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Order Reduction of LTI Systems Using Balanced Truncation and Particle Swarm Optimization Algorithm

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Abstract

This paper presents a novel hybrid model reduction method to simplify large-scale continuous-time single-input–single-output and multi-input–multi-output dynamic systems using the advantages of the balanced truncation method and the particle swarm optimization (PSO) algorithm. The balanced truncation method obtains the reduced model denominator coefficients to ensure stability. The PSO algorithm minimizes the integral square error between the step responses of the original system and the reduced model as much as possible. It leads to the optimal numerator coefficients. The advantage of the proposed approach is that, for optimizing reduced model numerator coefficients, the search space boundaries of the PSO algorithm are not entirely random. They are selected using the balanced truncated reduced model numerator. So, the suggested method avoids two significant problems with evolutionary algorithms: the arbitrary choice of search space and the longer simulation time. Four power system models and four numerical examples from the literature are considered to assess the effectiveness of the proposed reduction method. The step responses and Bode diagrams for the higher-order system and its corresponding reduced models are displayed. For comparison, statistics are tabulated based on the rise time, the settling time, the peak overshoot, the integral square error, and the root-mean-square error. The proposed method ensures stability and other features of the higher-order system in the reduced model. The performance measure values and the time domain characteristics demonstrate the ability and effectiveness of the proposed approach. All the case studies use a MATLAB simulation environment.

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Data Availability

For case studies involving a hydro-power system, a single area reheated hydro-thermal power system, and the SMIB power system, the corresponding author, is willing to provide block diagrams, basic mathematical descriptions, and operating points upon reasonable request.

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Authors and Affiliations

  1. Electrical Engineering Department, National Institute of Technology Rourkela, Rourkela, India

    Bala Bhaskar Duddeti, Asim Kumar Naskar & K. R. Subhashini

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  1. Bala Bhaskar Duddeti
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  3. K. R. Subhashini
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Correspondence to Bala Bhaskar Duddeti.

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Appendix

Appendix

Parameters used for the PSO algorithm:

The choice of PSO parameters has a significant impact on optimization performance. Considered the following assumptions while simulating the PSO algorithm:

No.

Parameters

values

Remarks

1

Swarm or population size (N)

100

The swarm's initial variety increases according to the number of particles in it. The search space can be covered in a larger area every iteration when there is an immense swarm. However, the search degrades to a similar random search as the number of particles increases and the computing cost of each iteration increases. In addition, more enormous swarms may require fewer iterations to arrive at a workable solution than smaller ones

2

Maximum number of Iterations (Tmax)

100

A small number of iterations may result in premature convergence. An excessive number of iterations results in unnecessarily increased computing complexity (provided that the number of iterations is the only stopping condition). It is particular to a given issue

3

Inertia weight (w)

1

The inertia weight regulates the influence of the initial velocity in a new direction. Exploration (which diverts the swarm's global search) is caused by a significant inertia weight, whereas a lower inertia weight causes exploitation (decelerates the particles). So, the balance between the exploration and exploitation phases is good when the inertia weight changes from iteration to iteration instead of staying the same. In each iteration, the “w” will be multiplied by a user-defined damping ratio and get smaller as the algorithm advances

Inertia weight damping ratio (wdamp)

0.9

4

Cognitive part acceleration coefficient (ccc)

2

The practical and precise search for the optimal global solution depends heavily on the appropriate regulation of the two acceleration factors. If ccc >> csp, each particle is considerably more drawn to its own personal best position, resulting in extreme wandering. On the other hand, if csp >> ccc, particles are more intensely enticed to the global best place, driving particles to pour prematurely towards optima. If ccc = csp, particles are allured towards the average of local best position Pbest, i, and global best

5

Social part acceleration coefficient (csp)

2

6

Solution/search space bounds

Select the bounds according to the criteria discussed in this article

7

Steady-state gain approximation

The steady-state value of the higher-order system is preserved closely in the simplified model, so the steady-state error is nearly zero

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Duddeti, B.B., Naskar, A.K. & Subhashini, K.R. Order Reduction of LTI Systems Using Balanced Truncation and Particle Swarm Optimization Algorithm. Circuits Syst Signal Process 42, 4506–4552 (2023). https://doi.org/10.1007/s00034-023-02304-7

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