proceedings of the American Control Conference Philadelphia, Pennsylvania June 1998

A Hybrid Robust Power System Control Design Combining System Identification and Genetic Algorithms

Flivia L. Titot Glauco N. Taranto§ Paulo C. Pellandat t Instituto Militar de Engenharia

PraGa General TibGcio, 80 - Rio de Janeiro, RJ 22290-270 - Brazil 5 Electrical Engineering Department

COPPE/UFRJ - Rio de Janeiro, RJ 21945-970 - Brazil

Abstract-This paper presents a hybrid control design method that combines features from a system identi- fication problem with meta-heuristics of Genetic Al- gorithms (GA). The main objective of the method is to find robust controllers that make the closed- loop system less sensitive to parametric uncertain- ties and also force the closed-loop spectrum to be lo- cated in a subregion that represents minimum damp- ing. Minimum-phase low-order decentralized dynamic- output-feedback controller structure is readily accom- plished in the proposed method. The approach is de- vised to be applied in the design of power system damp- ing controllers. However, the mathematical treatment employed is general to other engineering applications. li'eyzuords: Power Systems, Decentralized Control, Robust Control, Genetic Algorithms, System Identification.

I. INTRODUCTION

Damping enhancement in power systems has been sub- ject of study for many years. This problem tends to be aggravated as interlocking power grids become to be oper- ated closer to their capacity. Coordinated design of power system damping controllers turns out to be a key factor in the overall performance of the system when subject to disturbances.

The paper proposes a design method that makes use of system identification and search method concepts. The link between parametric identification and system stability was investigated in [l, 21. The straight relation between system stabilization and system identification becomes ev- ident in adaptive control. Roughly speaking, the worse the closed-loop parameter identification quality, the better the robustness of a controller against the uncertain parameter. The Parameter Robust Control by Bayesian Identification (PRCBI) synthesis provides an alternative robustness mea- sure that is based on the link between the poor quality of closed-loop parameter identification and controller robust- ness.

The PRCBI synthesis although yielding a robust con- troller against parametric uncertainties, does not guaran- tee closed-loop desirable performance specifications, such as minimum damping ratio. To circumvent this problem, we use a Genetic Algorithm that selects the controllers, among those which have nice parametric robustness prop- erties, that place the closed-loop poles inside a subregion

of the left-half side of the complex s-plane. The method is validated in the design of damping con-

trollers installed on two different FACTS devices in a small power system. A decentralized dynamic-output- feedback control structure is adopted for the design. Con- troller robustness is accomplished against transmission line impedance variations, and a minimum damping ratio of 9% is achieved at five different system operating conditions.

11. PROBLEM FORMULATION

Tuning of power system damping controllers typically uses a small-signal power system model represented by the state-space equations

i ( t ) = Az(t) + Bu(t)

Y(t ) = C z ( t ) + Du(t) (1)

where 2 is the vector of the state variables; U is the vector of the input variables and y is the vector of the measured variables.

Let Pi(.), i = 1,2, ..., m, in Figure 1, be a representa- tive set R of system operating conditions and Kd(s) be a diagonal matrix transfer function with p control channels. Defining the error signal as e = yref - y, the decentral- ized control design requires a control law U = Kd(s)e that stabilizes the system for all Pi(s) E R.

Fig. 1: Closed-Loop Setup

For each of the p individual controllers, it is assumed a classical control structure with the dynamic model con- sisting of a constant gain, a washout filter, and a double lead-lag stage as follows:

0-7803-4530-4198 $10.00 0 1998 AACC 3403

A. The PRCBI Synthesis

Based on the work in [l, 21, consider the following linear- time-invariant discrete-time realization

x k + i = A(0)zk 4- B ( 0 ) U k 4- E& gk = c ( 0 ) x k + D ( 0 ) U k + v k (3)

where 2 E %n is the state vector, U E %P is the input vector, y E is the output vector and A E % ( n x n ) , B E %(nxp), C E !J?(mxn), D E %(mxp) and E E !J?(nxp). Assume that the pair A, C is observable, and that the system noise E %P and the measurement noise r) E ?JF are uncorrelated jointly Gaussian signals satisfying the following conditions:

E{&) = E{r)k) = 0 (4)

where 6jk = 1 for j = k and 0 otherwise, E{.} denotes the expectation operator, and Q E %PXP with Q 2 0 and R E SmXm with R > 0 are covariance matrices.

From Bayes' Recursive Formulation, it was shown in [a] that

where 0 E ST is the unknown parameter vector, 8, is the nominal value parameter vector, p o ( 0 ) is the initial uni- form probability density which is known over the continu- ous parametric range, M is the output prediction covari- ance matrix, hc is a constant, Y k = {yo, y1 , . . . , yh} is the set of measurement until step k , A0 represents the param- eter uncertainties, MO is the output prediction covariance matrix computed by a Kalman filter designed on 8 , , AA4 is the output covariance matrix variation due to the paramet- ric uncertainty and Go, > 0 is the Bayesian identification quality matrix.

The steady-state equation (6) shows that the conditional probability density p ( 0 , + A61Yk) describes a Gaussian process with expectation 8, and covariance Po = Go,/k. Therefore, matrix Go, is proportional to the covariance es- timation matrix of 0 = 8, +A0, if measurement set Y k and the initial probability density p o ( 8 ) are known.

The parametric robustness criterion has been established as PI

Jprcbi = rr~in[Tr(G;~')] KC (7 )

where K, is the design variable representing the controller. The main point in the PRCBI synthesis is to find out

a controller K, that degrades the Bayesian identification quality expressed by matrix Go,, thus corresponding to improved robustness, as the closed-loop-pole sensitivities decrease with respect to the uncertain parameters.

111. GA OPTIMIZATION SETUP

The GA performance criterion function is defined to be the sum of the spectrum damping ratios for all operating

conditions in Q. The constraint set comprises the param- eter bounds plus the performance requirement for a mini- mum specified damping ratio. The latter constraint defines the allowed subregion on the left-half side of the complex plane. Therefore, the design problem can be formulated as the following optimization problem:

subject to,

Kiim,,, 5 Kii 5 Kii,,,== aim,,, 5 ai 5 aimaz

cmin 5 (C j ) i (9) where n is the system order and C is the closed-loop-pole damping ratio.

The problem defined in (8) and (9) is a complex op- timization problem with an implicit objective function, which depends on the evaluation of the eigenvalues of a matrix.

A. Modified GA Elements

The solution of the optimization problems defined in (7) and in (8) - (9) can be obtained using a modified version of the simple GA described in [3]. The main elements of the GA are defined as follows:

1) Fitness Function

The fitness function J p used by the GA is defined as

0 i fany (Cj)i 5 O ,& if all (cj)i > 0 and if any (Cj) i 5 <I ,& if all (cj)i > and if any ( c j ) i 5 <2 I( -

PK if all

F if all (Cj)i 2 Cmin

(&)i > ~ K - I and if any ( t ) i < Cmin

(10) where ,L$ < ,& < . . . < Plc are positive scalars specified to discriminate the stable solutions with performance con- straint not satisfied, firstly from the unstable solutions, and secondly from the stable solutions with all constraints sat- isfied, and 0 < (1 < & < . . . < C K - ~ < The fitness function is scaled linearly. The simple GA described in [3] and used in this work, max- imizes a function. Therefore, instead of using directly ( 7 ) , we do

Jr = max Jp;tbi (11)

JT = J , + J p (12)

The overall fitness function that balances the robustness cost J, and the performance cost J p is computed as follows:

with

J's and J - - Jr = - Jp' i = 1 , 2 , . . . , n p (13)

where J,.,,, and JPm,, are the maximum value of the ro- bustness cost and performance cost, respectively, evaluated at the current Keneration and no is the population size.

Jrmax - JPmax

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2) Parameter Encoding and Limits

The controller parameters are encoded as fixed-length binary strings using the concatenated, multiparameter, mapped, fixed-point coding described in [3]. The upper and lower limits on the design parameters are established based on engineering judgment.

3) Genetic Operators

Selection was performed using proportional selection (roulette wheel). An elitist strategy was also experimented. Two types of crossover operators were tested: one-point and uniform crossover. The crossover rate is kept con- stant during the optimization process. Mutation was im- plemented using a fixed mutation rate.

4) Initialization Procedure

The initial population is chosen at random. However, computational experiments have indicated that better re- sults are obtained if several individuals of the initial pop- ulation correspond to stable solutions for the tuning prob- lem] i.e., all ( t ) i 2 0. This is achieved by simply continu- ing the random generation process until a certain number of individuals (no) of the population satisfy the stability criterion. The computation time spent in this initializa- tion process is compensated by a greater chance of finding an acceptable solution.

5) Stopping Rule

The GA is stopped whenever a maximum number of gen- erations (ns) is reached.

B. Design Algorithm

In this section we describe all the steps involved in the design method.

Step 1: Randomly choose no << np sets of the parameters Kii and ai that at least stabilizes all Pi(s) E a. Step 2: Randomly choose np - no sets of the parameters Ii'ii and ai to complete the initial population. Step 3: For each controller (individual) compute J,., and

Step 4: At the end of the current generation, compute JT. Step 5 : Apply GA operators to form new population. Step 6: If maximumnumber of generation ng is not reached go to Step 3, otherwise stop.

F-

Figure 2 depicts the algorithm flow chart.

IV. ILLUSTRATIVE EXAMPLE

The method described in the previous section is illus- trated by the simultaneous tuning of an SVC and a TCSC damDine: controllers for the test system shown in Figure 3

Initial POP. IC = o I I 1

I

eVal( JT) L-r'

and firs; studied in [4]. 3405

Fig. 2: Algorithm Flow Chart

A. Example System Characteristics

All generators are modeled with 6 state variables with identical parameters [6]. All exciters are represented by 2 state variables with identical parameters [7]. The system is analyzed under two nominal power flow conditions. In the first condition, named the Nominal Direct-Flow System (NDFS), Machines 3 and 4 are exporting 640 MW and Ma- chines 5 and 6 are exporting 610 MW. Most of the exported active power is consumed by the load L3. In the second condition, the power flow on the TCSC path reverses due to a Load L3 decrease, and a Load L ~ o increase. Now Ma- chines l and 2 are exporting 710 MW and Machines 3 and 4 are exporting 650 MW. The latter power flow condition is named the Nominal Reversed-Flow System (NRFS).

In both nominal flow conditions the system presents ap-

La) Zeps-s 10 11 9 Swing

Zeqs-10

13 14 12 -5

4

- & 1- tL4 I Id I

TCSC

Fig. 3: Three-Area Six-Machine System

proximately the same mode shapes. There are two low- frequency electromechanical inter-area modes in the sys- tem. The first inter-area mode (Mode 1) consists of the machines of Area B oscillating against the machines of Ar- eas A and c, and the second inter-area mode (Mode 2) consists of the machines of Area A oscillating against the machines of Area C. Other three electromechanical modes are local modes of machine oscillations within the areas. To motivate the design, we apply an SVC on Bus 5 to en- hance the damping of Mode 1, and insert a TCSC in one of the tie lines between Buses 13 and 14 to enhance the damping of Mode 2.

For robust design, the controller must be able to pro- vide additional damping to all credible system conditions. Besides the two nominal conditions NDFS and NRFS, we consider three weaker operating conditions as described in Table 1. The Weak 1 system represents a weaker tie in the SVC transmission path, and the Weak 2 system represents a weaker tie in the TCSC transmission path when the sys- tem is in the NDFS condition. The Weak 3 system repre- sents a weaker tie in the TCSC transmission path when the system is in the NRFS condition. The frequencies f(Hz) and the damping ratios C(%) of the inter-area modes are also given in Table 1.

9 5 - Q e 4 -

3 -

B. Design Results

6 -

8 5 -

f4. 3 -

2 -

The objective is to design a TCSC and an SVC damping controller to enhance the damping of the inter-area modes in all five operating conditions. In addition to the objec- tive of low-frequency damping enhancement we seek the controllers that turn the closed-loop system less sensitive to the variation of transmission line impedances.

In the design process, reduced-order 10-state system models obtained from the full-order 48-state system models using Hankel norm [8] were used.

Absolute values of synthesized angles derived from local voltage and current measurements were used as feedback signals [9].

Table 2 presents the closed-loop damping ratio (%) of the inter-area modes and the value of Tr(Giol). The first row corresponds to the closed-loop performance with the con- troller designed when the fitness function JT considers only the performance criterion J p , meaning that the PRCBI syn- thesis was not taken into consideration. The second row of Table 2 corresponds to the closed-loop performance with the controller designed when the fitness function JT consid- ers only the robustness criterion J , , meaning that damping enhancement criterion was not taken into consideration. Finally, the third row shows the results when both criteria J p and J,. are taken into account. Note the trade-off occur- ring between robustness (minimum trace) and performance (higher damping).

Table 3 shows the controller parameters found by the design method. In all three cases a washout time constant of Tw = 10 sec. was considered. We assume that w1 and w2 are obtained from the NDFS operating condition, yielding w1 = 3.77 rad/sec and w2 = 7.60 rad/sec. Based on the results of [9] we set the bounds on the control parameters as:

-5.0 5 Kii 5 5.0 0.1 < cy; < 10.0

Figures 4-6 show the location of the closed-loop poles for different values of impedance Zeqs-6 when the system is in the NDFS operating condition (The points marked with an “x” correspond to the nominal-impedance system). In Figure 4 the controller utilized is the one optimized only against the performance criterion J p . Note that the con- troller forces the system to achieve a desirable minimum damping. All three figures show the line of 9% damping ratio.

In Figure 5 the controller utilized is the one optimized only against the robustness criterion J,. Note that for the same variations of the impedance Zeq5-s, used in Figure 4, the closed-loop poles are less spread out. However they are located in the forbidden region of less than 9% damping.

Finally, Figure 6 shows the close-loop poles with the con- troller designed when the objective function balances be- tween the performance and the robustness criteria.

! r . .

...a

42 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 l/Sec

Fig. 4: Closed-loop pole location for different values of impedance Zegs-6 with first controller

Q

J

! x

,..a 0 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2

1 /sec

Fig. 5: Closed-loop pole location for different values of impedance ZeqgW6 with second controller

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TABLE 1: OPEN-LOOP OPERATING CONDITIONS

System

NDFS NRFS

Weak 1 Weak 2 Weak 3

zeq5-6 Zeqs-10 L3 LlO Mode 1 PU PU (MW) (MVAr) (MW) (MVAr) f(Hz) C(%)

0 .08+j0 .8 0.04+j0.4 3000 1000 400 100 0.60 1.94 0.08 + j0.8 0.04 + j0.4 400 100 3000 1000 0.62 3.70

0.095 + j0.95 0.04 + j0.4 3000 1000 400 100 0.39 -8.63 0.08+j0.8 0.28+j2.8 3000 1000 400 100 0.44 -6.33 0.08 + j0.8 0.28 + j2.8 400 100 3000 1000 0.74 13.44

Mode 2 I

[ Optimized I NDFS

1.12 3.67 1.19 4.10 1.03 3.99 0.77 -4.93

NRFS I Weak 1 I Weak 2 1 Weak 3 I Trace I Criterion

JP Jr JT

Mdl Md2 Mdl Md2 Mdl Md2 Mdl Md2 Mdl Md2 x103 11.98 16.28 12.44 13.66 13.73 18.48 33.04 39.36 12.95 12.34 21.28 5.48 7.52 5.79 7.08 0.54 7.83 9.46 15.32 2.25 16.79 1.608 9.92 13.09 9.43 11.11 9.44 14.37 22.46 34.13 14.30 11.77 7.218

6 -

45- a* e 4 -

‘ 1

Fig. 6: Closed-loop pole location for different values of impedance 2e46--5 with third controller

V. CONCLUSIONS

The PRCBI method synthesizes robust controllers against parametric uncertainties in all system matrices by minimizing the trace of a covariance matrix. Constraints in the PRCBI-synthesized controller is not easily done. By applying Genetic Algorithms we are able to search for the controllers that have desirable performance among the con- trollers that are robust in a PRCBI sense. This hybrid design method was successfully applied in designing two

TABLE 3: CONTROLLER PARAMETEW

Optimized

Jr JT

power system damping controllers with a dynamic-output- feedback minimum-phase decentralized structure.

ACKNOWLEDGMENTS

We acknowledge Prof. G. M. P. Gomes (IME-Brazil) for introduc- ing us the concepts of the PRCBI synthesis, and Prof. D. M. Falcio (COPPE/UFRJ-Brazil) for introducing us the concepts of GA.

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