Optimal Model Approximation of Linear Time-Invariant Systems Using the Enhanced DE Algorithm and Improved MPPA Method

Abstract

In this paper, the authors propose a novel model order reduction method integrating evolutionary and conventional approaches for higher-order linear time-invariant single-input–single-output (SISO) and multi-input–multi-output (MIMO) dynamic systems. The proposed method makes use of a differential evolution algorithm with enhanced mutation operation for the determination of reduced order model (ROM) denominator polynomial coefficients. In addition, an improved multi-point Padé approximation method is used to determine the optimal ROM numerator polynomial coefficients. The optimum property of the ROM is measured by minimising the integral square of the step response error between the original high-order dynamic system and the ROM. In the case of the MIMO system reduction approach, an optimal ROM transfer function matrix is determined by minimising a single objective function. This objective function is defined by a linear scalarising of the multi-step error function matrix components $\left( {E_{ij} } \right)$. The proposed method guarantees the preservation of the stability, passivity and accuracy of the original higher-order system in the ROM. The proposed method is validated by applying it to a ninth-order SISO system, as well as to the tenth- and sixth-order linearised single-machine infinite-bus power system model with and without an automatic excitation control system. The simulation results and the comparison of the integral square error and impulse response energy values of the ROM demonstrate the dominance of the proposed method over the latest reduction methods available in the literature.

Appendix

The single-machine infinite-bus (SMIB) power system, which is considered in Example 2, is shown in Fig. 15.

Fig. 15

Single-machine infinite-bus (SMIB) power system

The system under study consists of a three-phase, two-pole 160-MVA turbogenerator with an automatic excitation control system (i.e. a standardised IEEE Type-I exciter with rate feedback and power system stabilising signals) supplying power via a step-up transformer and a high voltage transmission line to an infinite grid. In Fig. 15, $X_{T}$ and $X_{L}$ represent the reactance of the transformer and the transmission line, respectively, and $V_{T}$ and $V_{B}$ are the generator terminal and infinite-bus voltages, respectively.

A detailed block diagram of the SMIB power system [25] is shown in Fig. 16. The numerical values of the parameters that define the total system and its operating point come from [24, 25] and are given below.

Fig. 16

Block diagram representation of the SMIB power system

The nomenclature of the above system parameters is as follows:

$\delta , \omega , V_{T} , P, Q$	$\begin{aligned} {\text{Synchronous}}\;{\text{machine}}\;{\text{torque}}\;{\text{angle}},\;{\text{Speed}},\;{\text{Terminal }}\;{\text{voltage}},\;{\text{Active}}\; {\text{and }} \hfill \\ \;{\text{Reactive}}\;{\text{power}} \hfill \\ \end{aligned}$
$K_{1} , K_{2} , K_{3} , K_{4} ,K_{5} , K_{6}$	${\text{Synchronous machine linear model parameters}}$
$H, T_{a} , T_{e} , T_{m}$	${\text{Synchronous}}\;{\text{machine }}\;{\text{inertia }}\;{\text{constant}},\;{\text{Accelerating}},\;{\text{Electrical}}\;{\text{and}}\;{\text{Mechanical}}\;{\text{torques}}$
$R_{e} , X_{e}$	${\text{Equivalent}}\;{\text{resistance }}\;{\text{and}}\;{\text{Reactance }}\;{\text{of}}\; {\text{external }}\;{\text{system}}$
$E_{q} , E_{FD} , \tau_{d0}$	$\begin{aligned} {\text{Voltage }}\;{\text{proportional}}\; {\text{to}}\; d - {\text{axis}}\;{\text{flux}}\;{\text{linkages}},\;{\text{Field }}\;{\text{voltage}}\;{\text{and}}\;{\text{Open - circuit }} \hfill \\ {\text{time}}\;{\text{constant}} \hfill \\ \end{aligned}$
$K_{E} , S_{E} , \tau_{E}$	${\text{Self-excited}}\;{\text{field }}\;{\text{constant}},\;{\text{Saturation}}\;{\text{function}}\;{\text{and}}\;{\text{Time }}\;{\text{constant}}\;{\text{of}}\;{\text{an}}\;{\text{exciter}}$
$K_{A} , \tau_{A} , V_{R}$	${\text{Regulator }}\;{\text{gain}},\;{\text{Time}}\;{\text{constant}}\;{\text{and}}\;{\text{Output }}\;{\text{voltage}}$
$K_{F} , \tau_{F}$	${\text{Rate }}\;{\text{feedback}}\; {\text{gain}}\; {\text{and}}\; {\text{Time}}\; {\text{constant}}$
$K_{R} , \tau_{R}$	${\text{Transducer }}\;\left( {\text{or}} \right)\;{\text{Filter}}\;{\text{gain}}\;{\text{and}}\;{\text{Time}}\;{\text{constant}}$
$K_{0} ,\tau_{0} , V_{s}$	${\text{Speed}}\;{\text{gain}},\;{\text{Reset }}\;{\text{time-lag}}\;{\text{constant}}\;{\text{and}}\;{\text{Stabiliser}}\;{\text{output}}\;{\text{voltage}}$
$\tau_{1} , \tau_{2} , \tau_{3} , \tau_{4}$	${\text{Lead }}\;{\text{and}}\;{\text{Lag }}\;{\text{time }}\;{\text{constants}}\; \tau \; {\text{of}}\; {\text{the}}\; {\text{power}}\; {\text{system}}\; {\text{stabiliser}}$
$s$	Laplace operator
$\Delta$	Incremental (step) change of input

The numerical values of the system parameters and operating point are as follows:

${\text{Synchronous }}\;{\text{machine}}$	$3{\text{ - phase}}, 160\; {\text{MVA}},\; {\text{power}}\; {\text{factor}} = 0.894,x_{d} = 1.7, x_{q} = 1.6, x_{d}^{\prime} = 0.254 \;{\text{p}} . {\text{u}} . ,$ $\tau_{d0} = 5.9, H = 5.4\; {\text{s}}, \omega_{R} = 314 \;{\raise0.7ex\hbox{${\text{rad}}$} \!\mathord{\left/ {\vphantom {{\text{rad}} {\text{s}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\text{s}}$}}$
${\text{Type - I}}\; {\text{exciter}}$	$K_{A} = 50, K_{E} = - 0.17, S_{E} = 0.95, K_{F} = 0.04, K_{R} = 1,K_{0} = 1, \tau_{A} = 0.05,$ $\tau_{E} = 0.95, \tau_{F} = 1.0, \tau_{R} = 0.05, \tau_{0} = 10.0, \tau_{1} = \tau_{3} = 0.44, \tau_{2} = \tau_{4} = 0.092\;{\text{s}}$
${\text{External }}\;{\text{system}}$	$R_{e} = 0.02, X_{e} = 0.4 \;{\text{p}} . {\text{u}} .\left( {{\text{on}}\;160 \;{\text{MVA }}\;{\text{base}}} \right)$
${\text{Operating}} \;{\text{point}}$	$P_{o} = 1.0, Q_{o} = 0.5, E_{\text{FDo}} = 2.5128, E_{qo} = 0.9986, V_{to} = 1.0, T_{mo} = 1.0\;{\text{p}} . {\text{u}} . ,$ $\begin{aligned} \delta_{o} = 1.1966\; {\text{rad}}, K_{1} = 1.133, K_{2} = 1.3295, K_{3} = 0.3072, K_{4} = 1.8235, \hfill \\ K_{5} = - 0.0433, K_{6} = 0.4777. \hfill \\ \end{aligned}$

The system in Fig. 16 can be described in state-space form, as in Eq. (1). The state vector $X\left( t \right)$ is defined with state variables as $X^{\text{T}} \left( t \right) = \left[ {E_{q} \omega \delta v_{1} v_{R} E_{FD} v_{2} v_{3} v_{4} v_{5} } \right]$. The input and output vectors are given by $U^{\text{T}} \left( t \right) = \left[ {\Delta V_{\text{ref}} \Delta T_{m} } \right] \;{\text{and}}\; Y^{\text{T}} \left( t \right) = \left[ {\delta V_{t} } \right]$. For the system parameters and operating points described, the numerical values of the system, input, output and feedforward matrices $A, B, C \;{\text{and}}\; D$, respectively, are given below.

The system, input and output matrices of the SMIB without AECS (i.e. a standard IEEE Type-I exciter without rate feedback (RF) and without power system stabiliser (PSS)) are represented by $A^{\prime}, B^{\prime}\;{\text{and}}\; C^{\prime}$, respectively. The state-space representation of the SMIB system without AECS yields a sixth-order model of the following form:

$$ \left. {\begin{array}{*{20}c} {\frac{{{\text{d}}\tilde{X}\left( t \right)}}{{{\text{d}}t}} = A^{\prime} \cdot \tilde{X}\left( t \right) + B^{\prime} \cdot U\left( t \right)} \\ {Y\left( t \right) = C^{\prime} \cdot \tilde{X}\left( t \right) + D^{\prime} \cdot U\left( t \right)} \\ \end{array} } \right\} . $$

(33)

The state vector $\tilde{X}\left( t \right)$ is defined with state variables as $\tilde{X}^{\text{T}} \left( t \right) = \left[ {\tilde{E}_{q} \tilde{\omega } \tilde{\delta } \tilde{v}_{1} \tilde{v}_{R} \tilde{E}_{\text{FD}} } \right]$.The input and output vectors are given by $U^{\text{T}} \left( t \right) = \left[ {\Delta V_{\text{ref}} \Delta T_{m} } \right]\; {\text{and}}\; Y^{\text{T}} \left( t \right) = \left[ {\tilde{\delta } \tilde{V}_{t} } \right]$, respectively.