Methodology

This chapter considers the modelling and control issues of a SMIB system. Thus, the work is split into two sections. The first section deals with the reduced-order modelling of a SMIB system considered from the literature. The SMIB presents a higher-order system when modelled. However, to design its controller, a low order model is required due to implementation issues of the controller. The GWO is taken up to reduce the parent SMIB model. Both the numerator and denominator polynomial coefficients are considered as unknowns and are determined with the help of the GWO applying constraints to retain dc gain, minimum phase and stability of the higher-order model. The pseudo random binary sequence (PRBS) was used as the input for both the original and reduced-order models containing unknown numerator terms. The difference between the responses is considered as the error function. Sum of square error (SSE) was minimized to obtain the numerator polynomial. It is worth mentioning here that the GWO was developed to emulate the leadership hierarchy and the hunting activity of grey wolves found in the northern part of America. The detailed algorithm along with its pseudo code can be found in Ref. [9] for interested researchers. Once the reduced system is developed, the second section of this work involves the controller design where the three-term controller coefficients are determined using the GWO method. The concept of the AMM technique is followed for the controller synthesis. As many as seven heuristic techniques such as Ant lion optimization (ALO), Dragonfly algorithm (DA), Moth flame optimization (MFO), Multi- verse optimizer (MVO), Grasshopper optimization algorithm (GOA), Sine cosine algorithm (SCA) and Salp swarm algorithm (SSA) are taken up for comparison to justify the effectiveness of the proposed methodology.

Initially, the reduced order systems are developed with the said methods and the error functions are evaluated. Further, these algorithms are run at least 30 times to provide a statistical measure of the fitness function. Once again, the representations of the minimum, maximum, average and standard deviation are tabulated. Since multiple data sets are compared, the Kruskal Wallis test [44] is carried out to test the validity of the outcome. In addition to this, the /^-values are calculated by applying the rank-sum test of Wilcoxon [45], which is once again a non-parametric test. Usually, the /^-values are considered to be meaningful if they are less than 0.05 for 95% confidence interval. If the /г-values are found to be greater than 0.05, then the outcomes of the experiments are insignificant. Since the proposed method is compared with manifold metaheuristic techniques, the Holm-Bonferroni correction [46] is incorporated in the Wilcoxon test to get modified /^-values. Moreover, quite a sufficient number as well as relevant time-domain and frequency-domain parameters are assessed for the reduced models in comparison to the original system. As many as five benchmark error performance indices are evaluated to show comparison with some of the recently developed metaheuristic approaches. Thereupon, the controller parameters are estimated with the help of the GWO in the AMM framework. In the controller synthesis problem, the responses of the plant-controller cascade and the chosen reference model are compared, setting to minimize the integral of square error (ISE) in order to determine the controller gains. In addition, the minimum fitness value is also reported.

Experimental Results and Discussion

A SMIB model, considered from the literature [47], is represented as

Since an eighth-order model is presented, it is quite obvious that the system needs to be reduced to a relatively lower order in order to develop a suitable controller. Applying the GWO technique, the reduced-order model is obtained as follows

In the above model, dc gain, minimum-phase feature and stability are preserved. Several methods such as ALO, DA, MFO. MVO, GOA, SCA and SSA are used for comparison whose models are given in Table 10.1. Further, the least fitness function value, SSE in this case, is also reported in this table.

TABLE 10.1

Reduced Model Representations and Their Error Function

Method

Reduced-Order Models

SSE

Proposed

0.19142

ALO

3.4751

DA

3.3577

MFO

2.6270

MVO

2.7145

GOA

2.3921

SCA

1.3534

SSA

3.0516

The bold value represents the minimum error value.

It is seen from Table 10.1 that the reduced system obtained by the proposed approach produces the least SSE. Thus, it can be inferred that the GWO method generated the best reduced model amongst the methods compared. Further, statistical measures of the error function are also studied, and the results of which are shown in Table 10.2. The four most popular indices are considered for the study, namely the lowest, highest, mean and the standard deviation. The first two basically denote the span of the error function while the standard deviation accounts for the stability of the algorithm. The lower the value of the standard deviation, the more stable is the algorithm. The best results i.e. the least value obtained in each column of the table are marked with the aid of bold letters.

From Table 10.2, it is apparent that the GWO outperforms all the reported algorithms in terms of best, worst, average and standard deviation values and hence indicated by bold letters in the table. Only the results of SCA are close to those of the GWO method in terms of the minimum fitness value. The standard deviation of the ALO algorithm is nearly close to that of the proposed technique, indicating that the algorithm is stable enough. Usually, the standard deviation value is less than the average value as found in the table, validating the theoretical concept as well. Since multiple algorithms are used for comparison with our method, the Kruskal Wallis test is carried out as a measure of non-parametric statistical inference for the validity of the results obtained. The test diagram is shown in Figure 10.2.

The Kruskal Wallis test diagram shown in Figure 10.2 clearly indicates that out of the seven metaheuristic algorithms used for comparison, the suggested GWO method proved significant compared to the six algorithms. In only one algorithm, namely Group 7, there seems to be some closeness of the data set. To check further, a Wilcoxon test based on rank-sum was conducted on the data samples and the p-values are reported in Table 10.3. A value less than 0.05 will be considered to be meaningful while a value than 0.05 will be taken up as insignificant. This limit is considered for 95% confidence interval, a popular one amongst the different confidence intervals.

All the p-values in Table 10.3 are less than 0.05. Thus, the results obtained by the proposed technique are significant with respect to all other algorithms. The same p-values in the table are merely accidental. A value lower or higher than that may not

TABLE 10.2

Statistical Analysis of the Fitness Function, SSE

Methods

Lowest

Highest

Mean

Std. Deviation

Proposed

0.19142

0.2635

0.2133

0.0083

ALO

3.4751

3.9430

3.7604

0.2034

DA

3.3577

4.5830

3.9883

0.5096

MFO

2.6270

3.6789

3.2688

0.4876

MVO

2.7145

4.2911

3.4013

0.5802

GOA

2.3921

4.4221

3.9015

0.8536

SCA

1.3534

3.5973

2.3523

0.8322

SSA

3.0516

3.9563

3.6230

0.3685

Kruskal Wallis test for checking the significance of mean ranks

FIGURE 10.2 Kruskal Wallis test for checking the significance of mean ranks.

TABLE 10.3

Calculations Showing p-Values Using the Non-parametric Wilcoxon Rank-Sum Test

Algorithm

ALO

DA

MFO

MVO

Proposed

1.0496e-12

GOA

I.0496e-I2

SCA

1.0496e-12

SSA

1 0496e-12

Proposed

1.0496e-12

I.0496e-I2

1.0496e-12

-

TABLE 10.4

Modified Wilcoxon Test Results with Holm-Bonferroni Corrections

Algorithm

p-Values after Holm-Bonferroni Corrections

h-Values

Proposed

10-" x

[0.7347 0.7347 0.6298 0.5248 0.4198 0.3149 0.2099J

[l 1 1 1 1 1 lj

be achievable from the given data set. The p-values are further modified applying Holm-Bonferroni corrections and quoted in Table 10.4. The Л-values representing whether the test of hypothesis is true or false are marked in the table, l’s recorded against each h-value correspond to statistically significant results, whereas 0’s denote insignificant results.

It is observed from the p-values in the table that they are quite less than 0.05 and hence significant. This is also indicated by the h-values where the Fs represent meaningful outcomes, and 0’s on the other hand indicate that they are insignificant. Moreover, the important time and frequency domain parameters of the reduced systems are provided in Table 10.5 to make a fair comparison with the original system.

From the results shown in Table 10.5, it is clearly visible that the GWO method produces the closest match to the original higher-order model in terms of both time- domain and frequency-domain parameters. The other methods with which the comparison is carried out are only close in terms of overshoot, undershoot, gain and phase margins. The rise time and settling time of the ALO, DA, MFO, MVO, GOA and SCA are really huge, thereby suggesting sluggish response. Only the rise time and the settling time of the SSA method are slightly better. The phase margins of the MFO, GOA and SSA show a wide deviation from that of the original model. The step responses of the original system and the reduced test system are shown in Figure 10.3 to further validate the results shown in Table 10.5 in terms of time- domain specifications.

It is quite clear from the time response curves of the original and the reduced systems in Figure 10.3 that the reduced-order model produced by the suggested technique closely matches the parent model. The frequency response is further plotted in Figure 10.4 to check the closeness of the proposed model in terms of the magnitude and phase plot of the Bode diagram.

The Bode diagrams of the parent and reduced system models show a very close resemblance as observed in Figure 10.4. Some of the well-known errors widely popular in the literature of control are calculated for each of the reduced order test systems and their outcomes are enumerated in Table 10.6. The best rather than the least values reported for each of these errors are indicated with the help of bold letters for the proper understanding of the readers.

TABLE 10.5

Quantitative Time- and Frequency-Domain Measures and Their Comparison with the Original System

Test

System

Methods

Rise Time (s)

Settling Time (s)

Overshoot

(%)

Undershoot

(%)

Gain

Margin (dB)

Phase Margin (°)

Original

1.0692

1.5686

0.7852

0

Inf

90.8510

Reduced

Proposed

1.1967

5.1427

2.2227

0

Inf

92.7294

ALO

273.2268

486.4874

0

0

Inf

93.1089

DA

409.1763

728.0001

0

0

Inf

96.3845

MFO

104.1833

182.9161

0

0

Inf

156.4374

MVO

70.0453

140.6177

0

0

Inf

95.5164

GOA

1.3223e+03

2.3453e+03

0

0

Inf

109.8870

SCA

40.9063

336.3791

0

0

Inf

93.3197

SSA

17.3216

30.5515

0

0

Inf

135.1152

Step response matching of parent and reduced models using the GWO technique

FIGURE 10.3 Step response matching of parent and reduced models using the GWO technique.

Bode responses of the original and their reduced systems obtained by applying the proposed method

FIGURE 10.4 Bode responses of the original and their reduced systems obtained by applying the proposed method.

TABLE 10.6

Representation of Popular Error Indices Using Different Methods

Methods

IAE

ITAE

ISE

USE

Hin( Norm

Proposed

0.0677

0.0503

0.0046

0.0037

0.1136

ALO

2.2142

1.2810

4.5083

2.4893

3.3427

DA

2.2095

1.2789

4.4813

2.4750

3.3155

MFO

2.0110

1.1807

3.7716

2.1086

2.9623

MVO

1.0732

0.8342

1.1443

0.9601

1.6727

GOA

2.1770

1.2630

4.3310

2.3989

3.2233

SCA

0.3876

0.3214

0.1766

0.1672

0.8452

SSA

1.8383

1.0611

3.1013

1.7002

2.7011

From Table 10.6, it is observed that the proposed GWO technique surpasses all other algorithms in terms of the error indices considered. The least value in each column is thus indicated with the help of bold letters. A reference model is then selected as per [48] to constitute the controller design of the reduced SMIB model. Generally, the response of the plant and controller with unknown parameters, connected in cascade, is compared to the response of the reference model. The objective is set in such a way that the plant and controller combination follows the response of the reference model approximately. The ISE is minimized to determine the controller parameters whose input-output relationship is defined by

The controller gains are thus determined and the results are shown in Table 10.7 along with the fitness function values. Similar to the model order reduction problem, in the controller design problem, the proposed methodology is compared with

TABLE 10.7

Different Controller Gains and Their Fitness Values

Methods

K,

K,

Ко

fmin

Proposed

0.12589

0.048128

3.0368e-06

6.6008e—09

ALO

2.8592

1.0965

1.1993e—06

0.00015018

DA

5.0000

2.4788

3.3968e-06

0.00020168

MFO

0.095166

3.2931

5.0366e-06

1.741e—07

MVO

0.05926

0.014016

0.00063065

0.00017156

GOA

2.9233

5.0000

0.21605

0.38052

SCA

0.10152

0.075265

4.6343e-05

4.5961e-07

SSA

4.2953

0.78715

0.0010865

0.016981

some of the widely cited heuristic techniques. The least fitness values reported in this table give an indication of the best controller parameters obtained, marked in bold.

In Table 10.7, the GWO method reports the least ISE value and hence the best choice for the controller amongst all other compared techniques. MFO and SCA methods only provide close match. The methods such as ALO, DA and MVO also yield satisfactory results. The results of GOA and SSA are far away from the best reported results. The one producing the best fitness value is also expected to closely match the response of the reference model. Thus, the GWO technique successfully addresses the modelling and control of SMIB test systems. Multiple test runs could also be performed for the controller synthesis part. In that case, parametric and non-parametric tests could be conducted to get better assessments. Moreover, new optimisation techniques such as the whale optimisation algorithm (WOA), Harris hawks optimisation (HHO), equilibrium optimizer (EO), marine predator algorithm (MPA) etc. may be applied to get still better performance of modelling and control aspects of SMIB systems. Obvious variants such as chaotic form, opposition-based methods and new hybrid combinations of these algorithms could be employed for the reduced-order modelling and controller synthesis problem.

 
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