B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part II, LNCS 7077, pp. 278286, 2011. Springer-Verlag Berlin Heidelberg 2011

Reduced Order Modeling of Linear MIMO Systems Using Soft Computing Techniques

Umme Salma1 and K. Vaisakh2

1 Department of Electrical and Electronics Engineering, GITAM Institute of Technology, GITAM University, Visakhapatnam, AP, India

usalma123@gmail.com 2 Department of Electrical Engineering, AU College of Engineering, Andhra University,

Visakhapatnam, AP, India vaisakh_k@yahoo.co.in

Abstract. A method is proposed for model order reduction for a linear multivariable system by using the combined advantages of dominant pole reduction method and Particle Swarm Optimization (PSO). The PSO reduction algorithm is based on minimization of Integral Square Error (ISE) pertaining to a unit step input. Unlike the conventional method, ISE is circumvented by equality constraints after expressing it in frequency domain using Parsevals theorem. In addition to this, many existing methods for MIMO model order reduction are also considered. The proposed method is applied to the transfer function matrix of a 10th order two-input two-output linear time invariant model of a power system. The performance of the algorithm is tested by comparing it with the other soft computing technique called Genetic Algorithm and also with the other existing techniques.

Keywords: Reduced order model, Integral Square Error, Parsevals theorem, Particle Swarm Optimization, Genetic Algorithm.

1 Introduction

In real problems the analysis of high order systems (HOS) is costly and tedious. Hence simplification procedure for original HOS are generally employed to realize for simple models based on physical considerations or by using mathematical approaches. Numerous methods are available in the literature for order reduction for linear continuous systems in time domain as well in frequency domain such as step response, frequency response etc. The reduced order model must be a good approximation of original model and it should retain the physical characteristics of the system such as step response, stability etc. Further, numerous methods of order reduction are also available in the literature, which are based on the minimization of the integral square error (ISE) criterion. However, a common feature in these methods is that the values of the denominator coefficients of the low order system (LOS) are chosen arbitrarily by some stability preserving methods such as dominant pole, Routh Approximation Methods, Routh Stability Criterion etc. and then the numerator coefficients of the LOS are determined by minimization of the ISE.

Reduced Order Modeling of Linear MIMO Systems Using Soft Computing Techniques 279

In the recent years, Particle Swarm Optimization (PSO) technique appeared as a promising algorithm for handling the optimization problems. Particle swarm optimization (PSO) is a population based stochastic optimization technique developed by Dr. Eberhart and Dr. Kennedy in 1995, inspired by social behavior of bird flocking or fish schooling [1]. PSO shares many similarities with evolutionary computation techniques such as Genetic Algorithm (GA) with a population of random solutions and searches for optima by updating generations. Unlike GA, however, PSO has no evolution operators such as crossover and mutation. In PSO, the potential solutions, called particles, fly through the problem space by following the current optimum particles. Here PSO method for model reduction is proposed. PSO method is based on minimization of Integral square error (ISE) between the transient responses of original higher order model and reduced order model pertaining to a unit step input [2].

Another approach for calculation of ISE [3] is considered. Unlike the conventional method, ISE is alternatively expressed in frequency domain using Parsevals theorem and evaluated by considering a set of equality constraints involving the coefficients in the numerator and denominator of the original and reduced order transfer functions. Basically the method starts with the fixation of denominator coefficients of lower order system (LOS) by dominant pole method and determining the coefficients of numerator polynomials of each element of LOS transfer function matrix by minimizing the ISE in between the transient responses of original and LOS using PSO. The algorithm is described in detail in the following sections and is applied to a 10th order two input-two output linear time invariant practical power system. In the present paper, in addition to PSO and GA ten more existing methods for MIMO model reductions are also considered.

2 Problem Formulation

Let the transfer function of the higher order system (HOS) of order r having p inputs and m outputs be

=

(1)

The general form of of [G(s)] is taken as:

= =

(2)

(or) =

(3)

Where < ----- < are poles of the HOS. Let the transfer function matrix of the LOS of order having inputs and

outputs to be synthesized is:

280 U. Salma and K. Vaisakh

[G(s)] =

Or = , 1,2, , ; 1,2, , (4)

The general form of of is taken as

= =

(5)

or =

(6)

Where < ----- < are the dominant poles of the HOS. Depending on the order to be reduced to, the poles nearest to the origin are

retained. Therefore the denominator polynomial is given by [4]: = (7)

3 Particle Swarm Optimization

The PSO method is a population based search algorithm where each individual is referred to as particle and represents a candidate solution. Each particle treated as a point in a d-dimensional space and represented as X= (xi1, xi2 xid) and flies through the search space with an adaptable velocity that is dynamically modified according to its own flying experience and also the flying experience of the other particles.

Further, each particle has a memory and hence capable of remembering the best position in the search space ever visited by it. The best previous position of particle that corresponds with the fitness value represented as pbest = (pi1, pi2 pid) and the best position of all particles in the population is denoted as gbest. In each iteration the value of gbest and pbest are calculated. For the nth iteration velocity and particles position are updated as shown in the following equations respectively.

) (8)

= + (9)

Where, = pbest of particle I, =gbest of the group, w = inertia weight. , =cognitive and social acceleration respectively. , = random numbers uniformly distributed in the range (0, 1).

Reduced Order Modeling of Linear MIMO Systems Using Soft Computing Techniques 281

Fig. 1. Position updates in PSO for a two dimensional Parameters

In PSO, each particle moves in the search space with a velocity according to its own previous best solution and its groups previous best solution. The velocity update in PSO consists of three parts; namely momentum, cognitive and social parts. The balance among these parts determines the performance of a PSO algorithm [2].

4 PSO Model Order Reduction

The various steps involved in PSO model reduction are as follows:

Step1: Specify the parameters of PSO. Step2: Generate the initial population for the Particles. Step3: Find the fitness value ISE for the initial Population. Here objective function f(x) is fitness value ISE. Step4: The velocity and position of all particles are randomly set within pre-defined ranges. Step5: At each iteration, the velocities and positions of all particles are updated. Step6: Update the memory by updating and . When condition is met = pi if f (pi) > f (), = gi if f (gi) > f ( ) Step7: Stopping Condition The algorithm repeats steps 5 to 7 until pre-defined

number of iterations. It reports the values of gi, and f (gi,) as its solution. This technique can be extended to multi-input and multi-output systems.

Parameters used for PSO algorithm are Swarm Size=10; Max. Generations=100; c1 =0.2; c2=0.2; wstart=0.9; wend=0.4.

5 Integral Square Error Minimization Technique (ISE) The ISE of the unit-step response is given by

ISE = 2 dt (10) Where and denote the unit-step responses of original and reduced order systems respectively.

Using Parsevals theorem the ISE can alternatively be expressed in the frequency domain and can be evaluated as described by [3]:

ISE = || [h0cp-1Qp-1h0{cp-2Qp-2 - cp-3Qp-3---- +(-1)p-1c1Q1}+ (-1)p-1c0Q0] (11)

282 U. Salma and K. Vaisakh

6 Numerical Example The Phillips - Heffron model of Single-Machine Infinite Bus (SMIB) Power system is shown in fig.2. The numerical values of the parameters, which define the total systems as well as the operating point, are given in [4].

Fig. 2. Block diagram of Phillips-Heffron model of Single-Machine Infinite Bus (SMIB) Power system

The transfer function matrix of the 10th order two-input two-output [4] is given by

(12)

Where the common denominator D(s) is given by

Ds s 64.21s 1596s 1.947 10s 1.268 10s 5.036 10s 1.569 10s 3.24 10s 4.061 10s 2.095 10s 2.531 10 and

as 2298s 9.85 10s 1.38 10s 6.838 10s 6.1 10s 5.43 10

as 29.09s 1868s 4.61 10s 5.459 10s 3.185 10s 8.702 10s 1.206 10s 7.606 10s 6.483 10

as 85.23s 3651s 5.208 10s 2.98 10s 8.471 10s 3.105 10s 2.752 10s 2.45 10

as 1.26s 85.18s 2089s 2.568 10s 1.909 10s 7.123 10s 1.084 10s 2.972 10s 1.942 10

The poles of the above system [G(s)] are at:

Reduced Order Modeling of Linear MIMO Systems Using Soft Computing Techniques 283

0.1001 , , 0.2392 j3.2348 , , 0.8977 j1.3552 , 2.1375, 9.6454, 11.9632, , 19.0451 j2.4859

The reduced transfer function matrix after applying the proposed algorithm is:

[R(s)] = (13)

b11 (s) 7.07 18.24 2.49 b21(s) 0.78 7.9 1.2 b12 (s) 0.68 29.97 2.5 b22(s) 0.02 2.4 0.03 Where 0.5785 10.5690 1.0532 The poles of the LOS [R(s)] are at 0.1001, , 0.2392 j3.2348 The adequacy of the 3rd order reduced models obtained above is tested by comparing with the original 10th order system and also for GA by Parmar [4] by time responses of the outputs (i.e. and Vt)) for two distinct input Step changes in fig 3(a)-(b):

With 0.05 p.u and 0 With 0 and = 0.05 p.u.

Fig. 3(a) . 0.05 p.u and 0

Fig. 3(b). 0.05 p.u and 0

Fig. 3(c). 0 and 0.05 p.u

Fig. 3(d). 0 and 0.05 p.u

From the above simulation results, the time responses it is clear that the 3rd order reduced system obtained by the proposed algorithm is adequate coincide quite well with those of the original order system for the same input step change.

284 U. Salma and K. Vaisakh

7 Comparisons of Reduced Order Models

Comparison of the proposed algorithm with some well known existing order reduction techniques are also shown in Table 1.

Table 1. Comparison of reduced order models

r11 r12 r21 r22 I. Proposed Method PSO 7.07s2-18.24s-2.49 ISE: 3.0328

0.68s2+29.97s+2.5 ISE: 0.4272

-0.78s2+7.9s+1.2 ISE: 0.6558

-0.02s2-2.4s+ 0.03 ISE: 0.0580

Reduced Denominator: s3 + 0.5785s2 + 10.5690s + 1.0532 II. Parmar, G., et.al [4] 7.4s2 - 24s - 2.3 ISE: 3.6563

0.63s2+28.9s+ 2.67 ISE: 0.5287

-0.6s2+7.96s+1.03 ISE: 0.7850

-1.51s2-2.99s-0.081 ISE: 0.2263

Reduced Denominator: s3 + 1.877s2 + 3.313s + 0.327 III. Rama Jaya Lakshmi et.al. [5] -2.45s2-2.24s-0.19 ISE: 2.2542

4.32s2+2.79s+0.24 ISE: 10.3816

1.11s2+1.01s+0.09 ISE: 0.1102

-0.38s2-0.11s- 0.01 ISE: 0.0980

Reduced Denominator: s3+1.453s2+1.066s+0.092 IV. Bei-bei, Wu., & Chaun-qing, G.U. [6] -1.84s2-6.94s-0.7 ISE: 5.7421

6.66s2 +8.97s+ 0.84 ISE: 34.2366

0.85s2+3.13s+ 0.32 ISE: 0.0946

1.01s2 - 0.36s-0.02 ISE: 0.3224

Reduced Denominator: s3 + 1.877s2 + 3.313s + 0.327 V. Vishwakarma, C.B. and Prasad, R. [7] 8.8s2-64.35s-20.1 ISE: 1.0575

38.27s2 +88.78s+24 ISE: -3.1511

-3.03s2+29.0s+ 9.1 ISE: 0.1228

7.27s2-5.23s -0.72 ISE: -0.0091

Reduced Denominator: s3 + 10.03s2 + 32.28s + 9.374 VI. Prasad, R. et.al. [8] -2.43s2-2.24s-0.2 ISE: 1.9304x103

4.33s2 +2.79s+0.24 ISE: 6.5014x103

1.11s2-1.01s -0.089 ISE: 398.2047

-0.39s2-0.11s-0.03 ISE: 58.4842

Reduced Denominator: s3 +1.453s2 + 1.066s + 0.093 VII. Jayanta Pal and L. M. Ray [9] -1.83s2-6.94s -0.7 ISE: 68.3341

6.66s2 +8.7s+ 0.838 ISE: 34.2065

0.85s2+3.13s+ 0.32 ISE: 0.0947

1.01s2-0.35s -0.03 ISE: 0.3221

Reduced Denominator: s3 + 1.877s2 + 3.312s + 0.327 VIII. Shieh, L.S., and Wei, Y. J. [10] -2.22s2-5.92s- 0.57 ISE: 2.7234

-6.45s2+7.4s+0.68 ISE: 13.5551

1.02s2+2.67s-0.256 ISE: 0.0801

-0.89s2 -0.3s- 0.02 ISE: 0.1117

Reduced Denominator: s3 + 1.895s2 + 2.822s + 0.264 IX. Shamash, Y. [11] -2.22s2-5.92s-0.57 ISE: 2.7210

6.45s2 +7.4s+ 0.677 ISE: 13.5562

1.02s2-2.67s+0.256 ISE: 0.0800

-0.89s2 -0.2s-0.02 ISE: 0.1117

Reduced Denominator: s3 + 1.895s2 + 2.822s + 0.264

Reduced Order Modeling of Linear MIMO Systems Using Soft Computing Techniques 285

Table 1. (Continued)

An error index ISE known as Integral Square Error in between the transient parts of original and reduced order models are calculated to measure the accuracy of the LOS. The smaller the ISE, the closer is

to . In the above tabular form it has been observed that in most of the cases the value of ISE obtained from PSO is less with that of other methods. When compared with some methods regarding ISE values, the performance of PSO can be increased by the improvement of PSO variants, which is a future study of work.

8 Conclusion

In the present work, the author proposes an algorithm for multi-input multi-output model order reduction by PSO based on the minimization of Integral Square Error (ISE) pertaining to a unit step input. Here ISE is circumvented by equality constraints after expressing it in frequency domain using Parsevals theorem. In the proposed method the denominator coefficients of the low order system (LOS) are preserved by dominant pole method and then the numerator coefficients of the LOS are determined by minimization of the ISE. The proposed method is applied to a 10th order two-input and two- output linear time invariant model of a practical power system. The adequacy of lower order models obtained by the proposed method has been judged by comparing the output time responses to the corresponding ones of the original system model. The reduced order models are compared with the Genetic Algorithm method. These methods are also compared for their ISE values with the other ten existing methods to prove the validity of the proposed method. Here PSO has been proved as a promising algorithm for handling MIMO model order reduction problems. In future study the concentration has to be done towards improvement of PSO variants to increase the efficiency of PSO.

X. Nahid Habib et.al. [12] - 64.36s-20.12 ISE: 1.7905

88.79s + 24.01 ISE: 1.5121

29.02s+ 9.07 ISE: 0.1844

- 5.2286s -0.7193 ISE: 0.0585

Reduced Denominator: s3 +10.03s2 + 32.28s + 9.374 XI. Anurag Vijay Agrawal and Ankit Mittal [13] 13.2s2-175.9s-11.7 ISE: 1.9882

94.5s2 + 217s +13.9 ISE: -1.5094

-5.54s2+79.4s+5.3 ISE: 0.2056

-22.4s2 - 8.0s-0.42 ISE: 0.0038

Reduced Denominator: s3 + 19.26s2 + 83.3s + 5.454 XII. Liaw, C.M. [14] 5.76s2-22.13s+3.58 ISE: -2.2591

7.39s2+27.76s+ 2.69 ISE: 1.5106

-2.5s2+9.9s+1.02 ISE: 0.5484

-2.72s2-1.12s-0.08 ISE: 0.2481

Reduced Denominator: s3 + 0.5785s2 + 10.57s + 1.053

286 U. Salma and K. Vaisakh

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Reduced Order Modeling of Linear MIMO Systems Using Soft Computing TechniquesIntroductionProblem FormulationParticle Swarm OptimizationPSO Model Order ReductionIntegral Square Error Minimization Technique (ISE)Numerical ExampleComparisons of Reduced Order ModelsConclusionReferences