ with the controller matrices held constant, and an LMI in the controller matrices and γ with X, S, Q and T held constant. The following constraint, which forces the closed loop poles to have a decay rate greater than σ, is an LMI in X p with the controller matrices held constant, and an LMI in the controller matrices with X p held constant: 2σX p + A T clx p + X p A cl < 0 (5) The resulting system of inequalities (4) and (5) is thus bilinear, and constrains the system s closed loop H norm with respect to the structured uncertainty and its pole positions. The pole constraint at σ may be seen as a tuning parameter. A local minimum value of γ, γ and γ-minimising values of the controller, Lyapunov and scaling matrices may be found with the iterative scheme below, which is similar in form to the D-K iteration technique for µ-synthesis [2]: Initialisation: Find a feasible controller of the required order, i.e. one that has closed loop poles to the left of σ. For such a controller there will be a finite minimum value of γ, γk such that inequality (4) is satisfied, although it may be large. Methods of finding feasible initialising controllers are discussed in Section 6. XS-Step: For the given controller matrices find the values of X, S, Q and T and the corresponding minimising value of γ, γk such that inequality (4) is satisfied. Also find an X p that fulfills inequality (5). K-Step: For the Lyapunov matrices and scaling matrices from the XS-Step find the controller matrices and the corresponding minimising value of γ, γxs such that inequalities (4) and (5) are satisfied. Continuation Or Termination Step: Repeat the XS and K steps until γxs γ K is small. This scheme will thus always find a local minimum for γ and can be considered as a procedure for robustifying a controller of given order. Within the XS-Step finding X p is a decoupled feasibility problem. An appropriate optimisation problem for characterising X p is to maximise the decay rate of the inequality (5) by maximising α in 2σX p + A T clx p + X p A cl + αi < 0, (6) as this pushes X p deep inside its feasibility set. A variation of the above procedure which mirrors multi-objective LMI based design is to set X p = X in the XS step, which adds conservatism to the XS step, but this shaping of the Lyapunov matrix could provide more freedom of movement of K in the K step. A difficulty with this iterative scheme is that there is a non-convex constraint S 2 > Q in the XS-step. To overcome this difficulty, this constraint is replaced by the linear (convex) constraint S 2 o + ( S S o ) S o + S o ( S S o ) Q > 0 (7) that uses the previous value of S, S0. The fulfillment of this constraint is sufficient (but not necessary) for fulfillment of the non-convex constraint, so its use introduces additional conservatism into the design. 6 Methods For Generation Of Initialising Controllers Finding feasible low order output feedback controllers to initialise the iterative scheme is a non-trivial task because finding such controllers is a non-convex problem, as discussed in [8] and [0] for example. The results below are drawn from [8]. Finding low order controllers of order n c to fulfill a decay rate constraint can be formulated as a combination of a pair of LMIs characterising the decay rate constraint in two Lyapunov matrices X and Y, and the non-convex constraint [ ] X I n c = rank(m XY ),M XY = (8) I Y Several methods have been suggested for fulfilling this constraint: a simple method is to minimise the trace of the matrix M XY using LMI based optimisation, which often leads to a reduction in its rank. 7 Results A series of initialising controllers were generated, and robustified using the method described in Section 6. The LMIs were solved using the SeDuMi solver [9] via the Se- DuMi interface [6].

4 To analyse robustness of the resulting controllers the real structural singular value µ of the closed loop system T zw (K), with having diagonal structure 2d was calculated. It may be expected that γk would be similar to the structural singular value µ when the constraint X = X p is not included in the calculation of γk, but would be larger when this constraint is included. 7. Initialising Controllers The trace minimisation method introduced in Section 6 successfully generated a set of low rank controllers K L when different values of the decay rate constraint σ were used, although no rank reduction was guaranteed. Table 2 lists the following properties of these controllers: n c : controller order; σ s : position of slowest closed loop pole of T zw (K); H : H norm of T zw (K); µ : structured singular value of T zw (K); γk : minimising value of γ for T zw (K) under the constraint (4) (that is, without the constraint X = X p ). The table indicates that µ increases (i.e. the controllers becomes less robust) as σ and the controller order increase; this demonstrates a trade off between robustness and controller order on the one hand and performance on the other hand. Although no µ related constraints have been included in the controller formulations, the structural robustness of these controllers is very good. A notable feature of these results and those below is that γ K is much larger than µ and not significantly lower than H. To generate a set of initialising controllers of a given order n c, the low order controllers were augmented a number of times by lead-lag filters in each output channel with bandwidth 300Hz and maximum phase lag 2 o. Each such augmentation increased the order of the controller by 2, and the feasibility of the resulting controllers were confirmed by calculation of the closed loop eigenvalues. 7.2 Application Of The Iterative Scheme The results of applying the iterative scheme for each of the initialising controllers are presented in Table 3. The results were generated by setting the pole position constraint σ to 5, which was half the value that led to the control requirements being fulfilled in [2]. The constraint X = X p was included in the optimisation as this improved the numerical reliability of the solution scheme and identifier n c σ s µ H γ K K l K l K l K l Table 2: Low Order Controllers K L identifier K L n c σ s µ H γ K n iter K f K l K f2 K l K f3 K l K f4 K l K f5 K l K f6 K l K f7 K l K f8 K l Table 3: Controllers K F generated by iterative scheme gave final values of µ at least as good as those achieved without this constraint. The characteristics shown are for the the set of µ - minimising controllers K F reached by the iterative scheme. The table includes the initalising controller from K L and the number of iterations taken to reach this minimum, n iter. The key features of the progress of the iterative scheme are: -The improvement in µ provided by the iterative scheme was small for controllers K l0, K l and their augmentations, but significant for K l2, K l3 and their augmentations. This suggests that either K l0 and K l are close to local minima, or that the iterative scheme is not particularly efficient for these controllers. - The µ of the controller K l0 and all the controllers derived from it are smaller than those of any of the other controllers. - The augmentation of the low order controllers in K L with lead-lag filters did not significantly improve the robustness of the controllers generated by the iterative scheme. 7.3 Performance Against Control Objectives The low order controller K l0 is used as an example as it is of low order and has good robustness properties. For this controller µ =.08, so it is guaranteed to be stable over 92% of the radius of the uncertainty region, which covers only the operating points with reactance The controller was in fact stable for all the 52 linearised operating points, and a change of operating points that caused it to go unstable was not found, although neither of these is guaranteed by its value of γ K. The power and voltage responses of the non-linear plant model with controller K l0 to a power setpoint increase of 0.5pu for each of 3 operating points are shown in Figure 2: op, op 2 and op 3 refer to reactances of 0.05pu, 0.25pu and.0pu respectively; the responses at op and op 2 are virtually coincident. In terms of power rise time and worst-case overshoot these compare favourably with the responses to the same power

5 Power (pu) Voltage (pu) op op 2 op time (s) Figure 2: Responses of Non-linear Plant model at 3 Operating Points: step in power setpoint with controller K l0 Voltage (pu) Power (pu) op op 2 op time (s) Figure 3: Responses of Non-linear Plant model at 3 Operating Points: step in power setpoint with full order controller setpoint increase with the full order controller synthesised in [2], which are shown in Figure 3. 8 Conclusions The design process was successful because a number of low order controllers were found which were robust over the operating range of the VSC HVDC terminal and whose performance compared favourably with that of a full order controller. However, the best performing controller was generated by a process in which there was little control over the rank of controllers generated and little explicit specification of robustness or performance specifications. The greatest influence on the progression of the iterative algorithm and the performance achieved by the resulting robustified controllers was the originating low order controller. This is a practical demonstration of the fact that the iterative algorithm only finds locally γ-minimising controllers. These conclusions motivate further investigation underway to find low order controllers more systematically. Such methods include using genetic algorithms, which are well suited to finding the minimum value of the nonconvex functions that characterise low order controllers, in combination with LMI based analysis methods, which can efficiently measure robustness and different aspects of performance. 9 Acknowledgements The support of Areva T&D for this work is acknowledged. References [] B. Anderson. Topologies for VSC Transmission. In Seventh International Conference on AC-DC Power Transmission (IEE Conf. Publ. No.485), pages IEE, 200. [2] M. Durrant, H. Werner, and K. Abbott. Synthesis of multi-objective controllers for a VSC HVDC terminal using LMIs. Submitted for publication in CDC [3] M. Durrant, H. Werner, and K. Abbott. Model of a VSC HVDC terminal attached to a weak AC system. In CCA 2003 Proceedings. IEEE, June [4] K. Eriksson. HVDC light and development of voltage source converters. ABB review, [5] P. Kundur. Power System Stability and Control. McGraw-Hill, 994. [6] D. Peaucelle, D. Henrion, Y. Labit, and D. Peaucelle. User s guide for SeDuMi interface.04. Technical report, LAAS-CNRS Research Report No , [7] C. Scherer. Multiobjective output-feedback control via LMI optimisation. IEEE Transactions on Automatic Control, 42(7):896 9, July 997. [8] R. Skelton, T. Iwasaki, and K. Grigoriadis. A unified approach to linear control design. Taylor & Francis, London, 998. [9] J.F. Sturm. Using SeDuMi.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software, -2:625 53, 999. [0] V. Syrmos, C.T. Abdallah, P.Dorato, and K.Grigoriadis. Static output feedback: - a survey. Automatica, 33(2):25 37, 997. [] H. Werner, P.Korba, and Tai-Chen-Yang. Robust tuning of power system stabilizers using LMItechniques. IEEE Transactions on Control Systems Technology, :47 52, [2] K. Zhou, J.C. Doyle, and K. Glover. Robust and Optimal Control. Prentice-Hall, Inc., Upper Saddle River, NJ, 996.