Application of modified multi-verse optimization for temperature control in thermal power plant condensers.

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Learn more: PMC Disclaimer | PMC Copyright Notice Sci Rep . 2026 Mar 6;16:12409. doi: 10.1038/s41598-026-40559-7 Search in PMC Search in PubMed View in NLM Catalog Add to search Application of modified multi-verse optimization for temperature control in thermal power plant condensers Sweta Panda Sweta Panda 1 Department of Electrical Engineering, VSSUT Burla, Burla, Odisha India Find articles by Sweta Panda 1 , Soumya Ranjan Das Soumya Ranjan Das 2 Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, India Find articles by Soumya Ranjan Das 2, ✉ , Arun Kumar Sahoo Arun Kumar Sahoo 3 Department of Electrical Engineering, IIIT Bhubaneswar, Bhubaneswar, Odisha India Find articles by Arun Kumar Sahoo 3 , M UmaMaheswar Rao M UmaMaheswar Rao 4 Department of CGDT/ Design, IIT Hyderabad, Hyderabad, India Find articles by M UmaMaheswar Rao 4 , Norah Saleh Alghamdi Norah Saleh Alghamdi 5 Department of Computer Sciences, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia Find articles by Norah Saleh Alghamdi 5 , Wattana Viriyasitavat Wattana Viriyasitavat 6 Department of Statistics, Chulalongkorn University, Bangkok, Thailand Find articles by Wattana Viriyasitavat 6 , Gaurav Dhiman Gaurav Dhiman 7 Department of Computer Science and Engineering, Yuan Ze University, Tao Yuan, Taiwan 8 Centre of Research Impact and Outcome, Chitkara University, Punjab 140417 Rajpura, India Find articles by Gaurav Dhiman 7, 8 Author information Article notes Copyright and License information 1 Department of Electrical Engineering, VSSUT Burla, Burla, Odisha India 2 Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, India 3 Department of Electrical Engineering, IIIT Bhubaneswar, Bhubaneswar, Odisha India 4 Department of CGDT/ Design, IIT Hyderabad, Hyderabad, India 5 Department of Computer Sciences, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia 6 Department of Statistics, Chulalongkorn University, Bangkok, Thailand 7 Department of Computer Science and Engineering, Yuan Ze University, Tao Yuan, Taiwan 8 Centre of Research Impact and Outcome, Chitkara University, Punjab 140417 Rajpura, India ✉ Corresponding author. Received 2025 Oct 10; Accepted 2026 Feb 13; Collection date 2026. © The Author(s) 2026, corrected publication 2026 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ . PMC Copyright notice PMCID: PMC13083958  PMID: 41792253 Abstract The surface condenser exhibits nonlinear dynamics and inherent time delays, making precise temperature regulation essential for stable operation in thermal power systems. In this study, a Modified Multi-Verse Optimizer (MMVO) is employed to tune the parameters of a PID controller for improved temperature control of a shell-and-tube condenser. The methodology involves formulating the PID tuning task as an optimization problem, applying MMVO with defined search bounds, and evaluating its performance using 23 standard benchmark functions and repeated simulation runs. Statistical indicators including best, worst, average, and standard deviation values across 30 independent executions are used to assess robustness. Comparative analyses with Ziegler–Nichols (ZN), Genetic Algorithm (GA), and the original Multi-Verse Optimizer (MVO) demonstrate that the modified approach achieves lower integral error indices and reduced overshoot, while providing more consistent performance across trials. The results indicate that MMVO-based PID tuning offers enhanced control capability for systems with strong nonlinearities and delay characteristics. Keywords: Surface condenser, PID controller tuning, GA, MVO Subject terms: Energy science and technology, Engineering, Mathematics and computing Introduction Currently, surface condensers are commonly used in thermal power plants, processing plants, food production, oil refineries, and mechanical systems for the transfer of thermal power. The heat transfer process occurs within shell and tube heat exchangers. A surface condenser 1 , 2 typically converts steam from a gaseous form to a liquid form below the atmospheric pressure. In industrial settings, large heat exchanger 3 systems are employed to harness any wasted thermal energy. Shell and tube heat exchangers 4 , 5 are favored in industries due to their higher efficiency. The heat exchanger is a complex system characterized by high non-linearity and delay time that affect its working conditions. These time delays can also create undesired effects on the system, potentially resulting in instability. Consequently, the primary goal is to maintain the temperature of the process fluid at a specified set point that meets process requirements by regulating the control valve. Therefore, a robust method is essential to achieve optimal function from the heat exchanger. Initially, several studies have been conducted to obtain optimal response from heat exchangers 6 – 9 . Some traditional techniques, such as the PID controller, have been discussed in 10 . The parameters for these techniques are adjusted using the Z-N tuning approach 11 , 12 . However, the response still exhibits significant overshoot and requires more time to stabilize. To further reduce both overshoot and settling time, the PID controller is fine-tuned using the guaranteed dominant pole placement method 13 , 14 . Tuning the PID controller via the guaranteed dominant pole placement reduces overshoot but introduces a steady-state error in the process. In the past ten years, several nature-inspired algorithms 15 , 16 have been effectively developed to address these challenges. By incorporating optimization techniques into the system, we can achieve results that are closer to the ideal response. Some commonly used evolutionary algorithms include the GA 17 , 18 , Particle Swarm Optimization (PSO) 19 , and Moth Flame Optimization (MFO) 20 , 21 . Recent studies have highlighted the increasing use of advanced metaheuristic algorithms in optimization and control applications. In 22 , the authors have proposed the modified atom search optimizer which combines an adaptive gbest-guided mechanism to overcome weaknesses in the standard atom search optimizer. Through this proposed optimizer, time wastes that are related to poor local optimum and loss of control over exploration and exploitation processes are tamed, meaning that this idea is suited for difficult instances of the optimizer. Similar to this, the author of 23 suggests a novel metaheuristic-driven control method for temperature regulation of the continuous stirred-tank heater process by integrating the recently created starfish optimization algorithm with the two degrees of freedom-PID acceleration control. The combination of both gives a versatile and efficient technique for managing highly nonlinear systems, with important implications for industrial temperature regulation applications. In 24 , an integrated approach featuring a proportional–integral–derivative with N filter (PIDN) controller alongside the artificial rabbit’s optimization algorithm to enhance temperature control in electric furnaces, addressing key challenges in precision and response stability. The proposed PIDN controller incorporates adaptive tuning techniques designed to improve response accuracy and reduce overshoot, tailored specifically for the dynamic requirements of electric furnace applications. In 25 , the diligent crow search method is utilized to optimize a novel multistage controller, TDn (1 + PIDn). To handle system nonlinearities, external disturbances, and the intricacies of dynamic reactions in steam condensers, the suggested controller was created. To increase modeling accuracy, stability, and robustness, the current work builds on existing developments by introducing a modified Atom Search Optimizer specifically designed for IIR system identification 26 . These works collectively underline the usefulness of contemporary optimization methods in addressing complicated control and signal-processing difficulties. This paper focuses on optimizing PID controller parameters using GA and MVO algorithms. However, these algorithms can become trapped in local minima. Although various metaheuristic algorithms have been applied to PID tuning, the literature shows limited attention to optimization methods specifically adapted to the nonlinear and time-delay characteristics of surface condensers. Most existing studies rely on standard GA, PSO, or MVO approaches, which often face convergence issues and local-optima constraints in such systems. This gap highlights the need for a modified optimization strategy capable of improving tuning performance under these challenging dynamic conditions. First, a modified Multi-Verse Optimizer (MMVO) is formulated by altering the exploration and exploitation mechanism to improve performance in nonlinear and time-delay systems. Second, this modified optimizer is employed for tuning PID control parameters specifically for a surface condenser, a configuration that has received limited attention in prior optimization-based studies. Third, a comprehensive evaluation is performed using 23 benchmark functions and repeated simulation runs to assess robustness, with statistical indicators reported. Finally, the proposed tuning method is compared with Ziegler–Nichols, GA, and the original MVO to demonstrate consistent improvements in overshoot, settling behavior, and integral error metrics. The main objective in this research is: To maintain the temperature of the outgoing fluid of the shell and tube heat exchanger at a desired set point by adjusting steam flow rate using the PID controller. To overcome the problems associated with GA and MVO. To implement Modified Multi Verse Optimizer technique to optimize the PID controller parameters. To justify the efficiency of the proposed algorithm. To study the time response analysis of the process. The paper is organized as follows. Section 2 presents the mathematical model of the surface condenser and its process components. Section 3 introduces the PID controller and the tuning approaches used. Section 4 describes the Modified Multi-Verse Optimizer (MMVO) and its procedure. Section 5 outlines the benchmark setup, simulation framework, and statistical criteria. Section 6 discusses the comparative results, and Sect. 7 concludes the study with future research directions. Heat exchanger Almost all industrial facilities comprise the creation or inclusion of energy in the way of heat. In process industries, heat exchangers are basically employed for heat transfer from one fluid to another through a solid barrier 1 . They are primarily used in power plants for recovering waste heat. Various kinds of heat exchangers are utilized in industrial usages, but shell and tube heat exchangers are the most prevalent and are able to deal a broad range of temperatures and pressures 2 . These exchangers offer a greater surface area for heat transfer and can be manufactured in diverse sizes and configurations more easily (Fig. 1 ). Fig. 1. Open in a new tab Principle of heat exchanger. Mathematical model Different assumptions are made while establishing the mathematical model: i.e., Heat loss to the atmosphere is negligible. The physical properties density ( ), latent heat of vaporization of steam ( ) and specific heat ( ) do not vary significantly with temperature. Mass balance equation Rate of mass accumulation within the tank: . Mass balance equation is 1 As rate of mass input to the tank = rate of mass output to the tank and density is constant V = Constant Thus, mass balance equation concludes that system have a constant volume. Energy balance equation For this system, total energy is the sum of kinetic energy, internal energy and Potential energy. Rate of accumulation of energy only involves the internal energy. Liquid temperature changes with time along the axial direction z from the value T 1 at the inlet to value T 2 at the exit. It is a distributed parameter system. Energy balance equation with control volume AΔz The buildup of enthalpy over the time interval Δt is equal to the enthalpy entering during Δt minus the enthalpy exiting during Δt, plus the enthalpy transferred from the steam to the liquid during Δt. 2 Hence, temperature is the dependent variable and function of two independent variable (t, Z). System investigated The system under investigation consists of a heat exchanger system and chemical reactor. The storage container contains the fluid from the chemical reactor, which is then pumped to the heat exchanger where it is heated to a specified temperature by the boiler. The storage tank delivers the processed fluid to the heat exchanger system through a pump and a one-way valve. In this setup, steam flows through the tubes. After the steam transfers heat to the process fluid, the condensed steam exits the heat transfer system at 100℃. Various hypotheses have been proposed to mathematically represent the heat exchanger system. The following are enumerated below. 3 Heat storing capacity is immaterial. In this control loop for feedback process, the controller functions in a reverse-acting manner, while the valve operates on an air-to-open (fail-close) basis. Figure 2 illustrates a schematic figure of temperature control within the heat exchanger system. Fig. 2. Open in a new tab Schematic diagram of temperature control of heat exchanger system. Experimental data The components of the heat exchanger system, including the valve, actuator, and sensor, have been mathematically represented based on the experimental data summarized in Table 1 . Table 1. Parameter of process data 27 . Parameter Value Surface condenser reaction to the steam flow gain /(kg/sec) Time constants Surface condenser reaction to deviation of process fluid flow gain /(kg/sec) Surface condenser reaction to deviation of process temperature gain /°C Actuator capacity kg/sec of steam Actuator time constant Temperature sensor variation 50 °C to150°C Temperature sensor time constant 10 s Open in a new tab Controller and optimization techniques PID controller A PID controller was selected for this study because it represents the most widely adopted control strategy in process and thermal systems, including surface condensers. Its simplicity, ease of tuning, and strong disturbance-rejection capability makes it suitable for processes exhibiting nonlinear behavior and time delays. Moreover, PID-based regulation aligns with current industrial practice, enabling direct applicability of the optimized parameters without requiring additional hardware or structural modifications. The control signal is derived as using PID control is: 4 where, e(t) is the error signal, u(t) is the controller output, K c is the controller gain, τ i and τ d are integral gain and derivative gain respectively. The PID controller can be represented in Laplace domain as 5 The equation for real PID controller is denoted as 6 Here indicates the filter parameter. Equation ( 6 ) is narrated as: 7 By substituting = α Here α is the filter coefficient (Fig. 3 ). Fig. 3. Open in a new tab Block diagram of feedback control loop. Tuning of PID controller There are several traditional techniques for tuning a PID controller. One such technique is the Ziegler-Nichols (ZN) method. In this approach, gain is referred to as the critical gain . A drawback of the ZN method is that it tends to produce a significant maximum overshoot (Table 2 ). Table 2. Different closed loop oscillation based tuning methods. Type of tuning methods Z-N Tyreus-Luyben Open in a new tab The characteristic equation =0) is represented as: 8 By implementing Routh stability in Eq. ( 8 ) gives The Auxiliary equation is: 9 By replacing in Eq. ( 9 ), gives . Z-N tuning method The PID controller parameters ( , , ) are determined using closed loop oscillation tuning techniques such as the Zeigler-Nichols method and the Tyreus-Luyben method, which are outlined in Table 3 . Table 3. Parameters of PID Tuned Using Different Tuning Methods. Tuning methods Zeigler-Nichols 0 0 0 Tyreus-Luyben 0 0 0 Open in a new tab As a result, the PID controller that has been tuned is: 10 The Z-N tuning method is engaged to adjust the three parameters of a PID controller. The initial PID controller values determined by all methods should be regularly fine-tuned through computer simulations until the closed system’s desired response is realized. This drives the enhancement of advanced and smart instruments that can help engineers in reaching the optimal performance of the PID controller. The result indicates a significant value of 73%, which is not acceptable in a process plant. To reduce the maximum overshoot and settling time further, PID tuning is performed using Genetic algorithm. Figure 4 presents the step response for the PID-controlled surface condenser using Z-N tuning. Fig. 4. Open in a new tab Step response for PID-controlled surface condenser using Z-N tuning. Genetic algorithm Genetic Algorithms (GA) utilize the idea of survival of the fittest to discover optimal solutions through several iterations. As a result, GA is mainly applied to optimization challenges that focus on maximization. Objective function (OBF) In a heuristic optimization strategy based controller, the first step is to define the OBF according to the required specifications. The ITAE foundation is effective in reducing settling time, a goal that IAE or ISE based tuning is unable to achieve. Additionally, the ITAE foundation also minimizes peak overshoot. A controller based on ITSE responds significantly to sudden changes in the set point, which is not ideal from a controller design standpoint. It has been noted that ITAE serves as a superior objective function. The parameters of the PID controller are adjusted using the ITAE OBF in GA. OBF = min (ITAE). Parameters of GA Maximum population: 30. Maximum generations: 40. Error criteria = ITAE. No of variables: 3 The output indicates overshoot of 6.5% and a settling time of 92 s. Figure 5 displays the step responses for the surface condenser controlled by a GA-Based PID controller. Fig. 5. Open in a new tab Step responses for GA based PID controller. MVO algorithm The MVO algorithm is a population-centric computation that draws inspiration from stochastic processes. The algorithm’s approach is divided into two stages: exploration and exploitation. In the MVO algorithm, the concepts of black holes and white holes are employed to investigate the search spaces, while the concept of wormholes is utilized for exploiting those spaces. During the process of optimization, the subsequent principles are employed regarding the universes of MVO: A higher inflation rate corresponds to an increased likelihood of the existence of white holes. A higher inflation rate results in a decreased likelihood of the existence of black holes. Universes characterized by a higher inflation rate tend to transmit variables through white holes. Universes with a lower inflation rate are more likely to receive additional variables through black holes. Entities in every universe can travel freely to the optimal universe through wormholes. The theoretical model of the approached algorithm is represented in Fig. 6 . Fig. 6. Open in a new tab Conceptual prototype of the proposed MVO algorithm. A roulette wheel system is utilized to illustrate the transfer of variables between universes through the concept of a white/black hole. In each rotation, the universes are organized according to their rates of inflation, and one is selected by this method to represent a white hole. Assume that . Where ), 11 where , , and The exploration can ensured utilizing the roulette wheel mechanism. With a specific end goal to keep up the decent variety of universes and achieve exploitation, it has been consider that every universe has wormholes to transfer its variables over space haphazardly. In Fig. 6 , white circles are equal to transferred objects through the wormholes. Continuously formed wormhole tunnels connect each universe to the most optimal universe developed to date, allowing for localized modifications in every universe and increasing the likelihood of enhancing the inflation rate through the use of wormholes as detailed below: 12 where, X j = ,j th parameter of the best universe constructed so far. Wormhole existence probability (WEP) = coefficient, Travelling distance rate (TDR) = coefficient, WEP is characterized as the possibility of wormholes presences in the universes. It is incremented linearly to amplify the local search during the optimization method. 13 where , , TDR is defined as the rate at which a variable can be transferred between a universe and the optimal universe constructed to date using wormholes. To achieve precise local search around the optimal universe, TDR is reduced over iterations. 14 where, p illustrates the local search precision over the cycles. Modified MVO (MMVO) The MVO is likely to become stuck in local optima and to enhance the MVO’s global search capabilities and its ability to break free from nearby optima, the algorithm has been modified. The TDR value is raised throughout the cycles to enable a more accurate exploration/local search in the vicinity of the optimal universe achieved. In the MMVO algorithm, the TDR equation has been altered. 15 WEP and TDR are demonstrated in Fig. 7 . Fig. 7. Open in a new tab ( a ) WEP versus TDR of MVO Algorithm ( b ) WEP versus TDR of MMVO Algorithm. To isolate the contribution of the modified Travelling Distance Rate (TDR) schedule in MMVO, an ablation-style comparison was carried out. In this study, all algorithmic settings including population size, number of iterations, parameter bounds, WEP schedule, and initialization were kept identical, and only the TDR formulation was switched between the original MVO and the modified MMVO. This controlled comparison demonstrated that the improvement achieved by MMVO is primarily attributable to the revised TDR update. For clarity comparative plot has been included to illustrate their variation across iterations. In the standard MVO, the TDR value decreases gradually to promote local exploitation in later iterations, whereas in MMVO, the TDR is increased progressively to enhance exploration around promising regions. This distinction in TDR dynamics provides the underlying mechanism that enables MMVO to escape local optima more effectively (Fig. 8 ). Fig. 8. Open in a new tab Flow chart of MMVO algorithm. The MMVO algorithm is employed to adjust the gains of the PID controller to increase the performance under standard operating conditions, searches for the optimal combination of these parameters by minimizing an ITAE-based objective function, selected for its ability to penalize prolonged error and overshoot. At each iteration, candidate solutions (universes) are updated through the MMVO’s modified wormhole and TDR mechanisms, allowing exploration of the search space and refinement around promising regions. The process continues until convergence criteria are met, resulting in a set of PID gains that yield improved dynamic performance for the condenser system. The optimized parameters are kp = 2.3866, ki = 0.0538 and Kd = 10.0000. Results and comparisons To justify the efficiency of the proposed MMVO approach, 23 test functions are selected. They are exceptional and have been extensively grasped by various examiners. The standard benchmark functions are recorded in Tables 4 , 5 and 6 where illustrate the optimal value of the function. Except the test functions starting from has value of is zero. Function has nonzero values. The test functions can be isolated into three gatherings: Table 4. Unimodal test functions. Function Range [− 100,100] 0 [− 10,10] 0 [− 100,100] 0 [− 100,100] 0 [− 30,30] 0 [− 100,100] 0 [− 1.28,1.28] 0 Open in a new tab Table 5. Multi-modal benchmark functions. Function Range [− 500,500] 0 [− 5.12.5.12] 0 [− 32,32] 0 [− 600,600] 0 [− 50,50] 0 [− 50,50] 0 Open in a new tab Table 6. Fixed-dimension multimodal benchmark functions. Function Range [− 65.53, 65.53] 0.998004 [− 5,5] 0.0003075 [− 50,50] − 1.0316285 Lb = [− 5,10] Ub = [0,15] 0.398 [− 5,5] 3 [− 0,1] − 3.86 [− 0,1] − 3.32 [0,10] − 10.1532 [0,10] − 10.4029 [0,10] − 10.5364 Open in a new tab 28 29 [30]. Analysis of unimodal benchmark functions The dimensions of the benchmark functions f 1 –f 13 were kept at 5, 10, 20, and 30. The maximum number of iterations was set to 500, and the number of universes was fixed at 30. Each algorithm was executed 30 independent times to enable statistical analysis, and the best, worst, mean, and standard deviation values were computed for all runs. For verification purposes, the performance of the proposed MMVO algorithm was compared with that of the standard MVO. The results for the unimodal benchmark functions f 1 –f 8 are presented in Table 7 . From these results, it can be observed that the unimodal functions are well suited for evaluating local search capability, and the MMVO algorithm demonstrates superior performance compared to MVO. Table 7. Results of unimodal benchmark functions. Function (Dim) Optimizer Best Value Worst Value Avg Value Standard deviation F 1 (5) MVO 0.0001 0.0021 0.0009 0.0005 MMVO 10 − 4 *0.0437 10 − 4 *0.9831 10 − 4 *0.3005 10 − 4 *0.2019 F 1 (10) MVO 0.0059 0.0468 0.0159 0.0091 MMVO 10 − 3 *0.1630 10 − 3 *0.9128 10 − 3 *0.3834 10 − 3 *0.1641 F 1 (20) MVO 0.1205 0.3998 0.2552 0.0672 MMVO 0.0025 0.0099 0.0066 0.0020 F 1 (30) MVO 0.1231 0.4994 0.2537 0.0889 MMVO 0.0031 0.0132 0.0064 0.0023 F 2 (5) MVO 0.0033 0.0114 0.0067 0.0020 MMVO 0.0005 0.0033 0.0012 0.0006 F 2 (10) MVO 0.0026 0.0146 0.0056 0.0025 MMVO 0.0004 0.0017 0.0010 0.0003 F 2 (20) MVO 0.0030 0.0086 0.0058 0.0017 MMVO 0.0005 0.0020 0.0011 0.0004 F 2 (30) MVO 0.0025 0.0124 0.0060 0.0022 MMVO 0.0006 0.0023 0.0011 0.0004 F 3 (5) MVO 10 3 *3.4295 10 3 *9.1155 10 3 *5.9670 10 3 *1.3709 MMVO 10 3 *1.2005 10 3 *2.8681 10 3 *1.0427 10 3 *0.0129 F 3 (10) MVO 0.0129 0.2513 0.1023 0.0659 MMVO 0.0004 0.0071 0.0025 0.0018 F 3 (20) MVO 0.0148 0.3312 0.1111 0.0742 MMVO 0.0008 0.0085 0.0036 0.0018 F 3 (30) MVO 95.9624 539.2682 212.6539 101.6505 MMVO 4.8616 33.6298 15.6809 7.1723 F 4 (5) MVO 0.0092 0.0414 0.0210 0.0064 MMVO 0.0019 0.0064 0.0036 0.0012 F 4 (10) MVO 0.0436 0.1968 0.0936 0.0348 MMVO 0.0053 0.0246 0.0156 0.0051 F 4 (20) MVO 0.2524 1.2720 0.5299 0.2203 MMVO 0.0465 0.2475 0.0934 0.0465 F 4 (30) MVO 0.2853 1.6137 0.6114 0.3253 MMVO 0.0443 0.2058 0.0860 0.0334 F 5 (5) MVO 0.0209 218.8522 11.1746 39.4784 MMVO 0.0863 345.3149 27.4791 78.8079 F 5 (10) MVO 5.0332 917.1020 100.5175 199.3305 MMVO 0.5740 793.4167 102.8266 182.9023 F 5 (20) MVO 10 3 *0.0180 10 3 *2.7793 10 3 *0.5887 10 3 *0.9497 MMVO 10 3 *0.0119 10 3 *1.7405 10 3 *0.3186 10 3 *0.5285 F 5 (30) MVO 10 3 *0.0417 10 3 *2.5880 10 3 *0.4284 10 3 *0.5492 MMVO 10 3 *0.0214 10 3 *2.1279 10 3 *0.3091 10 3 *0.4918 F 6 (5) MVO 0.0002 0.0026 0.0010 0.0005 MMVO 10 − 4 *0.0295 10 − 4 *0.6011 10 − 4 *0.2999 10 − 4 *0.1551 F 6 (10) MVO 0.0038 0.0508 0.0166 0.0096 MMVO 10 − 3 *0.1539 10 − 3 *0.9812 10 − 3 *0. 4028 10 − 3 *0.1976 F 6 (20) MVO 0.0742 0.4881 0.2473 0.0826 MMVO 0.0034 0.0148 0.0073 0.0029 F 6 (30) MVO 0.4540 2.2909 1.2636 0.3915 MMVO 0.0218 0.0678 0.0356 0.0118 F 7 (5) MVO 0.0001 0.0039 0.0011 0.0010 MMVO 0.0001 0.0048 0.0015 0.0012 F 7 (10) MVO 0.0006 0.0092 0.0030 0.0020 MMVO 0.0013 0.0098 0.0044 0.0023 F 7 (20) MVO 0.0039 0.0293 0.0163 0.0072 MMVO 0.0056 0.0284 0.0144 0.0052 F 7 (30) MVO 0.0139 0.0665 0.0334 0.0123 MMVO 0.0126 0.0517 0.0305 0.0083 Open in a new tab Analysis of multi-modal benchmark functions The multi-modal test function includes a global optimum, while the number of local optima rises exponentially as the dimensionality rises. It serves well for assessing the exploratory abilities of different algorithms. The results of benchmark functions - . for various methods are demonstrated in Table 8 . The experimental analysis illustrates that the MMVO performs particularly well in terms of exploration and in escaping undesirable local optima. Table 8. Results of multimodal benchmark functions. Function(Dim) Optimizer Best Value Worst Value Avg Value Standard Deviation F 8 (5) MVO 10 3 *− 1.9765 10 3 *− 1.4040 10 3 *− 1.6973 10 3 *0.1558 MMVO 10 3 *− 1.9765 10 3 *− 1.1473 10 3 *− 1.6651 10 3 *0.1133 F 8 (10) MVO 10 3 *− 3.4595 10 3 *− 2.3113 10 3 *− 2.9420 10 3 *0.2930 MMVO 10 3 *− 3.7358 10 3 *− 2.4313 10 3 *− 2.9543 10 3 *0.0396 F 8 (20) MVO 10 3 *− 6.9775 10 3 *− 4.1524 10 3 *− 5.5929 10 3 *0.6782 MMVO 10 3 *− 6.8399 10 3 *− 4.1338 10 3 *− 5.4966 10 3 *0.6573 F 8 (30) MVO 10 3 *− 9.0881 10 3 *− 6.1626 10 3 *− 7.6955 10 3 *0.6722 MMVO 10 3 *− 9.7463 10 3 *− 6.8801 10 3 *− 7.8512 10 3 *0.7027 F 9 (5) MVO 0.0001 2.9865 1.2276 0.8132 MMVO 2.9849 33.8285 15.0239 7.0613 F 9 (10) MVO 6.9774 42.7912 18.4136 8.0775 MMVO 18.9043 78.6014 37.5432 15.5503 F 9 (20) MVO 24.0074 108.5279 64.5982 20.3457 MMVO 45.7708 160.1910 95.2538 26.9696 F 9 (30) MVO 65.1016 203.6829 123.1528 37.4187 MMVO 85.5832 234.8331 163.0262 37.6935 F 10 (5) MVO 0.0079 0.0398 0.0179 0.0074 MMVO 0.0014 0.0041 0.0027 0.0007 F 10 (10) MVO 0.0350 2.5927 0.6586 0.7950 MMVO 0.0051 2.0135 0.0748 0.3662 F 10 (20) MVO 0.1281 2.2181 1.0549 0.7047 MMVO 0.0146 17.5375 0.7859 3.2006 F 10 (30) MVO 0.6150 3.1470 1.7065 0.5775 MMVO 0.0376 19.0249 4.6842 7.3763 F 11 (5) MVO 0.0221 0.3327 0.1284 0.0803 MMVO 0.0100 0.1433 0.0602 0.0283 F 11 (10) MVO 0.1462 0.6980 0.3957 0.1468 MMVO 0.1197 0.5143 0.2745 0.1007 F 11 (20) MVO 0.3035 0.7756 0.5230 0.1296 MMVO 0.0143 0.1419 0.0642 0.0273 F 11 (30) MVO 0.6109 0.9375 0.8453 0.0719 MMVO 0.0669 0.3392 0.1511 0.0611 F 12 (5) MVO 10 − 3 *0.0131 10 − 3 *0.4604 10 − 3 *0.0925 10 − 3 *0.0981 MMVO 0.0000 3.1151 0.3113 0.7973 F 12 (10) MVO 0.0001 1.2741 0.0785 0.2510 MMVO 0.0000 8.5181 1.5847 2.3329 F 12 (20) MVO 0.0021 2.6268 0.9663 0.8879 MMVO 0.0001 7.4979 4.0245 1.9826 F 12 (30) MVO 0.1193 9.1042 2.0443 1.8611 MMVO 1.8271 9.3383 5.4021 1.7861 F 13 (5) MVO 0.0000 0.0017 0.0002 0.0003 MMVO 10 − 4 *0.0026 10 − 4 *0.2089 10 − 4 *0.0724 10 − 4 *0.0629 F 13 (10) MVO 0.0009 0.0190 0.0082 0.0061 MMVO 0.0000 0.0111 0.0004 0.0020 F 13 (20) MVO 0.0223 0.1589 0.0576 0.0253 MMVO 0.0016 0.0334 0.0099 0.0080 F 13 (30) MVO 0.0723 0.5130 0.2187 0.1053 MMVO 0.0016 0.0334 0.0099 0.0080 Open in a new tab Analysis on fixed-dimension multimodal benchmark functions In comparison to functions f 8 –f 13 , the functions f 14 –f 23 are more straightforward due to their reduced dimensionality and a smaller number of local minima. The performance of benchmark functions f 14 through f 23 across various algorithms is presented in Table 9 . The functions f 14 –f 23 are capable of easily achieving the global optimum. According to Table 9 , it is evident that MMVO yields superior solutions compared to other algorithms, with the exceptions of f 14 and f 18. Table 9. Outcomes of benchmark functions with fixed dimensions across multiple modalities. Function(Dim) Optimizer Best value Worst value Avg value Standard deviation F 14 (2) MVO 0.9980 0.9980 0.9980 0.0000 MMVO 0.9980 20.1535 9.1525 6.1129 F 15 (4) MVO 0.0007 0.0631 0.0095 0.0137 MMVO 0.0003 0.0204 0.0052 0.0079 F 16 (2) MVO − 1.0316 − 1.0316 − 1.0316 0.0001 MMVO − 1.0316 − 0.2155 − 0.9500 0.0000 F 17 (2) MVO 0.3979 0.3979 0.3979 0.0000 MMVO 0.3979 0.3979 0.3979 0.0000 F 18 (2) MVO 3.0000 84.0000 8.4000 20.5504 MMVO 3.0000 84.0000 16.5000 25.3183 F 19 (3) MVO − 3.8628 − 3.8628 − 3.8628 0.0000 MMVO − 3.8628 − 3.8628 − 3.8628 0.1411 F 20 (6) MVO − 3.3220 − 3.1965 − 3.2817 0.0579 MMVO − 3.3220 − 3.2025 − 3.2862 0.0556 F 21 (4) MVO − 10.1531 − 2.6304 − 7.2943 3.0109 MMVO − 10.1531 − 2.6304 − 5.5583 3.2217 F 22 (4) MVO − 10.4028 − 1.8376 − 7.9625 3.3542 MMVO − 10.4028 − 1.8376 − 6.1228 3.0514 F 23 (4) MVO − 10.5364 − 2.4273 − 8.5842 3.1123 MMVO − 10.5364 − 1.6766 − 4.3841 3.0223 Open in a new tab Comparison of statistical result with MFO Performance of the MMVO is compared with the MFO in terms of best value for 40 dimensions, 60 no of universe and 600 iteration. For the comparison involving the Moth-Flame Optimization (MFO) algorithm, the experimental settings were intentionally selected to differ from those used in the evaluations of MVO and MMVO. MFO has been reported in prior literature to achieve its most stable performance when higher dimensionality, larger population sizes, and increased iteration limits are employed. Tables 11 and 12 shows the statistical results of unimodal and multimodal benchmark functions respectively. Table 11. Comparison result of multimodal benchmark function with MFO. Function MMVO(Best Value) MFO(Best Value) F 8 10 4 *− 1.2512 10 4 *− 1.358 F 9 112.4057 117.6214 F 10 0.0327 1.1029 F 11 0.0827 1.0064 F 12 1.8754 3.7687 F 13 0.0017 17.2521 Open in a new tab Table 12. PID controller parameters using various tuning techniques. Tuning method Z-N Tuning 14.28 0.9920 51.26 GA 2.713 0.067 9.903 MVO 2.1854 0.0514 9.9194 MMVO 2.3866 0.0538 10.0000 Open in a new tab From Table 10 , it can be observed that the MMVO outperforms than MFO except in F 3 . Table 10. Comparison result of unimodal benchmark function with MFO. Function MMVO (Best Value) MFO (Best Value) F 1 0.0210 2.253 F 2 0.0946 0.6584 F 3 0.9115 *9.8165 F 4 0.2856 51.6426 F 5 *0.328 549.6379 F 6 0.0210 3.4127 F 7 0.0082 0.9999 Open in a new tab From Table 11 , it can be observed that the MMVO gives satisfactory result than MFO. Simulation results The effectiveness of various techniques is assessed through time response analysis and standard error metrics. The step response analysis for the optimization methods (MVO and MMVO) can be seen in Fig. 10 , while the step response analysis for all methods is presented in Fig. 11 . To ensure consistency between the numerical results and the graphical responses, the controller parameters and operating conditions used in Figs. 10 and 11 have been directly aligned with those presented in Tables 12 , 13 and 14 . The simulations were executed over several independent runs for each tuning method. Across these runs, the time-response characteristics exhibited stable and repeatable trends, with no significant deviation from the representative responses shown in the figures. The displayed curves therefore reflect the typical system behavior observed during the repeated trials, and their correspondence with the tabulated performance indices confirms the reliability of the comparative analysis. Tables 12 , 13 and 14 outline the controller parameters, performance metrics, and error criteria, respectively. To evaluate performance, peak time, maximum overshoot, and settling time are calculated. The reduction of error is achieved by employing ISE, IAE, ITSE, and ITAE criteria. Fig. 10. Open in a new tab Step Responses of MVO and MMVO. Fig. 11. Open in a new tab Step responses of all methods. Table 13. Comparison of different parameters indices. Parameters MMVO MVO PID(GA) PID(Z-N Tuning) ITSE 194.6 215.2 225.2 243.1 ITAE 759.2 783.4 801.3 1141 ISE 16.85 17.7 17.5 12.81 IAE 24.75 25.96 27.21 27.88 Open in a new tab Table 14. Analysis of time response parameters. Parameters MMVO MVO PID(GA) PID(Z-N Tuning) Settling Time(s) 53 60 92 152 Maximum Peak Overshoot (%) 1 0.9 6.5 73.5 Peak Time(s) 1.01 1.009 1.065 1.735 Open in a new tab 16 17 18 19 MMVO-based optimization yields satisfactory outcomes with reduced overshoot and settling time, even though the rise time is somewhat longer compared to MVO optimization. The error factor are also reduced in the MMVO optimization. The proposed MMVO algorithm was found to deliver the lowest ITAE, ITSE, and IAE values, whereas the Ziegler–Nichols method resulted in a smaller ISE under the present operating conditions. As a result, Fig. 11 indicates that the response obtained with the MMVO-tuned PID controller is improved in comparison with the other methods. The PID controller utilizing the MMVO algorithm yields the lowest error indices. From Table 14 , it is observed that the tuning of PID using Z-N Tuning method shows high overshoot on the range of 73.5% and settling time 152s. To compensate the high peak overshoot, PID controller tuned using GA. It gives 6.5% peak overshoot and 92s settling time. PID controller tuned using MVO algorithm shows settling time 60s. Then three parameters of PID are tuned by using MMVO algorithm. In this case, it takes 53s to settle down and shows overshoot in the range of 1%. Conclusion and future scope This paper examines the performance of a PID controller using various tuning methods and optimization techniques. The main objective is to maintain the temperature of the outward liquid of a gas to liquid type warmth exchanger framework to a coveted temperature in the less conceivable time and least or no overshoot independent of changes in the load and process disturbances. Based on the observations, it’s evident that the PID controller optimized with MMVO delivers impressive outcomes with reduced overshoot and settling time, outperforming the PID controller tuned via the Z-N method, as well as the GA-tuned PID parameter optimization. PID controller with MMVO optimization offers overshoot in the limit of 1% and settling time 53s. According to the best value, worst value, standard deviation and average value of the benchmark functions, MMVO algorithm outperforms than other methods. Although the PID controller with MMVO optimization technique gives better result than that of the tuning of PID controller by Z-N method, but improvised controller with improvised optimization technique will be used to handle the system with increase in nonlinearities. This study has several limitations that define its current scope. The condenser system is modeled using a simplified process representation, and real plant nonlinearities such as fouling, varying steam quality, and measurement noise are not fully captured. The optimization is performed using a single ITAE-based objective function, and multi-objective trade-offs were not explored. In addition, the evaluation is limited to simulation results, and no hardware or real-time validation has been conducted. The performance of the proposed MMVO method has been assessed under fixed operating conditions, and its adaptability to rapidly changing industrial scenarios remains to be investigated. These limitations indicate opportunities for further development and experimental verification. Abbreviations IAE Integral absolute error ISE Integral squared error ITAE Integral of time multiplied absolute error ITSE Integral of time multiplied squared error GA Genetic algorithm MFO Moth flame optimization MVO Multi verse optimizer MMVO Modified multi verse optimizer PID Proportional plus integral plus derivative Z-N Ziegler-Nichols Appendix See Tables 4 , 5 and 6 . Author contributions Sweta Panda: Writing – original draft, Methodology. M. UmaMaheswar Rao: Investigation, Conceptualization. Arun Kumar Sahoo: Methodology, Data curation. Soumya Ranjan Das: Validation, Investigation. Norah Saleh Alghamdi and Wattana Viriyasitavat: Writing – review & editing, Funding acquisition. Gaurav Dhiman: Writing – review & editing, Methodology. Funding Open access funding provided by Manipal Academy of Higher Education, Manipal Data availability No datasets were generated or analysed during the current study. Competing interests The authors declare no competing interests. Footnotes Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Change history 6/11/2026 The original online version of this Article was revised: The original version of this Article erroneously omitted Affiliation 8, “Centre of Research Impact and Outcome, Chitkara University, Rajpura-140417, Punjab, India” from the Affiliation list. The Affiliation has been add and assigned to Gaurav Dhiman. The Article has been corrected. References 1. Panda, S., Panigrahi, T. K., Das, S. R. & Rao, M. U. Application of optimization techniques for temperature control of surface condenser. 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