Solving the Combined Heat and Power Economic Dispatch Problem in Different Scale Systems Using the Imperialist Competitive Harris Hawks Optimization Algorithm

The aim of electrical load dispatch (ELD) is to achieve the optimal planning of different power plants to supply the required power at the minimum operation cost. Using the combined heat and power (CHP) units in modern power systems, increases energy efficiency and, produce less environmental pollution than conventional units, by producing electricity and heat, simultaneously. Consequently, the ELD problem in the presence of CHP units becomes a very non-linear and non-convex complex problem called the combined heat and power economic dispatch (CHPED), which supplies both electric and thermal loads at the minimum operational cost. In this work, at first, a brief review of optimization algorithms, in different categories of classical, or conventional, stochastic search-based, and hybrid optimization techniques for solving the CHPED problem is presented. Then the CHPED problem in large-scale power systems is investigated by applying the imperialist competitive Harris hawks optimization (ICHHO), as the combination of imperialist competitive algorithm (ICA), and Harris hawks optimizer (HHO), for the first time, to overcome the shortcomings of using the ICA and HHO in the exploitation, and exploration phases, respectively, to solve this complex optimization problem. The effectiveness of the combined algorithm on four standard case studies, including 24 units as a medium-scale, 48, 84, units as the large-scale, and 96-unit as a very large-scale heat and power system, is detailed. The obtained results are compared to those of different algorithms to demonstrate the performance of the ICHHO algorithm in terms of better solution quality and lower fuel cost. The simulation studies verify that the proposed algorithm decreases the minimum operation costs by at least 0.1870%, 0.342%, 0.05224%, and 0.07875% compared to the best results in the literature.


Introduction
Today, energy saving and reducing pollutants are the most important concerns, mainly in industrialized societies.Therefore, technologies that play an effective role in this regard are given serious attention.The amount of fossil energy is finite and its use causes significant environmental pollution; therefore, the optimal management of these resources is very important in electrical energy production.One of the attractive methods for optimizing energy consumption is the use of the combined heat and power (CHP) unit, which simultaneously produces electricity and heat in a single system [1].In the CHP unit, due to the simultaneous production of two types of energy, the energy efficiency increases up to 90% [2], which significantly increases the efficiency of about 30% of traditional thermal power plants [3].Also, in CHP units, the operating costs are reduced by 10-40% [4], and the amount of pollution production is decreased up to 13-18% [5].
Reducing operational costs due to the high costs of building new power plants and transmission lines is always one of the main and important issues in the operation, and planning of power systems.The planners, and operators of this largest human-made system are always looking to reduce various operating costs by using efficient optimization techniques.Therefore, power system optimization, especially in the power generation sector, considering new facilities, such as CHP technology, is an important and interesting topic.By using more powerful optimization methods, and finding lower operating costs, the cost of power generation will be significantly reduced, and cost savings can be used to develop new projects and replace old equipment with new, and modern equipment.
In an integrated framework, where the use of thermal power plants is inevitable, using the CHP units is a suitable solution to increase power generation efficiency.This gives rise to the combined heat and power economic dispatch (CHPED) problem [6].In this new framework, thermal power plants, thermal boilers, and CHP units are considered to supply electric and thermal loads [7].The CHPED problem aims to minimize the total cost of operating the ONN units, satisfying all equality and inequality constraints [8].The main challenges that complicate the CHPED concept, include the valve point loading effect (VPLE) of thermal power units [9] and the power and heat independency of CHP units, known as the feasible operating region (FOR) [10].

A Brief Review of the Proposed Methods
In [8], a comprehensive review of some heuristic optimization algorithms applied to CHPED problem is presented, by providing the comparative results for 4-unit, 5-unit, 7-unit, 24-unit, and 48-unit test systems.Although the presented review contains useful material and directions for future research, no classifications are given, and the scope of the study was limited to use only some heuristic optimization algorithms.
Many studies have been done on the CHPED issue by implementing different approaches and techniques.They are mainly classified into three main categories, namely classical, or conventional methods, stochastic search-based techniques (evolutionary or heuristic algorithms), and hybrid approaches.As an important note, and despite a vast diversity of optimization algorithms proposed and used to solve the CHPED problem, finding better results for this problem in terms of accuracy and run-time, remains a very challenging issue and is the subject of ongoing research.

Classical, or Conventional Methods
The first category includes the mathematical methods like Lagrangian relaxation (LR) [11], two-layer LR [12], benders decomposition (BD) [13], nonlinear mixed-integer programming (NLMIP) [14], branch and bound (B&B) algorithm [15], semidefinite programming (SDP) method [16], dual partial-separable programming method [17].Such methods are fast and robust, and give almost similar cost values, but the corresponding burdens and runtimes are different [18].These methods are derivative-based techniques and are highly sensitive to the starting point and nature of the objective function.So the results obtained may not be global or even close to the global optimal solution [19].

Stochastic Search-Based Techniques
The second category is stochastic search-based techniques (evolutionary or heuristic algorithms), which are widely used in optimization problems, including the CHPED problem.These methods can address the complexities of the CHPED problem, such as value point loading effects (VPLE) [20].
The presented methods/algorithms in this category can be classified in evolutionary algorithms (EAs), swarm intelligence-based algorithms, human-based algorithms, and physics-based algorithms, as follows:
Also, some improved or enriched versions of EAs applied to the CHPED problem fall into this category.These new algorithms are proposed to prevent the convergence of the original EAs to local optima and increase the convergence speed [20].Biogeography-based learning particle swarm optimization (BLPSO) [5], PSO algorithm with time-varying coefficients [48], improved PSO (IPSO) [49], selective particle swarm optimization (SPSO) [50], time-varying acceleration coefficients PSO (TVAC-PSO) [51], improved group search optimization (IGSO) [20], improved marine predators algorithm (IMPOA) [37], improved Mühlenbein mutation (IMM) [52], improved GA (IGA) [53], self-adaptive real-coded genetic algorithm (SARGA) [54], improved artificial bee colony (IABC) algorithm [55], societybased grey wolf optimizer (SGWO) [56], cuckoo optimization algorithm with penalty function (PFCOA) [57], and effective cuckoo search algorithm (ECSA) [58] are some examples of this type of optimization techniques which were applied to the CHPED problem.As these techniques are derivative-free, they do not need a good starting point, and can escape from local minima solutions [53].These algorithms cannot guarantee finding the optimal solution, do not provide meaningful measurement regarding the distance from the global optima, and suffer from premature convergence [18].

Hybrid Optimization Methods
Optimally solving the CHPED problem using purely EAs or classical techniques is very difficult or even impossible, especially by considering the different objective functions, and various constraints.As an effective and appropriate solution, two significant categories of hybrid methods, including hybrid classical and EAs, and hybrid EAs (two or more EAs) have been addressed in the literature.
It should be noted that there are other heuristic methods, known as hyperheuristics, were suggested to deal with the complex optimization problems.They comprise a set of methods that are motivated (at least in part) to automate the design of heuristic methods to solve the hard computational search problems, and refer to a search technique or learning mechanism to select or generate heuristics to solve computational search problems [76].The main hyper-heuristic categories are heuristic selection and heuristic generation.Some early approaches developed before 2000 are automated heuristic sequencing, automated planning systems, automated parameter control in evolutionary algorithms, and automated learning of heuristic methods [77].Further details on this subject can be found in [76][77][78].

The Constraints of the CHPED Problem, and Case Study Systems
In this sub-section, the constraints of the CHPED problem, and different case study systems are addressed.

Problem Constraints
As mentioned, the CHPED is a complex and very nonlinear problem that includes different equality and inequality constraints.The main inequality constraints which are mainly related to thermal power plants consist of VPLE, prohibited operating zones (POZs), and ramp-rate limits (RRLs).Also, the FOR is considered a very important inequality constraint of the CHPED problem.On the other hand, modeling the transmission losses (TLs) is a more challenging issue.Table 1, presents a brief summary of this subject for some limited cases.

Case Study Systems
The simulated case study systems are mainly categorized into four different sizes small, medium, large, and very large scales.The small-scale systems are including 4, 5, and 7 units.The 11-unit, and 24-unit test systems are considered as medium-scale systems.The 48-unit, and 84-unit systems are grouped as large-scale test systems, and finally 96, and more units' test systems are considered very large-scale systems.Table 1 presents some detailed data on this issue.

Paper Contributions
The two main challenges of this problem are modeling the VPLE of POUs, and dependence of heat and power generation in CHP units.These cases create multiple local minima, and turn the problem into a highly nonlinear, non-convex, and non-smooth constrained optimization problem.The literature confirms that the vast majority of optimization methods can inherently handle only unconstrained problems [82].Also, gradient-, or derivativebased techniques, usually become easily trapped in local minima.On the other hand, the use of heuristic methods has disadvantages, such as, difficulty in initializing the initial population, high run-time, less guaranteed convergence, a large number of setting parameters, high sensitivity to setting parameters, and the need for many iterations for convergence [19].Considering the operation of these units for most of the year, any slight reduction in the final solutions of the problem will lead to cost savings in the range of thousands and even millions of dollars per year.All these cases require us to use more powerful algorithms to optimize the problem.
The ICA has already been used to solve the CHPED problem, in small-, medium-, and large-scale heat, and power systems, where the obtained results were superior to other algorithms.However, the literature confirms the poor performance of this algorithm in the exploitation phase.To overcome this weakness, the combination of this algorithm with the HHO algorithm has been used.
In this work, for the first time, the ICHHO algorithm, as the combinatorial version of ICA, and HHO is applied to solve the aforementioned problem in medium-, large-, and very large-scale combined heat and power systems.It should be noted that in our previous research [81], the ICHHO was introduced and applied mainly to multi-zone power and heat systems, on some small-and large-scale systems, and the performance of the algorithm was tested on the standard cases of five, seven, and forty-eight units in the multi-zone combined heat and power systems.The main objective of that reference was to determine the optimal operation points of the CHPED problem in the small multi-zone combined heat, and power systems, which was initially used to test the efficiency of the proposed ICHHO algorithm.
However, the main contribution of this new research is to solve the CHPED problem in single-zone combined heat and power structures, in different scales of medium, large, and very large, to verify the effectiveness and performance of the proposed algorithm.
The studied systems in this new work are larger than the previously analyzed systems in terms of variables, which can completely challenge the ability of the proposed algorithm in solving large-scale, nonlinear, and complex problems.The most important innovation of this paper is reducing the generation costs in different complex heat and power systems, which has significantly reduced the annual operation costs compared to other research.For this purpose, this research investigates the CHPED problem on the 24, 48, 84, and 96units in single-zone systems, by using some comparative analysis.To attain this goal four different case studies are considered, modeling the VPLE of power-only units (POUs) in all cases.Initially, to investigate the ICHHO algorithm performance, a 24-unit as medium-scale system which consists of 13 POUs, 6 cogeneration units, and 5 heat-only units (HOUs) is investigated.The results from these systems show the satisfying accomplishment of the suggested algorithm to handle the CHPED issue.In addition, two standard large-scale test systems, including 48, and 84 units, are also studied.The 48-unit system includes 12 CHP units, 26 POUs, and 10 HOUs.The second large-scale system, 84-unit, includes 40 POUs, 24 CHP units, and 20 HOUs.Also, the very large-scale system, 96-unit is including 52 POUs, 24 CHP units, and 20 HOUs.The results of the ICHHO algorithm show that this algorithm is predominant from the aspect of operation cost to other algorithms.
The main novelties of this research are as follows: • Providing a short review of the proposed methods to solve the CHPED problem.

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Proposing the ICHHO algorithm, to overcome the shortcomings of ICA, and HHO in the exploitation, and exploration phases, respectively to increase the performance of the hybrid algorithm.

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Utilizing the ICHHO, for the first time to medium-, large-and very large-scale combined power and heat systems, by modeling the VPLE, and FOR of CHPs.

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Investigating the algorithm performance on the studied cases, and comparing the obtained results with other techniques in the literature.

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Confirming the algorithm's ability to find the optimum points of the CHPED problem in large-scale systems.

Paper Structure
The rest sections of this research are structured as follows.In Section 2, the mathematical modeling of the problem is explained.Section 3 addresses the ICHHO algorithm, and the applied formwork to the CHPED problem.The results of the ICHHO application on different case studies and their comparison with other algorithms are addressed in Section 4. Ultimately, the main conclusions are detailed in Section 5.

Mathematical Formulation of CHPED Problem
The CHPED problem involves determining the optimal generations of POUs, HOUs, and CHP units, satisfying all of the practical constraints to minimize the operation costs.The problem should be considered by modeling the VPLE, generation capacity limits of different units, and the interdependence of heat and power of CHP units.In this section, the mathematical modeling of the CHPED problem is presented.

Objective Function
The CHPED problem is an optimization problem, aiming to minimize the fuel cost of committed units in terms of ($/h) and is expressed as Equation (1): N p is the number of POUs, N c is the number of CHP units, and N h is the number of HOUs.The amounts of power generated in terms of MW by ith POU and ith CHP unit are P i and O i , respectively.H i and T i represent the amount of output heat of the i-th unit of CHP and HOU in MWth, respectively.The total cost function in Equation ( 1) consists of the sum of the cost functions of the POUs, CHPs, and HOUs.Also, C pi (P i ), C ci (O i , H i ), C hi (T i ) represent the operation costs (all in $/h) of POUs, CHP units, and HOUs, respectively.
The cost function of POUs is shown by considering the VPLE of thermal power units with C pi .This term adds a sinusoidal component to the cost function and makes it uneven.Equation ( 2) presents the cost function of POUs, considering the VPLE constraint, as follows: a i ($/MW 2 h), b i ($/MWh), and c i ($/h) are the fuel cost factors of the ith POU, d i ($/h) and e i (rad/MW) are the fuel cost factors related to the VPLE of the thermal unit i.Also, P min i is the minimum generated power of the thermal unit i.The cost function of cogeneration units is modeled as [83]: α i , β i , γ i , δ i , ε i and ξ i are the cost factors of i-th CHP unit and are in $/MW 2 h, $/MW h, $/h, $/MWth 2 h, $/MWth h, and $/MW MWth h, respectively.
Furthermore, the fuel cost of the HOU is formulated as [84]: where η i , θ i , and λ i are the cost factors of the i-th boiler and are in terms of $/MWth 2 h, $/MWth h and $/h, respectively.

Constraints 2.2.1. Equality Constraints
These equalities are as follows: P d and H d are the demands for electric and thermal power in terms of MW and MWth, respectively.Equation ( 5) describes the equilibrium constraint of active power [35].This constraint balances the electrical power generated by the POUs and the CHP units with the total electrical power demand of the system.In addition, Equation (6) shows the thermal power constraint [40], which balances the total heat generated by CHP units and HOUs with the system heat demand.

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Capacity limitations of Power-only units The amount of generated capacity by POUs should be limited within a permissible range and is expressed as: P max i and P max i are the upper and lower limits of the i-th thermal power plant in the system.

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Permissible limits of boilers The amount of heat that a boiler unit can produce is in a certain bound, which is modeled as follows: T max i and T max i are the upper and lower bounds of the i-th boiler of the system.

• Permissible limits of CHP units
Active power generated by POUs and the heat produced by HOUs is limited by the relevant minimum and maximum limits.Also, the generated heat and power by CHP units are limited to the FOR, described by Equations ( 9) and (10).
O min i and O max i are the upper and lower bounds of the generated power by CHP units, respectively, and H min i and H max i are the lowest and highest possible produced heat by the CHP units, respectively.

The ICHHO Structure
Generally, the hybrid metaheuristics (HMHs) are divided into low, and high levels.In low-level HMHs, a given function of one metaheuristic method is replaced by another one, and there is no direct relationship to the internal working of a metaheuristic at the high level.All HMHs in both low, and high levels are implemented by relay, or parallel processing.In relay hybridization, a set of metaheuristics are applied one after another, while parallel type provides cooperative optimization models.In general view all of the HMHs, can be classified into homogeneous, or heterogeneous; global, or partial; and general, or specialist [85].
The ICHHO, is a type of parallel, high-level and heterogeneous hybrid algorithms with partial search.The ICHHO algorithm can pursue several optimum solutions using the multi-swarm structure of the ICA, and its revolution mechanism to diversify solutions.Also, the time-varying randomized nature of HHO, due to parameters of Levy Flightbased search, and escaping energy patterns, helps the ICHHO avoid being trapped in local minimums [86].
In the proposed ICHHO, for the combination of ICA and HHO algorithms, the ICA algorithm is referred to as the base algorithm.The ICA has an operator called assimilation operator, which is removed in ICHHO and replaced with HHO algorithm instead.
The ICHHO, was firstly introduced in [86], as a combination of the ICA with the HHO.The superiority of the ICHHO algorithm in solving different mathematical benchmark functions, include unimodal, and multimodal optimization problems were completely proved over GA, PSO, ICA, and HHO, in terms of the average (Ave), minimum (min), and the standard deviation (SD) of obtained results [86].
The HHO was initially introduced in 2019 [87], concludes with three main stages an exploration phase, exploitation phase, and transition between these two phases.Interested readers are referred to [81] for further descriptions.
The ICA is adapted from the evolution process of global communities, and it is very popular due to its high speed and accuracy in finding solutions to optimization problems [88].In ICA, several initial populations are randomly generated.Each member of the population is called a "country".An arbitrary number of the most powerful countries are considered colonizers and the rest colonies.In ICA, each country is a solution to the optimization problem.
The main weakness of the HHO algorithm is the exploration phase, which is due to the weakness of its search mechanism, in such a way that in the HHO algorithm, the parameter E (escaping energy of the rabbit) is a variable to determine the phase of the algorithm, exploration or exploitation.For this purpose, the following formula is used: E 0 is the initial energy of the rabbit, and in each iteration, it is updated in the interval of [−1,1].t and T are the iteration number and maximum iteration of the algorithm, respectively.For HHO to be in the exploration phase, E must be greater than 1.According to Equation (11), when the algorithm reaches the second half of the iteration (that is, t >= T/2), the value of E cannot be greater than 1, and this means that the search mechanism of the algorithm is lost in all iterations of t >= T/2.The strength of the HHO algorithm is its exploitation phase, which prevents the algorithm from getting trapped in local optima.This is due to the existence of the levy flight (LF) function in the algorithm.Parameter E, which is mainly placed in the exploitation phase, is also effective in this subject.On the other hand, ICA suffers from premature convergence.Because during the colonial competition of the colonizer countries, the number of empires decreases and when the number of empires reaches 1, the new calculations are finished before the algorithm reaches the maximum number of iterations.Therefore, fast convergence occurs and the algorithm gets trapped in local optima.
The literature [86], confirms that the HHO has a weak search mechanism, but it is powerful in the exploitation phase, because of its time-varying nature due to random parameters.These prevent the HHO algorithm to trap in local optima.On the other hand, ICA, has a powerful search mechanism, which diversifies solutions.The hybrid ICHHO algorithm performs well, and the above features ensure that this algorithm is protected from premature convergence and entanglement in local optima.In addition, the mechanism of the ICHHO algorithm has been fully and comprehensively explained in the authors' previous research (ref.[81]).For more details, the interested readers are referred to mentioned reference, to avoid repetition.

The Flowchart of the ICHHO Algorithm
Figure 1 shows the flowchart of the ICHHO hybrid algorithm.The general steps of the ICHHO to solve the optimization problems, are explained briefly below.
Step 2. Generating the hawks randomly.
Step 3. Moving all the hawks of all groups in the direction of the rabbit of their group.
Step 4. Applying the revolution factor.
Step 5. Shifting and moving the to the rabbit, based on the specified strategy.
Step 6. Calculation of the cost of all groups, as: where TC i is the total cost of i-th group, and ξ is a number between zero and one.
Step 7. Electing several hawks from the groups with the lowest power.
Step 8. Eliminating the weak groups.
Step 10.Stop, and print the results.The general steps of the ICHHO to solve the optimization problems, are explained briefly below.
Step 2. Generating the hawks randomly.
Step 3. Moving all the hawks of all groups in the direction of the rabbit of their group.
Step 4. Applying the revolution factor.
Step 5. Shifting and moving the to the rabbit, based on the specified strategy.
Step 6. Calculation of the cost of all groups, as: where  is the total cost of i-th group, and  is a number between zero and one.
Step 7 .Electing several hawks from the groups with the lowest power.
Step 8. Eliminating the weak groups.
Step 10.Stop, and print the results.

The Main Steps of Solving the CHPED Problem by the Proposed Algorithm
Here, the main steps for applying the ICHHO algorithm to the CHPED problem are described.
It should be noted that in the CHPED problem, the decision variables to be optimized are the output powers of POUs, and CHP units (i.e.,  and  , respectively), and the output heats of HOUs, and CHP units (i.e.,  and  , respectively).
Step 1. Setting algorithm parameters: These parameters are the number of hawks (NHawks), number of groups (NGroups), number of algorithm iterations (T), revolution probability, and .NHawks, NGroups, and T are considered different values for different

The Main Steps of Solving the CHPED Problem by the Proposed Algorithm
Here, the main steps for applying the ICHHO algorithm to the CHPED problem are described.
It should be noted that in the CHPED problem, the decision variables to be optimized are the output powers of POUs, and CHP units (i.e., P i and O i , respectively), and the output heats of HOUs, and CHP units (i.e., H i and T i , respectively).
Step 1. Setting algorithm parameters: These parameters are the number of hawks (NHawks), number of groups (NGroups), number of algorithm iterations (T), revolution probability, and ξ.NHawks, NGroups, and T are considered different values for different test systems, and their values are specified in related simulation sections.Also, the number of groups, revolution probability, and ξ for all test systems are equal to 10, 0.6, and 0.2, respectively.
Step 2. Determining the primary position of hawks; the power and heat produced by the units are the variables of the problem, which are supposed as the hawks in ICHHO.Each hawk is a vector that includes the powers of POUs and CHP units, as well as the heat of CHP units and boilers.Therefore, every hawk is a solution to the CHPED problem.The positions of the hawks are generated through Equations ( 13) to ( 16) [45], to satisfy the inequality constraints of the problem, as: Step 3. Calculation of the cost of all hawks; In step 2, the hawks position was generated randomly.Here, the cost of each hawk is calculated.To satisfy the equality constraints, two methods have been used consecutively.First, the method of correcting the answers is used.In this way, the values of P error and H error are determined as follows: Here, sum(P k ) and sum(O k ) are the summations of produced power by entire POUs and CHP units, respectively.Likewise, sum(T k ) and sum(H k ) are summations of produced heat by entire CHP units and HOUs.
To accurately match the total amounts of power and heat generation with consumption values and not to violate these very important constraints, the power and heat production values are changed until P error and H error are equal to zero, and the iteration process continues.This issue will ensure complete and accurate compliance with the equality constraints of production with the consumption of power and heat generation values.In addition to this method, to fully ensure the satisfaction of the equality constraints, the penalty function method has been used by assuming a weight of 10 6 .
Step 4. Determining the rabbit and groups.
Step 5. Moving the hawks of any group toward its rabbit.
Step 6. Changing the hawk position by using the revolutionary operator if possible.
Step 7. Replacing a hawk with a rabbit in the group, if that was better.
Step 8. Updating the cost of hawks and rabbits.Then between the groups, colonial competition is applied.During this competition, if a group is without a hawk, it will be eliminated.
Step 9. Repeat steps 5 to 8 until reaching maximum iterations.Otherwise, go to stage 10.
Step 10.Print the position and cost of the strongest group's rabbit as the best solution found by the algorithm.

The Complexity of the ICHHO Algorithm
The computational complexity of metaheuristic algorithms is depending on the numbers of steps they call cost function, which is shown by O().By assuming the T, N, and D as the number of iterations, the size of the population, and dimensional size of the problem, the upper, and lower bounds of complexity for HHO are O(N × T × D + 0.75 × N × T × D), and O(N × T × D).It should be noted that in the HHO algorithm, it is assumed that a maximum of 75% of all hawks will participate in update phase of positions.This imposes 0.

Simulation Results
In this section, the efficiency of the ICHHO algorithm to solve the CHPED problem in four different case studies is analyzed.In all test systems, the VPLE of POUs is considered.Also, the quality of the solutions in terms of the operation cost value obtained for each test system by the proposed algorithm is compared with the other algorithms in this field.It should be noted that for some algorithms in the literature, the actual value of objective function is calculated according to the generation values of the units, which have been addressed in that reference.This may be different from the results reported for the cost function of that reference.
To make a fair comparison between the results of the proposed algorithm and other algorithms, for each test system, the number of iterations was exactly selected equal to the what considered in previous researches.Therefore, the number of iterations for test systems 1, and 2 (24, and 48 units) is assumed to be 500 [20,37,41,45,46,70,90], and for test systems 3, and (84, and 96 units), it is assumed to be 1000 [30,37,41,62,91].Also, the size of the population affects the run-time of the algorithm.For each test system, to improve the optimal solutions compared to previous researches, in the form of trial and error the method was run several times with different population sizes (the number of hawks).Finally, a population was allocated that is optimally obtain better solutions than previous researches in reasonable run-time.It means that the population size is equal to 2000 for test system 1 (24-unit), and 300 for test systems 2-4 (48-unit, 84-unit, and 96-unit).
To compare the results of the ICHHO algorithm with other methods (in the test systems of 24 and 48 units), due to the number of used algorithms to solve the mentioned problem, the algorithms that had better results, was selected.For this reason, the compared algorithms for these two test systems are different.Also, the algorithms used for solving the problem in very large-scale systems (84, and 96 units) are completely different with the medium, and large-scale systems.There are only a few algorithms that have been used to solve very large-scale systems.Consequently, different algorithms were inevitably used for comparative studies.It should be noted that the ξ is selected 0.2 for all test systems.The detailed data of all case studies are presented in Appendix A.

Test System 1
The first test system is a medium-scale system with 24 units, including 13 POUs, 6 CHP units, and 5 HOUs.The power and heat demand of this system is 2350 MW and 1250 MWth, respectively.The data of this test system is extracted from the ref.[45].The results obtained from the ICHHO algorithm are presented in Table 2, and compared with the GSA [45], HBOA [41], IGSO [20], ICA, and MICA [90] algorithms.However, it should not be forgotten that for all reported algorithms, one, or both of the power/heat balances are violated and they are not fully satisfied.This raises serious doubt about the better results obtained by those algorithms in terms of mean and standard deviation than the ICHHO.This unacceptability is much more fundamental in the case of the results of some algorithms in which the constraints on power or heat generation are violated.So, it can be concluded that the results obtained by the ICHHO algorithm, in this case, are certainly better than the other reported results.
The total fuel costs obtained by different algorithms are depicted in Figure 2. The results show that the ICHHO algorithm performs better than other methods, in terms of lower cost.For example, assuming a constant annual load, the ICHHO algorithm decreases the annual operating cost by about $950,109 and $13,540,857 compared to HBOA [41], and ICA [90], respectively.Also, the comparative results confirm that, the total costs in the ICHHO algorithm are reduced by about 0.4057%, 0.1870%, 0.5721%, and 2.6008% compared to GSA [45], HBOA [41], IGSO [20], and ICA [90], respectively.Since the heat and power balances are violated with the application of MICA [90], they may not be compared in this case with the performance of the proposed ICHHO algorithm, where there is no violation balance is equal to 0.0000 (see Table 2).Furthermore, the convergence curve of the ICHHO algorithm is shown in Figure 3.The interested readers are referred to references [20,41,45], and [90] to analyze the convergence curve of the other algorithms.* For more details on the validity of the reported results, please study Section 4.1.** There is no meaningful compression between the ICHHO algorithm, and MICA [90], in which the power, and heat balances were violated equal to −0.0009, and are not acceptable, compared to the ICHHO algorithm.

Figure 2.
Comparison of obtained operating costs by different algorithms in a 24-unit system.In MICA [90], the power, and heat balances were violated equal to −0.0009, and are not acceptable, compared to the ICHHO algorithm.In MICA [90], the power, and heat balances were violated equal to −0.0009, and are not acceptable, compared to the ICHHO algorithm.

Test System 2
Test case 2 is a 48-unit large-scale system with 26 POUs, 12 CHP units, and 10 HOUs.The electric and thermal loads of this system are 4700 MW, and 2500 MWth.The data of this system is extracted from [45].Table 3 compares the results of the proposed algorithm and the operation costs obtained by the algorithms, including TVAC-GSA-PSO [70], MPOA and IMPOA [37], HTS [46], ICA, and MICA [90].The results show the superiority The CHPED problem has already been solved and investigated with the ICA algorithm, and its variant versions, whose solutions, and our detailed investigations for some cases are addressed based on ref [90].Regarding the HHO algorithm, its solutions for the studied problem are very weak, have very high operating costs, and it cannot compare and compete with other algorithms.As an example, for the first test system, the results of applying the ICA, and HHO, were presented, separately which confirm the inability of these algorithms compared to the combined version.To summarize, for other studied systems (except for the second system for ICA), the results of the above algorithms have been avoided.

Test System 2
Test case 2 is a 48-unit large-scale system with 26 POUs, 12 CHP units, and 10 HOUs.The electric and thermal loads of this system are 4700 MW, and 2500 MWth.The data of this system is extracted from [45].Table 3 compares the results of the proposed algorithm and the operation costs obtained by the algorithms, including TVAC-GSA-PSO [70], MPOA and IMPOA [37], HTS [46], ICA, and MICA [90].The results show the superiority of the ICHHO performance over the reported algorithms.The comparative results confirm that, the total costs in ICHHO algorithm are reduced by about 0.3684%, 1.1448%, 0.5796%, 0.342%, 2.9580%, and 0.4859% compared to TVAC-GSA-PSO [70], MPOA [37], IMPOA [37], HTS [46], ICA [90], and MICA [90] respectively.Furthermore, the annual operating savings of the proposed method compared to IMPOA, and MICA is about $5,922,254, and $4,960,941, respectively.Figure 4 shows the cost of operation by different algorithms for a 48-units system.The ICHHO convergence curve for test system 2 is shown in Figure 5.

Test System 3
This case, as a large-scale system, consists of 84 units; 40 POUs, 24 CHP units HOUs.The power and heat demands are 12,700 MW and 5000 MWth, respectiv data of this system is extracted from Ref. [30].Table 4 addresses the generated po heat by applying the ICHHO algorithm.Also, Table 5, provides the total operati compared with the other algorithms.It should be noted that the proposed ICHH CHPED problem is developed and programmed in Matlab R2019b environment plemented on an Intel(R) Core(TM) i5-6200U CPU @ 2.30 GHz, 2.40 GHz, 4 GB R bit operating system, x64-based processor PC (Acer, Aspire E5-575 series, N16Q China).The results confirm decreasing the total cost of the ICHHO algorithm co to other algorithms in the range of 0.052241-3.116385%(HECS [91], HS [62] respe Figure 6, also provides a comparison of total annual operating costs by differe rithms, assuming the constant load curves over the year.The results verify that

Test System 4
This very large-scale system consists of 96 units, including 52 POUs, 24 CHP units and 20 HOUs.The power demand is 9700 MW and the required heat is 5000 MWth.Th data of this system are available in Ref. [30].Table 6 presents the output power or hea obtained by ICHHO and Table 7 compares the operating costs of the ICHHO algorithm with other algorithms, including WOA [30], HBOA [41], MPOA [37], and IMPOA [37] The results confirm decreasing the total cost of the ICHHO algorithm compared to othe algorithms in the range of 0.07875-0.75271%(HBOA [41], WOA [30] respectively).Th results show a better performance of the ICHHO than different algorithms.The annua capital savings compared to WOA, HBOA, MPOA, and IMPOA are $ 15607353.81622012.9,11962754.9,and 3003026.9,respectively.Figure 8 shows a comparison of th costs of different algorithms.The ICHHO convergence curve for system four is shown in Figure 9.

Test System 4
This very large-scale system consists of 96 units, including 52 POUs, 24 CHP units, and 20 HOUs.The power demand is 9700 MW and the required heat is 5000 MWth.The data of this system are available in Ref. [30].Table 6 presents the output power or heat obtained by ICHHO and Table 7 compares the operating costs of the ICHHO algorithm with other algorithms, including WOA [30], HBOA [41], MPOA [37], and IMPOA [37].The results confirm decreasing the total cost of the ICHHO algorithm compared to other algorithms in the range of 0.07875-0.75271%(HBOA [41], WOA [30] respectively).The results show a better performance of the ICHHO than different algorithms.The annual capital savings compared to WOA, HBOA, MPOA, and IMPOA are $15,607,353.8,1,622,012.9,11,962,754.9, and 3,003,026.9,respectively.Figure 8 shows a comparison of the costs of different algorithms.The ICHHO convergence curve for system four is shown in Figure 9.

Sensitivity Analysis for 24-Unit System
In this section, a simple sensitivity analysis is presented to describe the dependence of the final solutions on the setting parameters.For this purpose, in the 24-unit system (Test System 1), the effect of the parameters on the final solutions is evaluated in three different cases.In the first one, the revolution probability, and  are selected equal to 0.6, and 0.2 (the relevant results are depicted in Table 2), respectively.Then these parameters are changed to 0.5, and 0.2; and finally selected as 0.4, and 0.1, respectively.The obtained results are depicted in Table 8.

Sensitivity Analysis for 24-Unit System
In this section, a simple sensitivity analysis is presented to describe the dependence of the final solutions on the setting parameters.For this purpose, in the 24-unit system (Test System 1), the effect of the parameters on the final solutions is evaluated in three different cases.In the first one, the revolution probability, and ξ are selected equal to 0.6, and 0.2 (the relevant results are depicted in Table 2), respectively.Then these parameters are changed to 0.5, and 0.2; and finally selected as 0.4, and 0.1, respectively.The obtained results are depicted in Table 8.Table 8 confirms that by changing the setting parameters of the algorithm, the total cost of the problem, will be increased seriously.This clarifies the significant impacts of optimal parameter setting on final results.

Main Findings
The purpose of this work is to investigate the performance of the ICHHO algorithm to obtain better solutions to the CHPED problem.In the simulated case studies, none of the problem constraints are violated (see for example Table 2, the power, and heat balances are equal to zero), and the obtained solutions are better than the other reported techniques.We should take in mind that for some of the reported algorithms, as indicated in Table 2 the power or heat balance constraints are not satisfied.This raises some doubt about the better results reported by those techniques, compared to the ICHHO algorithm.In Sections 4.1-4.4,four case studies are analyzed to confirm the performance of the presented algorithm in solving complex power system problems.

•
One of the most significant subjects that should be mentioned by the researchers is the validation of the results reported by different references, which can be easily checked by analyzing the total operation costs based on reported power and heat values • In test system 1, a 24-unit system, as a medium-scale system is simulated and the results confirmed superior of the ICHHO algorithm in finding total cost, compared to other techniques in the range of 0.1870% to 2.6008%.

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In test system 2, a 48-unit system, as a large-scale system, is simulated and the results proved superior of the ICHHO algorithm in finding total cost, compared to other methods in the range of 0.342% to 2.9580%.

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In test system 3, an 84-unit system, as a large-scale system, is simulated and the results verified superior of the ICHHO algorithm in finding total cost, compared to other algorithms in the range of 0.052241-3.116385%.

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In test system 4, a 96-unit system, as a very large-scale system, is simulated and the results confirmed the superiority of the ICHHO algorithm in finding total cost, compared to other techniques in the range of 0.07875-0.75271%.

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Similar to all comparative studies conducted in previously published research in this field; the improvement of the results of the studied problem by applying the proposed algorithm is very low (less than half a percent).It should be noted that the results obtained are based on the standard period of defining the CHPED problem of one hour, which by assuming the constant power and heat profile in a one year, the amount of cost saved will be very significant, and in the range of thousands, or even millions of dollars.Based on this, the cost reductions by the application of the proposed algorithm compared to the best solutions in the literature are equal to $852,173; $270,714; $1,864,543; and $1,622,013 for the four studied systems, respectively.

•
Finally, the proposed algorithm is a combination of two strong meta-heuristic algorithms that simultaneously take the advantage of the good features of the two algorithms.It is a strong algorithm in terms of speed and has an acceptable mechanism in terms of changing from the exploration phase to the exploitation and vice versa.On the other hand, this algorithm can find the optimal points of the CHPED problem, as one of the most complex and non-linear problems in power system engineering.Therefore, it can be claimed that the ICHHO algorithm can be applied to other engineering optimization problems.

Conclusions
The CHPED problem is an essential concept in power system operation studies, aiming to minimize the total cost of generation, while satisfying different types of constraints and limitations.In this paper, a short review of applied algorithms to handle the CHPED problem in three main categories namely classical, or conventional methods, stochastic search-based techniques (evolutionary or heuristic algorithms), and hybrid approaches are presented.Also, some details on problem constraints, and different case studies classified in the small, medium, large, and very large scales are provided.Then, the combined ICHHO algorithm, as a combination of ICA and HHO algorithms, is applied to the CHPED problem.The mentioned problem is solved considering the VPLE of the POUs, and the generation limits of POUs, HOUs, and CHP units.In addition, the interdependence of heat and power in CHP units, which causes the complexity of the problem is modeled.As shown, ICHHO can find better solutions to the CHPED problem in different case studies of 24-unit as a medium-scale system, 48-unit, and 84-unit as large-scale systems, and 96unit as a very large-scale system.Specifically, in the 24-unit test system, the amount of reduction in operating costs using the ICHHO algorithm compared to the GSA, HBOA, IGSO, and ICA algorithms is 0.4057%, 0.187%, 0.5721%, and 2.6008%, respectively.It saves the amounts of $235.81,$108.46,$333.09, and $1545.76 per hour.In the 48-unit test system, the cost reduction percentages of ICHHO compared to TVAC-GSA-PSO, MPOA, IMPOA, HTS, ICA, and MICA algorithms are 0.3684% ($428.8599/hoursaving), 1.1448% ($1342.9565/hoursaving), 0.5796% ($676.0565/hoursaving), 0.342% ($397.9565/hoursaving), 2.958% ($3534.8006/hoursaving), and 0.4859% ($566.3175/hoursaving), respectively.In the large test system of 84 units, the ICHHO algorithm causes lower operation costs compared to the HECS, MPHS, IHSPSO and CS algorithms, equal to $150.2452 per hour (−0.0522%), $706.6984 per hour (−0.2452%), $745.8681 per hour (−0.2588%), and $967.9478 per hour (−0.3356%).In the very large test system of 96 units, the results verify the lower operation costs of ICHHO algorithm compare to the HBOA, IMPOA, MPOA and WOA algorithms equal to: $185.1613 per hour (−0.07875%), $342.8113/hour(−0.14571%), $1365.6113/hour(−0.57795%), and $1781.6614/hour(−0.75271%).By examining the runtime of the ICHHO algorithm for large-, and very large-scale systems, it is evident that the proposed algorithm has a suitable and acceptable performance.As a suggestion for future research, the above algorithm can be applied to the multi-objective CHPED problem.Also, to bring the situation closer to reality, other practical constraints, such as prohibited operation zones of POUs, and the impacts of different uncertainties can be included.

Figure 1 .
Figure 1.The general flowchart of the ICHHO algorithm.

Figure 1 .
Figure 1.The general flowchart of the ICHHO algorithm.
75 × N × T × D further computational complexity.The upper, and lower bounds for ICA, are O(N × T × D + (N − I) × T × D), and O(N × T × D), respectively, where I is the number of imperialists.Considering that the ICHHO is established on the ICA framework, the lower, and upper bounds of ICHHO are estimated as O(N × T × D), and O(1.75 × N × T × D + (N − I) × T × D), respectively [88,89].

Figure 2 .
Figure 2.Comparison of obtained operating costs by different algorithms in a 24-unit system.In MICA[90], the power, and heat balances were violated equal to −0.0009, and are not acceptable, compared to the ICHHO algorithm.

Figure 2 .
Figure 2.Comparison of obtained operating costs by different algorithms in a 24-unit system.In MICA[90], the power, and heat balances were violated equal to −0.0009, and are not acceptable, compared to the ICHHO algorithm.

Figure 4 .Figure 4 .Figure 5 .
Figure 4. Comparison of operating costs obtained by different algorithms for test system 2.

Figure 6 .
Figure 6.The comparison of the costs for different algorithms for an 84-unit system.

Figure 6 . 3 Figure 7 .
Figure 6.The comparison of the costs for different algorithms for an 84-unit system.Biomimetics 2023, 8, x FOR PEER REVIEW 19 of 3

Figure 7 .
Figure 7.The convergence curve of the ICHHO algorithm for test system 3.

Figure 8 .
Figure 8.The comparison of the operating costs by different algorithms for the system 4.

Figure 8 .
Figure 8.The comparison of the operating costs by different algorithms for the system 4.

Figure 9 .
Figure 9.The convergence curve of the ICHHO algorithm for test system 4.

Figure 9 .
Figure 9.The convergence curve of the ICHHO algorithm for test system 4.

Table 1 .
A taxonomy of the proposed algorithms to solve the CHPED problem in terms of constraints, and case studies.
[90]r more details on the validity of the reported results, please study Section 4.1.**Thereis no meaningful compression between the ICHHO algorithm, and MICA[90], in which the power, and heat balances were violated equal to −0.0009, and are not acceptable, compared to the ICHHO algorithm.

Table 3 .
The obtained results for test system 2.

Table 4 .
The generated power and heat by applying the proposed algorithm on test system 3.

Table 5 .
The comparison of operating costs obtained by ICHHO with different algorithms for test system 3.

Table 6 .
The generated power and heat by applying the proposed algorithm on test system 4.

Table 6 .
The generated power and heat by applying the proposed algorithm on test system 4.

Table 7 .
The comparison of operating costs obtained by ICHHO with different algorithms for test system 4.

Table 7 .
The comparison of operating costs obtained by ICHHO with different algorithms for test system 4.

Table 8 .
The comparison of the obtained operating costs of test system 1 by ICHHO, for different setting parameters.

Table 8 .
The comparison of the obtained operating costs of test system 1 by ICHHO, for different setting parameters.