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: +4940428782112† 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 con-trollers for a VSC HVDC terminal which are robust overa range of operating points is presented. An uncertaintyregion is used to characterise the operating range as de-scribed by a non-linear model of the terminal. An LMIbased formulation is then used to synthesize low ordercontrollers which maximize the size of the structured un-certainty region within which closed loop stability is main-tained. The method uses an iterative scheme with afeasible low order controller as the initialising controller.The improvement provided by the iterative scheme andthe performance of the resulting controllers is comparedfor different initialising controllers using the non-linearmodel.

1 Introduction

VSC HVDC transmission (Voltage Source Converter HighVoltage Direct Current transmission) is an electrical trans-mission technology that has received considerable atten-tion in recent years due to the development of high powertransistor technology [1] [4]. A VSC HVDC transmissionsystem connects two AC networks using two AC-DC ter-minals and a DC link. In such systems it is necessary tocontrol power flow between the terminals under differentAC network operating conditions, and while the networkmoves between these conditions, while at the same timecontrolling the terminal AC and DC voltages. This is arobust control problem which becomes increasingly chal-lenging as the impedance of the network increases [3].

With some conservatism, LMI based multi-objective out-put controller synthesis [7] allows full-order controllers tobe designed which maximise the range of uncertain param-eters over which stability is maintained while maintainingperformance requirements for a nominal model. These areachieved respectively by minimisation of a H∞ norm andfulfillment of closed loop performance constraints such asH2 performance or pole positions. A potentially attrac-tive feature of the design method is that the controllersdesigned are quadratically stable across the stability re-

gion found, so the controller can withstand arbitrarily fastchanges in operating conditions within this region.

In [2] this method was adapted to allow the structure ofthe uncertainty to be exploited using a D-K type iterativescheme. This iterative scheme was then used to design afull order power and voltage controller for one terminal ofa VSC HVDC transmission system.

This paper describes an alternative adaption of thismethod that generates low order controllers in additionto exploiting the structure of the uncertainty, and its ap-plication to the same problem. The motivation for this isthat (high) full order controllers may not be acceptable inpractical applications.

The paper is organised as follows. The analytical modelof the terminal is described in Section 2. The control ob-jectives are outlined in Section 3 and a controller and un-certainty structure to achieve these is described in Section4. The controller synthesis is formally formulated and aniterative procedure to solve the resulting non-convex op-timisation problem as a series of LMIs are described inSection 5. In the remaining sections methods for gener-ating initialising controllers are discussed, and the perfor-mance and robustness achieved with different initialisingcontrollers are described.

2 VSC HVDC Model

As is common for 3 phase power circuits [5], physical quan-tities in this paper are represented in the dimensionless perunit (pu) form, and phasor quantities are represented in d

and q (direct and quadrature) units relative to a rotatingreference frame (RRF ).

A non-linear model of VSC HVDC terminal attached to anAC network has been developed and successfully validatedagainst a rigorous model including realistic converter be-haviour [3]. This model is represented schematically inFigure 1. The converter itself is represented by a volt-age source, the associated transformer is represented byRc and Lc and the filter used for suppression of converterswitching harmonics is represented by Cf . The AC net-work is represented by a fixed voltage source vn and an

Control 2004, University of Bath, UK, September 2004 ID-101

impedance that may vary, consisting of Ln and Rn. TheRRF used is the voltage phasor of the converter terminalvoltage, which is measured using a phase locked loop. Thecontrol inputs used are vcld and vclq, the voltages acrossthe converter impedance as measured in the RRF . Thedynamic behaviour between these inputs and the power(P ) and converter terminal voltage magnitude (|vl|) de-pends on the operating point defined by Rc , Lc, P andvl.

3 Control Objectives

The controller objectives for the system are:i) To control power P and converter terminal voltage |vl|using inputs vcld and vclq with no steady state error be-tween P and |vl| setpoints and measurements.ii) To maintain closed loop stability at 52 representativeoperating points as detailed in Table 1 and while movingbetween them. The operating points are defined by powerflow and AC system reactance.iii) To minimise the response time to power setpoint stepsfor a given controller order.iv) During power setpoint steps of 0.5pu, to achieve ter-minal voltage variation |vl| < 0.1pu, input variationvcld < 0.5pu, vclq < 0.5pu and power overshoot of lessthan 25%.The first objective was achieved using the controller struc-ture discussed in Section 4. The trade-off between thesecond and third objectives is formalised using LMIs inSection 5.

4 Uncertainty And Controller

Structure

To explicitly allow consideration of the operating rangeduring controller synthesis, the non-linear model was lin-earized at each of the operating points.

The members of the resulting set of linearised models eachhave 8 states, the ith member having state space matricesAmi, Bmi and Cmi; each of these matrices varies as afunction of i.

To allow offset free tracking of control setpoints, integral

VSC

PLL

AC Network vcd, vcqvl

RRF

Cf

Rc Lc

icd, icq

vn

ind, inq

LnRn

P

Figure 1: One terminal of VSC HVDC and AC Network

action was added to each input channel of the model. Thishas the added advantage of also embedding the variationof Bmi in a set of augmented Ai matrices. To embed theuncertainty in the Cmi matrices in augmented Ai matri-ces, each output channel of the model was augmented bya first order filter with a fast time constant (10−3s); thesefilters can also be seen as representing the filtering be-haviour of the P and |vl| measuring devices. The resultingaugmented system {Ai,B,C} is 12th order and only hasvariation in the operating point A matrices, denoted byAi. The nominal plant matrix A0 is defined as the ma-trix of the Ai matrix element averages across the rangeof operating points; the operating points are then repre-sented by the deviations of the augmented plant matricesAi from A0, Aδi. The method of [11] was then used todetermine matrices Bw and Cz of suitable size such that:

Aδi = Bw∆iCz ||∆i|| < 1, i = 1 . . . 52 (1)

where the constant, real matrix ∆i takes on different val-ues at the different operating points i. The matrices Bw

and Cz therefore characterise the uncertainty.

As detailed and explained in [2], a large fraction of theuncertainty can be described by ∆i being a 2 × 2 matrixwith structure

∆2d = {∆ = δI2×2, δ ∈ R}, (2)

This structure is taken advantage of in the controller for-mulation below.

5 Controller Formulation

The matrices Bw and Cz from Section 4 are first usedto create a generalised plant model, with the uncertaintyconnected between z and w:

x = A0x + Bu + Bww

y = Cx

z = Czx

w = ∆z

(3)

The closed loop transfer function Tzw(K) from w to z isa function of the controller K connected between y andu. From the Small Gain Theorem, if ||∆|| < 1

γthe system

is closed loop quadratically stable if ||Tzw(K)||∞ < γ.Hence, the smaller the value of ||Tzw(K)||∞, the greaterthe fraction of plants covered by the uncertainty that isquadratically stabilised by the controller.

Network Low High Operating |vl|Reactance Power Power Points

0.05 0.05 1.0 21 1.00.25 0.05 1.0 21 1.01.0 0.05 0.5 10 1.0

Table 1: VSC HVDC Terminal Operating Points

Control 2004, University of Bath, UK, September 2004 ID-101

From a modification of the Bounded Real Lemma as de-scribed in [2] , ||Tzw(K)||∞ < γ for an uncertainty ∆ witha structure ∆ if the following inequality and constraintson the scaling matrices S, T and Q are satisfied:

ATclX∞ + X∞Acl X∞Bclw + CT TT CclzS

BTclwX∞ + TC −Q + TD + DT TT DT

clzwS

SCTclz SDclzw −γ2I

< 0

S2 ≥ Q > 0∀∆ ∈ ∆ : T∆ = ∆T T, S∆ = ∆S, Q∆ = ∆Q

(4)

Where Acl, Bclw ,Cclz ,Dclzw are the state space matricesof Tzw(K). Note that Bclw and Cclz contain Bw and Cz,and S is the square root of the standard scaling matrix S.

For the diagonal structured uncertainty set ∆ = ∆2d thestructural conditions on the scaling matrices are satisfiedif S = ST , T = −TT and Q = QT .

If the state space matrices of K,Ak, Bk, Ck and Dk forany order nc are directly substituted into the matrices Acl,Bclw, Cclz and Dclzw in inequality (4), this constraint isan LMI in the variables X∞, S, Q, T and γ with thecontroller matrices held constant, and an LMI in the con-troller matrices and γ with X∞, S, Q and T held constant.

The following constraint, which forces the closed looppoles to have a decay rate greater than −σ, is an LMIin Xp with the controller matrices held constant, and anLMI in the controller matrices with Xp held constant:

2σXp + ATclXp + XpAcl < 0 (5)

The resulting system of inequalities (4) and (5) is thusbilinear, and constrains the system’s closed loop H∞ normwith respect to the structured uncertainty and its polepositions. The pole constraint at −σ may be seen as atuning parameter.

A local minimum value of γ, γ∗ and γ-minimising valuesof the controller, Lyapunov and scaling matrices may befound with the iterative scheme below, which is similar inform to the D-K iteration technique for µ-synthesis [12]:

Initialisation: Find a feasible controller of the requiredorder, i.e. one that has closed loop poles to the left of −σ.For such a controller there will be a finite minimum valueof γ, γ∗

K such that inequality (4) is satisfied, althoughit may be large. Methods of finding feasible initialisingcontrollers are discussed in Section 6.

XS-Step: For the given controller matrices find the val-ues of X∞, S, Q and T and the corresponding minimisingvalue of γ, γ∗

K such that inequality (4) is satisfied. Alsofind an Xp that fulfills inequality (5).

K-Step: For the Lyapunov matrices and scaling matri-ces from the XS-Step find the controller matrices and thecorresponding minimising value of γ, γ∗

XS such that in-equalities (4) and (5) are satisfied.

Continuation Or Termination Step: Repeat the XS

and K steps until γ∗

XS − γ∗

K is small.This scheme will thus always find a local minimum for γ

and can be considered as a procedure for ’robustifying’ acontroller of given order.

Within the XS-Step finding Xp is a decoupled feasibilityproblem. An appropriate optimisation problem for char-acterising Xp is to maximise the decay rate of the inequal-ity (5) by maximising α in

2σXp + ATclXp + XpAcl + αI < 0, (6)

as this pushes Xp deep inside its feasibility set. A varia-tion of the above procedure which mirrors multi-objectiveLMI based design is to set Xp = X∞ in the XS step,which adds conservatism to the XS step, but this ’shap-ing’ of the Lyapunov matrix could provide more freedomof movement of K in the K step.

A difficulty with this iterative scheme is that there is anon-convex constraint S2 > Q in the XS-step. To over-come this difficulty, this constraint is replaced by the lin-ear (convex) constraint

S2o + (S − So)So + So(S − So) − Q > 0 (7)

that uses the previous value of S, S0. The fulfillment ofthis constraint is sufficient (but not necessary) for fulfill-ment of the non-convex constraint, so its use introducesadditional conservatism into the design.

6 Methods For Generation Of

Initialising Controllers

Finding feasible low order output feedback controllers toinitialise the iterative scheme is a non-trivial task becausefinding such controllers is a non-convex problem, as dis-cussed in [8] and [10] for example. The results below aredrawn from [8].

Finding low order controllers of order nc to fulfill a decayrate constraint can be formulated as a combination of apair of LMIs characterising the decay rate constraint intwo Lyapunov matrices X and Y , and the non-convexconstraint

nc = rank(MXY ),MXY =

[

X I

I Y

]

(8)

Several methods have been suggested for fulfilling this con-straint: a simple method is to minimise the trace of thematrix MXY using LMI based optimisation, which oftenleads to a reduction in its rank.

7 Results

A series of initialising controllers were generated, and ro-bustified using the method described in Section 6. TheLMIs were solved using the SeDuMi solver [9] via the Se-DuMi interface [6].

Control 2004, University of Bath, UK, September 2004 ID-101

To analyse robustness of the resulting controllers the realstructural singular value µ∆ of the closed loop systemTzw(K), with ∆ having diagonal structure ∆2d was cal-culated. It may be expected that γ∗

K would be similarto the structural singular value µ∆ when the constraintX∞ = Xp is not included in the calculation of γ∗

K , butwould be larger when this constraint is included.

7.1 Initialising ControllersThe trace minimisation method introduced in Section 6successfully generated a set of low rank controllers KL

when different values of the decay rate constraint −σ wereused, although no rank reduction was guaranteed. Table2 lists the following properties of these controllers:

nc: controller order; −σs: position of slowest closed looppole of Tzw(K); H∞: H∞ norm of Tzw(K); µ∆: struc-tured singular value of Tzw(K); γ∗

K : minimising value ofγ for Tzw(K) under the constraint (4) (that is, withoutthe constraint X∞ = Xp).

The table indicates that µ∆ increases (i.e. the controllersbecomes less robust) as −σ and the controller order in-crease; this demonstrates a trade off between robustnessand controller order on the one hand and performance onthe other hand. Although no µ∆ related constraints havebeen included in the controller formulations, the struc-tural robustness of these controllers is very good.

A notable feature of these results and those below is thatγ∗

K is much larger than µ∆ and not significantly lower thanH∞.

To generate a set of initialising controllers of a given ordernc, the low order controllers were augmented a number oftimes by lead-lag filters in each output channel with band-width 300Hz and maximum phase lag 12o. Each such aug-mentation increased the order of the controller by 2, andthe feasibility of the resulting controllers were confirmedby calculation of the closed loop eigenvalues.

7.2 Application Of The Iterative SchemeThe results of applying the iterative scheme for each ofthe initialising controllers are presented in Table 3. Theresults were generated by setting the pole position con-straint −σ to 5, which was half the value that led to thecontrol requirements being fulfilled in [2]. The constraintX∞ = Xp was included in the optimisation as this im-proved the numerical reliability of the solution scheme and

identifier nc −σs µ∆ H∞ γ∗

K

Kl0 0 6.5 1.08 27.0 23.2Kl1 1 10.0 1.2 27.5 23.7Kl2 2 16.5 1.6 28.3 24.4Kl3 3 18.9 2.0 28.3 24.4

Table 2: Low Order Controllers KL

identifier KL nc −σs µ∆ H∞ γ∗

K niter

Kf1 Kl0 2 6.5 1.07 26.7 23.0 2

Kf2 Kl0 4 7.7 1.05 26.6 22.9 8

Kf3 Kl1 1 9.6 1.2 27.5 23.7 7

Kf4 Kl1 5 10.5 1.13 26.7 22.9 2

Kf5 Kl2 2 12 1.27 27.4 23.6 17

Kf6 Kl2 6 11.1 1.16 26.4 22.7 13

Kf7 Kl3 3 10.1 1.3 27.7 23.8 34

Kf8 Kl3 7 9.7 1.21 26.5 22.9 32

Table 3: Controllers KF generated by iterative scheme

gave final values of µ∆ at least as good as those achievedwithout this constraint.

The characteristics shown are for the the set of µ∆-minimising controllers KF reached by the iterative scheme.The table includes the initalising controller from KL andthe number of iterations taken to reach this minimum,niter. The key features of the progress of the iterativescheme are:

-The improvement in µ∆ provided by the iterative schemewas small for controllers Kl0, Kl1 and their augmenta-tions, but significant for Kl2, Kl3 and their augmentations.This suggests that either Kl0 and Kl1 are close to localminima, or that the iterative scheme is not particularlyefficient for these controllers.

- The µ∆ of the controller Kl0 and all the controllers de-rived from it are smaller than those of any of the othercontrollers.

- The augmentation of the low order controllers in KL withlead-lag filters did not significantly improve the robustnessof the controllers generated by the iterative scheme.

7.3 Performance Against Control

ObjectivesThe low order controller Kl0 is used as an example as it isof low order and has good robustness properties. For thiscontroller µ∆ = 1.08, so it is guaranteed to be stable over92% of the radius of the uncertainty region, which coversonly the operating points with reactance 0.05. The con-troller was in fact stable for all the 52 linearised operatingpoints, and a change of operating points that caused itto go unstable was not found, although neither of these isguaranteed by its value of γ∗

K .

The power and voltage responses of the non-linear plantmodel with controller Kl0 to a power setpoint increase of0.5pu for each of 3 operating points are shown in Figure2: op1, op2 and op3 refer to reactances of 0.05pu, 0.25puand 1.0pu respectively; the responses at op1 and op2 arevirtually coincident.

In terms of power rise time and worst-case overshoot thesecompare favourably with the responses to the same power

Control 2004, University of Bath, UK, September 2004 ID-101

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Pow

er (

pu)

0 0.2 0.4 0.6 0.8 10.95

0.975

1

1.025

1.05

time (s)

Vol

tage

(pu

)

op1

op2

op3

Figure 2: Responses of Non-linear Plant model at 3 Op-erating Points: step in power setpoint with controller Kl0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Pow

er (

pu)

0 0.2 0.4 0.6 0.8 10.98

0.99

1

1.01

1.02

time (s)

Vol

tage

(pu

)

op1

op2

op3

Figure 3: Responses of Non-linear Plant model at 3 Op-erating Points: step in power setpoint with full order con-troller

setpoint increase with the full order controller synthesisedin [2], which are shown in Figure 3.

8 Conclusions

The design process was successful because a number oflow order controllers were found which were robust overthe operating range of the VSC HVDC terminal and whoseperformance compared favourably with that of a full ordercontroller. However, the best performing controller wasgenerated by a process in which there was little controlover the rank of controllers generated and little explicitspecification of robustness or performance specifications.

The greatest influence on the progression of the iterativealgorithm and the performance achieved by the resultingrobustified controllers was the originating low order con-troller. This is a practical demonstration of the fact thatthe iterative algorithm only finds locally γ-minimisingcontrollers.

These conclusions motivate further investigation under-way to find low order controllers more systematically.

Such methods include using genetic algorithms, which arewell suited to finding the minimum value of the non-convex functions that characterise low order controllers,in combination with LMI based analysis methods, whichcan efficiently measure robustness and different aspects ofperformance.

9 Acknowledgements

The support of Areva T&D for this work is acknowledged.

References

[1] B. Anderson. Topologies for VSC Transmission. InSeventh International Conference on AC-DC PowerTransmission (IEE Conf. Publ. No.485), pages 298–304. IEE, 2001.

[2] M. Durrant, H. Werner, and K. Abbott. Synthesis ofmulti-objective controllers for a VSC HVDC terminalusing LMIs. Submitted for publication in CDC 2004.

[3] M. Durrant, H. Werner, and K. Abbott. Model of aVSC HVDC terminal attached to a weak AC system.In CCA 2003 Proceedings. IEEE, June 2003.

[4] K. Eriksson. HVDC light and development of voltagesource converters. ABB review, 2002.

[5] P. Kundur. Power System Stability and Control.McGraw-Hill, 1994.

[6] D. Peaucelle, D. Henrion, Y. Labit, and D. Peaucelle.User’s guide for SeDuMi interface 1.04. Technical re-port, LAAS-CNRS Research Report No. 02333, 2002.

[7] C. Scherer. Multiobjective output-feedback controlvia LMI optimisation. IEEE Transactions on Auto-matic Control, 42(7):896–911, July 1997.

[8] R. Skelton, T. Iwasaki, and K. Grigoriadis. A unifiedapproach to linear control design. Taylor & Francis,London, 1998.

[9] J.F. Sturm. Using SeDuMi 1.02, a MATLAB toolboxfor optimization over symmetric cones. OptimizationMethods and Software, 11-12:625–53, 1999.

[10] V. Syrmos, C.T. Abdallah, P.Dorato, andK.Grigoriadis. Static output feedback: - a sur-vey. Automatica, 33(2):125–137, 1997.

[11] H. Werner, P.Korba, and Tai-Chen-Yang. Ro-bust tuning of power system stabilizers using LMI-techniques. IEEE Transactions on Control SystemsTechnology, 11:147–52, 2003.

[12] K. Zhou, J.C. Doyle, and K. Glover. Robust and Opti-mal Control. Prentice-Hall, Inc., Upper Saddle River,NJ, 1996.

Control 2004, University of Bath, UK, September 2004 ID-101