Introduction

Usually, LSS when modeled will turn out into HOS because of large parameters involvement. Analysis of such HOS is tedious, complex, time-consuming, costlier, and requires more storage capacity. Over the last few decades, MOR techniques are developed to address the above issues. Hence, MOR techniques have been paid much attention in the field of control engineering to simplify large-scale practical systems like nuclear reactors, jet air engines, boilers, and generators connected to infinite bus systems. The biggest advantage of the MOR technique is the developed ROM provides the same behavior as its original HOS. Many MOR techniques have been proposed by several researchers in both time and frequency domains. The benchmark method that was developed by using Routh’s table is presented in [1]. This method was realized as the prominent MOR technique in the field of control systems and it states that the ROM derived from the HOS is always stable when the original HOS is stable, and ROM can also be derived by using impulse energy approximation. Later, another method that uses both Routh's table and Pade approximation is proposed in [2]. This model gives exact approximations without altering the characteristics of the original HOS of both SISO and MIMO systems. After that, a paper that describes a detailed narration of the MOR is outlined in [3]. The authors in [4] proposed a method in which the reduced transfer function of the HOS is determined from the Routh stability array by using an eighth-order model. Later, an improved version of the Pade approximation is depicted in [5, 6]. A paper that focuses on the derivations of transfer functions from the HOS is presented in [7]. A hybrid method of using Routh's stability equation method and Pade approximation is proposed in [8] which is simple and gives effective ROM for nonlinear systems also. In addition to the above, several classical MOR techniques [9,10,11,12,13,14,15,16,17,18,19,20,21,22] are reported in the literature. The application of MOR techniques to various practical systems is explored in [23,24,25,26,27]. However, involvement of more mathematical calculations, complexity and time consummation are the major drawbacks observed in classical techniques. In recent years, bio-inspired algorithms have emerged as powerful tools for solving complex problems. They were proven to address the issues listed above. Many heuristic algorithms are developed for the MOR techniques. Particle Swarm Optimization (PSO) [28, 29], Differential Evolution (DE) [30], Cuckoo Search Algorithm (CSA) [31,32, 33], Fire-Fly Algorithm (FFA) [33], Bat Algorithm (BA) [34], Big-Bang Crunch Algorithm (BBCA) [35], Enhanced DE[36, 37], PSO-DV[38] are some of the algorithms that are developed for the MOR. Though several methods are available, yet, an effective MOR technique for complex large-scale practical systems is still indispensable for getting the robust operation. Moreover, the above methods suffer from various drawbacks like premature convergence, not having good balance between exploitation & exploration stages, involvement of more design parameters, and steps that makes the optimization process complex and fails in evolving global best solutions. This motivates the authors to propose an effective method to address the above issues. Recently, IWOA [39] is proven its potential in solving complex engineering problems and addressed majority of the issues mentioned above. Hence, IWOA is considered to derive the ROM of some complex LSSs. The remainder of the paper is as follows. The second section depicts the statement of the problem, while the third section illustrates an overview of WOA. The fourth section explains the proposed IWOA for MOR and the performance analysis of IWOA on the test functions of CEC14 and CEC17. The fifth section presents the simulation results of three examples and the conclusion part is given in section six.

Problem statement

In this work, the authors considered some of the practical LSSs as examples to test the worth of the proposed IWOA method. The power system formulated by a modified single generator connected to an infinite bus system popularly known as Single Machine Infinite Bus System (MSMIB) and an air core of the transformer has been considered as the practical test examples to test the proposed IWOA technique. The MSMIB system is realized as a MIMO system, and the air core transformer linear section is realized as a SISO system. The IWOA is used to determine the coefficients of ROM of original higher-order MIMO and numerator and denominator coefficients of original higher-order SISO systems. Both the SISO and MIMO systems are represented as given below.

Consider a \(n^{th}\)- order SISO can be represented by

$$ G_{n} (s) = \frac{Y(s)}{{U(s)}} = \frac{{\sum\limits_{i = 0}^{n - 1} {a_{i} s^{i} } }}{{\sum\limits_{j = 0}^{n} {b_{j} s^{j} } }} $$
(1)

where the order of numerator and denominator polynomial are i = 0,1,2,3,….n-1 & j = 0,1,2,3,….n, respectively. The equivalent \(r\;th\) (r < n) ROM order to be derived, is represented by

$$ R_{r} (s) = \frac{{Y_{r} (s)}}{{U_{r} (s)}} = \frac{{\sum\limits_{i = 0}^{r - 1} {c_{i} s^{i} } }}{{\sum\limits_{j = 0}^{r} {d_{j} s^{j} } }} $$
(2)

where the order of numerator and denominator polynomial are i = 0,1,2,3,….r-1 & j = 0,1,2,3,….r, respectively. And any MIMO plant, G having the order of \(n < \infty\), can be described by set of equations

$$ \mathop X\limits^{.} = {\text{AX}} + {\text{BU}} $$
(3)
$$ \,Y = {\text{CX}} + {\text{DU}} $$
(4)

where \(A\) is \(m \times m\), \(B\) is \(m \times 1\), \(C\) is \(m \times r\), and D is \(m\) × \(m\) matrices, respectively.

The equivalent ROM of the original MSMIB system is assumed as Gr, having the order of nr < n. The ROM to be determined has the same features as G, represented by the set of equations

$$ \mathop {X_{r} }\limits^{.} = A_{r} X_{r} + B_{r} U $$
(5)
$$ \,Y_{r} = C_{r} X_{r} + D_{r} U $$
(6)

where \(A_{r}\) is \(r \times r\) matrix, \(B_{r}\) is \(r \times 1\) matrix, \(C_{r}\) is \(1 \times r\), \(D_{r}\) is \(r \times r\) matrices of the ROM.

Overview of WOA

The WOA [40] was developed in the year 2016 after seeing the attacking behavior of the Humpback whales. These whales attack the prey in a nine-shaped path known as a bubble-net feeding mechanism. The whale generates a lot of bubbles along the nine-shaped path to form a net-type configuration to reach the target. The WOA is simple to understand and easy for the development of coding. The number of design parameters is less when compared to other algorithms. The rate of convergence is fast and takes minimum simulation time for getting optimum solutions (Fig. 1).

Fig. 1
figure 1

The hunting strategy of Humpback Whales

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Steps involved in the WOA algorithm

Initially, initial parameters like population size, maximum iterations, minimum and maximum values of control variables, and other parameters of WOA are selected. Here, the ROM coefficients are considered as control variables. The initial solutions are generated randomly on a given objective function. The following steps depict various steps involved in determining the optimum solutions using WOA.

Step 1 A Shrinking encircle mechanism for the position update of hunting agent

As the initial solutions are generated randomly, the current solutions are considered as best solutions and the position of the whale is updated by using this mechanism. To attack the target, the whale follows a path which is defined by the equation below.

$$ \mathop S\limits^{ \to } = R = R\;\overrightarrow {P} * (t) - \overrightarrow {P(t)} $$
(7)
$$ \overrightarrow {P}_{(t + 1)} = \overrightarrow {P}^{ * } (t) - A\;\overrightarrow {S} $$
(8)

All the hunting agents change their position by using the above equations. Here, \(\mathop S\limits^{ \to }\) depicts the distance between the whale and the target, R and A are the random numbers that are varied randomly during the optimization process, \(t\) be the current iteration, \(P^{ * }\) corresponding to the optimum solution attained until now, \(\mathop P\limits^{ \to }\) stands for the position vector, represents the absolute value. Here,\(A\) \(R\) are characterized as

$$ A \, = \, 2a \, r \, - \, a $$
(9)
$$ R = 2r $$
(10)

Here,\(\mathop A\limits^{{}}\) is the arbitrary number that alters between 2 and 0 and \(\mathop r\limits^{{}}\) is an arbitrary number that varies between (0,1). In every iteration process, the values of \(a,A,R\) are updated for each hunting agent. The present hunting agents update their location by Eq. 7 if the value of \(A\) is less than 1, and they follow the below equation otherwise.

$${{S = R\;\overrightarrow {P}_{{{\text{rand}}}} - }} \overrightarrow {P}$$
(11)
$$ \overrightarrow {P} (t + 1) = \overrightarrow {P}^{*} (t) - A\overrightarrow {S} $$
(12)

where \(\overrightarrow {\text{P}} \;_{{{\text{rands}}}}\) is the random position vector which is selected in during the process from the present populations?

Step 2

A spiral mechanism for the position update of the hunting agent.

Hunting agents or ROM coefficients follow both shrinking cycle and spiral-shaped path during the search for their prey. To simulate the spiral-shaped track between the whale and target, a spiral equation is formulated. All the hunting agents update their position based on the formulated equation given below.

$$ \overrightarrow {P} (t + 1)\; = \;\overrightarrow {S} e^{bl} \cos (2\pi l) + \overrightarrow {P}^{ * } (t) - \;A\;\overrightarrow {S} $$
(13)

where

$$ \overrightarrow {S} = \overrightarrow {P}^{*} (t) - \overrightarrow {P} (t) $$
(14)

Here 'l' is a stochastic limit that diverges between 0 and 1. Both spiral itinerary and spiral itinerary are merged by giving a 50% probability to both of them to update the positions. Finally, all the hunting agents follow the path described by the equation below.

$$\vec{P}(t + 1) = \left\{ {\begin{array}{*{20}ll} {\vec{P}^{*} (t) - A{\mkern 1mu} \vec{S}} & {if\;\delta < 0.5} \\ {\vec{S}e^{{bl}} \cos (2\pi l) + \vec{P}^{*} (t) - A{\mkern 1mu} \vec{S}} & {f\delta \ge < 0.5} \\ \end{array} } \right.$$
(15)

where \(\delta\) is an arbitrary value that alters between 0 to 1.

IWOA for the MOR

The exploration and the exploitation stages are two important phases of any heuristic search algorithm. Maintenance of a good balance between these stages finds global solutions for complex engineering problems. Every particle finds a local best solution in the exploitation stage and a global best solution in the exploration stage. S. Mirjalili who developed the WOA has mentioned in his recent paper [31] that WOA fails to find the global optimum solutions as the solution to be updated in WOA for the next iteration depends upon the previous solution or current best solution. If the obtained current solution is not found to be the best solution, then the remaining solutions follow the current solution, and then there may be a chance of failure of getting global solutions. It is also mentioned that by improving the exploitation rate of the WOA, the strength of WOA could be improved. The convergence rate of WOA depends upon the best solutions attained so far. Hence, if the current best solutions obtained so far are not accurate then the algorithm may fail to find the global best solutions. This is the major disadvantage of the WOA. To address these issues, IWOA has been developed in this paper. In the IWOA, a weighting factor is added to the updating agent to accelerate the exploitation rate of the algorithm and thence to get the improvement in searching capability. The value of this factor is selected in such a way that it varies between 0.9 and 0.4 during the optimization process. The key function of the factor is to control the current best solutions so that they do not fall in local optima. It provides a good equilibrium between the stages of both exploration and exploitation. And finally, the algorithm can find the global optimum solutions. It was the change done to the WOA to attain IWOA. All the hunting agents update their positions using the following equations.

$$ \overrightarrow {S} = Rw\overrightarrow {P}^{ * } (t) - \overrightarrow {P} (t) $$
(16)
$$ \overrightarrow {P} (t + 1) = \overrightarrow {wP}^{ * } (t) - A\overrightarrow {S} \, $$
(17)

Here, \(w\) is the weighing factor which is described as,

$$ \left\{ \begin{gathered} w = \beta + \alpha \times {\text{rand}}\;(0,1) \hfill \\ \beta = \beta_{\min } + (\beta_{\max } - \beta_{\min } ) \times {\text{rand}}\;(0,1) \hfill \\ \end{gathered} \right. $$
(18)

where \(\beta_{\min } ,\;\beta_{\max } \;{\text{and}}\;\beta\) are the lower, upper, and mean values of the weighing factor. Here, α represents a constant number and the value is selected as 0.4 and rand is a numeral value that alters between 0–1 arbitrarily. During the optimization process, the hunting agents get minimum and maximum values of weights to fiddle with the weighting factor for the global optimization process. The spiral mechanism for the IWOA is modeled as

$$ \vec{P}(t + 1) = \vec{S}{\mkern 1mu} e^{{bl}} \cos (2\pi l) + w\vec{P}^{*} (t) - A{\mkern 1mu} {\mkern 1mu} \vec{S} $$
(19)
$$ {\text{Where}}\; \overrightarrow {S} = w\overrightarrow {P}^{ * } (t) - \overrightarrow {P} (t) $$
(20)

In the IWOA, the whales update their positions by following both shrinking encircle and spiral mechanisms to reach the target mechanism. Hence, to account for this, a 50% probability has been assigned to both the mechanisms. The whales update their positions by using the below equation if A is < 1; otherwise, they update their positions by using Eq. 11.

$$ \overrightarrow {P}_{(t + 1)} = \left\{ {\begin{array}{*{20}ll} {\left\{ w \right.\overrightarrow {P}^{*} (t) - A\overrightarrow {S} } & {if\;\delta < 0.5} \\ {\overrightarrow {S} e^{bl} \cos (2\pi l) + w\overrightarrow {P}^{*} (t) - A\overrightarrow {S} } & {if\;\delta \ge < 0.5} \\ \end{array} } \right.\, $$
(21)

Performance analysis of IWOA

Before deriving the ROMs, the worth of IWOA is tested with the standard benchmark test functions of CE14 and CEC17. These test functions are highly nonlinear and getting a minimum fitness value is a challenging task. A total of thirty-three test functions are considered and tested with IWOA. The IWOA is applied to all test functions to investigate its potential of the IWOA. The test results in terms of minimum fitness value for all the test functions are presented in Tables 1 and 2, respectively. The test results of CEC 14 and CEC17functions are displayed in Tables 1 and 2, respectively. The results are compared with PSO, DE, and WOA-based results to show the superiority of the proposed IWOA. It is observed from the results that the fitness values obtained for the functions f1(x), f2(x), f3(x,), f4(x,), f7(x), f8(x) of CEC14 and test functions f1(x), f3(x), f4(x), f5(x), f6(x), f7(x), f9(x), f10(x), F11(x) of CEC17 are minimum in case of IWOA when compared to other algorithms. Hence, it is realized that the IWOA is succeeded in finding the optimum solutions for the majority of the test functions when compared to other algorithms. Therefore, IWOA is utilized to design the ROM coefficients for the HOSs. The flowchart for the IWOA to get ROMs is depicted in Fig. 2. The convergence plots of some of the test functions are shown in Figs. 3 and 4. From the plots, it is proven that the IWOA finds the optimum solutions when compared to other techniques.

Table 1 Test results of CEC14 test functions
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Table 2 Test results of CEC17 test functions
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Fig. 2
figure 2

Flowchart to determine the ROM coefficients

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Fig. 3
figure 3

Convergence plots of various CEC 14 test functions

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Fig. 4
figure 4

Convergence plots of various CEC 14 test functions

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Implementation of IWOA for the MOR

Here, IWOA is used to determine the elements of \(A_{r}\), \(B_{r}\), \(C_{r}\), and \(D_{r}\) of the MHP model. The following are various steps involved to get ROM using Improved WOA.

Step1: Initialization

The algorithm starts with the initialization process. Here, elements of the original HOSs are chosen as design parameters and are optimized to get desired ROM that preserves most of the dynamic behavior of the original HOS. Here, a fourth-order ROM is to be derived from their HOS using IWOA. The initial values of control variables are arbitrarily generated by using the expression given below.

$$ Z_{i + 1} = Z_{i}^{{{\text{min}}}} + {\text{rand}}. (Z_{i}^{{{\text{min}}}} - Z_{i}^{{{\text{max}}}} ) $$
(22)

where Z defines the control variable which is to be updated in every iteration, rand is an arbitrary number that varies between 0 to 1. Here \(Z_{i}^{{{\text{min}}}}\) and \(Z_{i}^{{{\text{max}}}}\) are the minimum and maximum values of the control variables, respectively. The total number of iterations is taken as 100; population size is selected as 50.

Step 2

Evaluating the objective function.

Here, to determine the ROM coefficients of the MSMIB system, ISE is used as the objective function. The reason behind the selection of this function is it is the most popular and efficient measure of the dynamic execution of the system. The below equation depicts the expression for ISE.

$$ J_{1} = \int\limits_{0}^{t} {(e)^{2} } = \int\limits_{0}^{t} {\left[ {y(t) - y_{r} (t)} \right]}^{2} $$
(23)

Here, ‘e’ is the error between the step responses of the HOS and reduced order system (ROS). Here \(y(t)\) and \(y_{r} (t)\) are the step responses of the original HOS and ROMs, respectively. IWOA is used to minimize the objective function \(J_{1}\) so that the derived ROM can retain the maximum dynamic response of the original HOS.

Step 3

Shrinking encircle mechanism for the position update of hunting agent.

Here, the ROM coefficients which are to be optimized are selected as search agents. All the ROM coefficients follow Eqs. 16–18 to update their positions.

Step 4

Spiral mechanism for the position update of the hunting agent.

All the ROM coefficients to be optimized follow Eqs. 19–20 to update their positions. IWOA has been run several times until the optimized ROM coefficients are obtained. Figure 2 shows the flow chart for IWOA to optimize the ROM coefficients.

Numerical examples and results

To investigate the performance of the proposed IWOA-based MOR technique, three practical examples are considered; out of them, two are SISO systems and one is a practical MIMO system (MSMIB system).

Example 1 single input single output (SISO) system

To start with, a seventh-order HOS is taken as one of the test examples collected from the literature. Here, the coefficients of the numerator and denominator of the HOS are assigned as the tunable elements to minimize the fitness function. The boundary values of tunable elements are selected and the process of optimization is started with the generations of random solutions. The original HOS of the SISO system is given by

$$ G_{7} (s) = \frac{\begin{gathered} 2000s^{6} + 121700s^{5} + 1.21 \times 10^{6} s^{4} + 7.454 \times 10^{6} s^{3} + 5.527 \times 10^{7} s^{2} \hfill \\ + 3.156 \times 10^{7} s + 2.075 \times 10^{6} \hfill \\ \end{gathered} }{{s^{7} + 65.85s^{6} + 984.2s^{5} + 12130s^{4} + 97300 + 429400 + 2.0188 \times 10^{6} s + 999500}} $$
(24)

The optimized second-order ROM of the original HOS by using IWOA is shown below..

$$ R_{2} (s) = \frac{45s + 9.8332}{{0.1s^{2} + 0.035s + 3.712}} $$
(25)

The second-order ROMs of classical methods are compared with the proposed IWOA-based ROM (IWOA-ROM) and are depicted in Table 3. The ISE values are also presented in the table. Table 3 represents the comparison of ROMs of classical and heuristic techniques with the proposed ROM.

Table 3 Comparison of ROMs with proposed IWOA
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The step responses of original HOS, IWOA-ROM and ROMs of other heuristic and classical techniques are depicted in Fig. 5. Figure 6 depicts the bode plot of original HOS, IWOA-ROM, and ROMs of other heuristic and classical techniques. From the simulation results, it is proven that the IWOA-ROM exactly matches with original HOS in the majority of the regions, i.e., at both transient and steady states. The phase margin and gain margin of the IWOA-ROM closely coincide with the original system when compared to other algorithms. Hence, it is observed that the IWOA gives optimum ROM than other techniques in both time and frequency domains. The convergence plots of the IWOA, WOA, DE, and PSO algorithms are shown in Fig. 7.

Fig. 5
figure 5

Comparison of IWOA-ROM with ROMS of other techniques for the step input

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Fig. 6
figure 6

Comparison of IWOA-ROMs with ROMs of other techniques with a bode plot

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Fig. 7
figure 7

Convergence plots of IWOA and other techniques

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Example 2

To investigate the worth of the IWOA, here another example of a practical case study is considered. The linearized section of the air-core transformer, which consists of ten segments, is considered as a second example. The transfer function of the air core transformer is obtained as.

$$ G(s) = \frac{\begin{gathered} (s^{9} + 510.1s^{8} + 1.106e5 \, s^{7} + 1.33 \, e7 \, s^{6} + 9.691 \, e8 \, s^{5} + 4.393 \, e10 \, s^{4} + \hfill \\ 1.223 \, e12 \, s^{3} + 1.981 \, e13 \, s^{2} + 1.652 \, e13 \, s \, + 5.211 \, e14) \hfill \\ \end{gathered} }{\begin{gathered} (s^{10} + 529.8s^{9} + 120200s^{8} + 1.527 \, e7 \, s^{7} + 1.191 \, e9 \, s^{6} + 5.892 \, e10 \, s^{5} + 1.842 \, e12 \, s^{4} \hfill \\ + 3.513 \, e13 \, s^{3} + 3.784 \, e14 \, s^{2} + 1.965 \, e15 \, s + 3.32 \, e15) \hfill \\ \end{gathered} } $$
(26)

The IWOA has been run several times until the optimized second-order ROM is attained. The optimized ROM in the second-order form using IWOA is depicted below.

$$ R_{2} (s) = \frac{0.6318s + 7.112}{{0.6419s^{2} + 16.58s + 45.35}} $$
(27)

The derived second-order IWOA-ROM is compared with other ROMs which were determined using WOA, DE, PSO, and classical methods that existed in the literature. The details of ROMs are reported in Table 4. Table 4 presents the comparison of the IWOA-ROM with the ROMs of the other techniques which are available in the literature. The ISEs of all the techniques are also depicted in the table. It is examined from the results that the IWOA-ROM has provided the smallest amount of ISE value when contrasted to other methods. The step responses of original tenth-order HOS, IWOA-ROM and other ROMs are shown in Fig. 8. It is proven from the responses that the proposed IWOA-ROM exactly matches with the original HOS at all the regions, i.e., at both transient and steady states. Hence, it can be concluded that the proposed IWOA has succeeded in finding an optimum ROM when compared to other techniques (Figs. 9 and 10).

Table 4 Comparison of IWOA-ROMs with some familiar ROMs
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Fig. 8
figure 8

Comparison of IWOA-ROM with ROMS of other techniques for the step input

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Fig. 9
figure 9

Bode response comparisons of original HOS and Proposed ROMs

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Fig. 10
figure 10

Convergence plots of proposed and other algorithms

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Example 3

MSMIB system (MIMO system).

This MSMIB system gives the ninth order when it is modeled and is taken as the third example to derive its equivalent fourth-order ROM. The main objective of reducing the ROM into the fourth order is to retain the majority of the dynamic performance of the original HOS. Under this example, three typical operating conditions have been considered and subsequent ROMs are derived to examine the strength of the IWOA technique. Table 5 shows the operating conditions taken for deriving G-constants. The block diagram of the MHP model for the MSMIB system is depicted in Fig. 11. The MHP model is formulated using the G-constants, which can be determined from the following expressions (Fig. 12).

Table 5 Operating conditions for MSMIB system
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Fig. 11
figure 11

The block diagram Heffron-Phillip's model

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Fig. 12
figure 12

Convergence plots of IWOA, WOA, DE, and PSO techniques

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For any operating condition, the G-constants are represented as

$$ G_{1} = \frac{{V_{{s_{o} }} E_{{q_{o} }} {\text{si n}} \delta_{s } }}{{X_{q} + X_{t} }} + \frac{{X_{q} - X^{\prime}_{d} }}{{X_{t} + X^{\prime}_{d} }}V_{{s_{{}} }} {\text{sin}} \delta_{s} $$
(28)
$$ G_{2} = \frac{{X_{q} + X_{t} }}{{X_{t} + X^{\prime}_{d} }} i_{q} $$
(29)
$$ G_{3} = \frac{{X_{t} + X^{\prime}_{d} }}{{X_{d} + X_{t} }} $$
(30)
$$ G_{4} = \frac{{X_{d} - X^{\prime}_{d} }}{{X_{t} + X^{\prime}_{d} }} V_{s} E_{{q_{{}} }} {\text{si n}} \delta_{{s_{{}} }} $$
(31)
$$ G_{5} = \frac{{X_{q} V_{{d_{{}} }} V_{s} {\text{cos}} \delta_{s} }}{{(X_{q} + X_{t} ) V_{t} }} - \frac{{X^{\prime}_{d } V_{{s_{{}} }}\; {\text{sin}} \delta_{s} }}{{(X_{t} + X^{\prime}_{d} ) V_{t} }} $$
(32)
$$ G_{6} = \frac{{X_{t} }}{{X_{t} + X^{\prime}_{d} }}\frac{{V_{{q_{{}} }} }}{{V_{{t_{{}} }} }} $$
(33)
$$ G_{{v_{1} }} = \frac{{E_{{q_{o} }} {\text{sin}} \delta_{s} }}{{(X_{t} + X_{q} )}} - \frac{{(X_{q} - X^{\prime}_{d} ) I_{{q_{{}} }} {\text{cos}} \delta_{s} }}{{(X^{\prime}_{d} + X_{t} )}} $$
(34)
$$ G_{{v_{2} }} = - \frac{{(X_{d} - X^{\prime}_{d} )\cos \delta_{{s_{o} }} }}{{(X^{\prime}_{d} + X_{t} )}} $$
(35)
$$ G_{{v_{3} }} = \frac{{X_{q} V_{d} {\text{sin}} \delta_{s} }}{{X_{q} + X_{t} }} + \frac{{X^{\prime}_{d} V_{{q_{{}} }} {\text{cos}} \delta_{s} }}{{(X_{t} + X^{\prime}_{d} ) V_{{t_{{}} }} }} $$
(36)

where \(E_{{q_{o} }} = E^{\prime}_{{q_{o} }} - (X_{q} - X^{\prime}_{d} )i_{d}\). For any operating condition of the MSMIB system, \(V_{S}\) is the transformer secondary bus voltage, \(\delta_{s }\) is the load angle measured to transformer secondary voltage,\(E_{q}^{^{\prime}}\) is the field flux transient emf, \(X_{q}\) is the reactance of q-axis, \(X_{d}\) is the reactance of d-axis, \(X_{d}^{^{\prime}}\) is the reactance of q-axis, id is the stator current of d-axis, iq is the stator current of q-axis, \(E_{fd}\) is the field voltage, \(T_{e}\) is the time constant of exciter, \(K_{e}\) is gain of exciter, \(V_{ref}\) is the reference voltage, \(V_{pss}\) is the input of the PSS, \(V_{t}\) is the terminal voltage of the generator, \(X_{t}\) is the reactance of the transformer. The K-constants for the three operating conditions are determined by using the above expressions and listed in Table 6. Table 6 shows the G-constants determined for the considered operating conditions. Using the K- constants and the machine data, the state space representations of the MHP model for all three operating conditions have been derived and are shown in Appendix. The fourth-order ROMs of HOS in state space for three operating conditions using the IWOA technique are represented form in Table 7. Table 7 depicts the state space representation of IWOA-ROMs.

Table 6 K- constants for operating conditions of MSMIB system
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Table 7 Optimized IWOA-ROMs in state space representation
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Comparison plots of HOS and ROM for IWOA, WOA, DE, and PSO-based algorithms for the three loading conditions are shown in Fig. 13. Plots displayed in Fig. 13a–c) give the step responses of the loading condition. Plots in Fig. 13d–f depict the step responses of loading condition 2, and the plots for the third loading condition are shown in Fig. 13g–i. The dark blue color depicts the response of the original HOS for the step input, the pink color stands for the response of IWOA-ROM for the step input, the red color plot gives the step response of the WOA-ROM, and the green color corresponds to the response of the PSO-ROM for the step input. All step responses are compared on a common time scale to test their performances. Simulation results clearly show that the behavior of the ROM obtained from IWOA closely matches the behavior of the original HOS when compared to the other ROMs at transient and steady-state conditions for loading condition one. Similarly, ROMs second and third loading conditions also exhibited the same type of performance when compared to the other algorithms. Hence, it is concluded from the results that the ROMs derived from the IWOA technique retain most of the dominant behavior of its original higher-order plant at all the loading conditions when compared to the ROMs derived from WOA, DE, and PSO algorithms.

Fig. 13
figure 13

Step response comparisons of original HOSs and ROMs for the three Step loading conditions

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Further, to show the efficacy of the proposed work, bode plots of the obtained ROMs and the original HOSs are shown in Figs. 14, 15 and 16. Plots of four important states: \(\Delta \delta ,\;\Delta \omega ,\;\Delta Eq,\;\Delta E_{fd}\) of original HOS and ROMs are presented in the figures. The black color plot represents the response of the original ninth-order MSMIB system, the dark pink color represents the response of the proposed IWOA-ROM, the blue color plot represents the response of the WOA-ROM, the red color plot depicts DE-ROM and the response of PSO- ROM is represented in green color. The results prove that the phase margin and gain margins of the proposed IWOA- ROM of four states match with the original HOSs’ states in a better way when compared to other ROMs.

Fig. 14
figure 14

Bode plot responses of loading condition 1

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Fig. 15
figure 15

Bode plot responses of loading condition 2

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Fig. 16
figure 16

Bode plot responses of loading condition 3

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Conclusions

IWOA-based MOR technique is proposed on three typical complex LSSs. ISE is considered as an objective function to optimize the coefficients of ROMs of the original HOS systems. SISO and MIMO systems are considered to study the comparative analysis. The MHP model, which is having ninth order, air core of the transformer of the order of tenth order, and seventh-order systems are taken as test cases to examine the performance of the proposed IWOA-based MOR technique. Performance of the ROMs derived using WOA, DE, and PSO and other classical techniques has been contrasted with IWOA- ROM. The test results conclude that the IWOA-ROM exhibited better performance when compared to other methods.