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A meliorated Harris Hawks optimizer for combinatorial unit commitment problem with photovoltaic applications
Journal of Electrical Systems and Information Technology volume 8, Article number: 5 (2021)
Abstract
Conventional unit commitment problem (UCP) consists of thermal generating units and its participation schedule, which is a stimulating and significant responsibility of assigning produced electricity among the committed generating units matter to frequent limitations over a scheduled period view to achieve the least price of power generation. However, modern power system consists of various integrated power generating units including nuclear, thermal, hydro, solar and wind. The scheduling of these generating units in optimal condition is a tedious task and involves lot of uncertainty constraints due to time carrying weather conditions. This difficulties come to be too difficult by growing the scope of electrical power sector day by day, so that UCP has connection with problem in the field of optimization, it has both continuous and binary variables which is the furthermost exciting problem that needs to be solved. In the proposed research, a newly created optimizer, i.e., Harris Hawks optimizer (HHO), has been hybridized with sine–cosine algorithm (SCA) using memetic algorithm approach and named as meliorated Harris Hawks optimizer and it is applied to solve the photovoltaic constrained UCP of electric power system. In this research paper, sine–cosine Algorithm is used for provision of power generation (generating units which contribute in electric power generation for upload) and economic load dispatch (ELD) is completed by Harris Hawks optimizer. The feasibility and efficacy of operation of the hybrid algorithm are verified for small, medium power systems and large system considering renewable energy sources in summer and winter, and the percentage of cost saving for power generation is found. The results for 4 generating units, 5 generating units, 6 generating units, 7 generating units, 10 generating units, 19 generating units, 20 generating units, 40 generating units and 60 generating units are evaluated. The 10 generating units are evaluated with 5% and 10% spinning reserve. The efficacy of the offered optimizer has been verified for several standard benchmark problem including unit commitment problem, and it has been observed that the suggested optimizer is too effective to solve continuous, discrete and nonlinear optimization problems.
Introduction
Machine learning and artificial intelligence and so many problems are related to real world which have continuous and discrete behavior and constrained and unconstrained in nature. For this kind of attributes, there are a few challenges to handle a few sorts of issues utilizing traditional methodologies with scientific techniques [1]. A few sorts of research have tried that these all strategies are insufficient viable or effective to bargain numerous kinds of noncontinuous problem and nondifferentiable problem and furthermore in such huge numbers of realworld problem. In this way, metaheuristic algorithm is considered and it is used to handle such a significant number of problems which are generally basic in nature and easily executed. Nowadays, the recent developed optimizer is Harris Hawks optimizer [HHO] [2]. Original version of Harris Hawks optimizer (HHO) had highlights that can in any case be improved as it might insight convergence problems or may effectively get caught in neighborhood optima [3]. Many variants had been developed which are discussed in Table 1 (a), which are used to improve or upgrade the existing HHO, so that the efficiency of the optimization techniques will be enhanced [4]. The procedure of optimization technique is proceeded till this can fulfill the most extreme iteration. In the present days, developing mindfulness and enthusiasm for effective, economical and fruitful utilization of such kinds of metaheuristic calculation is under current examination. Modified version of existing algorithm was also upgraded by mutation to solve the realworld optimization problem also [5]. Nonetheless, after no free lunch theorem (NFL) [6], wide range for optimization dependent through enhancement methods prescribed and showed normal equal execution on the off chance that it is applied to every likely sort of errands dependent on optimization technique [7].
In the recent year, the electrical power sector is classified as huge proportions, vastly interconnected and highly nonlinear as dimension of electric power system is rising continuously due to huge electrical power demand in all the essential segment like commercial, agriculture, residential and industrial region. On electricity grid, the influence of overloading occurs due to rising the propensities in electrical load demand, privatization and deregulation taking place on electrical grids. For this condition, it needs progress of electric grid as the same step, as the rise in electric load demand and efficient power generation scheduling and commitment has the ability to regulate the time varying electrical load demand which is run for utilization of available grid [8]. Nowadays, recent power sector has some various sources of electrical power locations containing hydro, thermal and nuclear power generation system; during a whole day, the electric power demand fluctuates with various peak values [9]. Thus, it is essential to determine that power generating units should be turned on, when necessary in power system network and the preparation or order in which the generating unit should kept in turn off condition is by considering the efficiency of cost for turn on and shut down for the respective power generating units. The whole procedure of constructing these assessments is known as unit commitment (UC) [10].
The main novelty of the proposed research work includes the hybrid variant of Harris Hawks optimizer, i.e., hybrid Harris Hawks–Sine–Cosine algorithm (hHHOSCA) has been developed. The exploration phase of the existing Harris Hawks optimizer has been improved. A recently invented hybridized optimizer using memetic algorithm approach is used to solve unit commitment problem of power system. This paper offers the resolution of unit commitment optimization problems of the power system by using the hybrid algorithm, as UCP is linked optimization as it has both binary and continuous variables; the strategy adopted to tackle both variables is different. In this paper, the proposed sine–cosine algorithm searches allocation of generators (units that participate in generation to take upload) and once units are decided, allocation of generations (economic load dispatch) is done by Harris Hawks optimizer. The feasibility and efficacy of operation of the hybrid algorithm are verified for small, medium power systems and large system considering renewable energy sources in summer and winter and the percentage of cost saving for power generation is found. The results for 4 generating units, 5 generating units, 6 generating units, 7 generating units, 10 generating units, 19 generating units, 20 generating units, 40 generating units and 60 generating units are evaluated. The 10 generating units are evaluated with 5% and 10% spinning reserve.
Survey of literature
In the field of research area, the optimization method is the vastest region of research through which the research works are effectively moving forward. Nowadays, researchers are working with multiple works for various problems using different techniques and they are capable of measuring the output successfully. To discover the new algorithms, the research work is on successfully running condition and to mitigate the drawbacks of present existing techniques.
Some of the research works in the field of optimization include ant colony optimization (ACO) algorithm [11], ant lion optimizer (ALO) [12], adaptive gbestguided search algorithm (AGG) [13], bat algorithm (BA) [14], biogeographybased optimization (BBO) [15], branch and bound (BB) [16], binary bat algorithm (BBA) [17], bird swarm algorithm (BSA) [18], bacterial foraging optimization algorithm (BFOA) [19], backtracking search optimization (BSO) [20], and binary gravitational search algorithm (BGSA) [21], colliding bodies optimization (CBO) [22], cuckoo search algorithm (CS) [23], chaotic krill herd algorithm (CKHA) [24], cultural evolution algorithm (CEA) [25], dragonfly algorithm (DA) [26], dynamic programming (DP) [27], earthworm optimization algorithm (EOA) [28], elephant herding optimization (EHO) [29], electromagnetic field optimization (EFO) [30], exchange market algorithm (EMA) [31], forest optimization algorithm (FOA) [32], fireworks algorithm (FA) [33], flower pollination algorithm (FPA) [34], gravitational search algorithm (GSA) [35], genetic algorithm (GA) [36], firefly algorithm (FFA) [37], grasshopper optimization algorithm (GOA) [38], gray wolf optimizer (GWO) [39], human group optimizer (HGO) [40], Hopfield method [41], interior search algorithm (ISA) [42], imperialist competitive algorithm (ICA) [43], krill herd algorithm (KHA) [44], invasive weed optimization (IWO) [45], lightning search algorithm (LSA) [46], league championship algorithm (LCA) [47], multiverse optimizer (MVO) [48], mixed integer programming (MIP) [49], mine blast algorithm (MBA) [50], mothflame optimization (MFO) [51], simulated annealing (SA) [52], monarch butterfly optimization (MBO) [53], particle swarm optimization (PSO) [54], random walk gray wolf optimizer (RWGWO) [55], optics inspired optimization (OIO) [56], runnerroot algorithm (RRA) [57], sine–cosine algorithm (SCA) [58], shuffled frogleaping algorithm (SFLA) [59], stochastic fractal search (SFS) [60],seeker optimization algorithm (SOA) [61], teaching–learningbased optimization (TLBO) [62], symbiotic organisms search (SOS) [63], search group algorithm (SGA) [64], salp swarm algorithm (SSA) [65], and whale optimization algorithm (WOA) [66], weighted superposition attraction (WSA) [67], virus colony search (VCS) [68], water wave optimization (WWO) [69], Tabu search (TS) [70], water cycle algorithm (WCA) [71], winddriven optimization (WDO) [72], modified sine–cosine algorithm [mSCA] [73], and improved sine–cosine algorithm [ISCA] [74]. Leadership quality was improved by Levy flight (LF) search and gray wolf optimizer [GLF–GWO] [75], greedy differential evolution–gray wolf optimizer [gDEGWO] [76], memorybased gray wolf optimizer [mGWO] [77], and memoryguided sine–cosine algorithm [MGSCA] [1].
Faisal Rahiman Pazheri et al. presented scheduling of power station with energy storage facility. Utilities of power are stimulated by converting the present conventional power plant into hybrid power plant by installing available energy storage facilities and renewable electric power unit to come across the sudden increase in the power demand. Facility of energy storage maintains a level of the penetration of renewable power to 10% of required load demand throughout the period of operation for hybrid power plant [136]. Chandrasekaran et al. proposed FF algorithm to get solution of the SUC problem for thermal/solar power sector considering issues regarding smart grid. The research paper included some critical review on reliable impacts of major resources of smart grid considering demand response (DR) and solar energy. Thus, it was essential to implement method for an integration of thermal and solar generating system [144]. Selvakumar et al. implemented a new strategy for solving unit commitment problem for thermal units integrated with solar energy system. There would be changes in the cost of power generation considered solar energy. The main objective was reduction of total production price for the electricity generating unit, and this paper also explained the variances by considering solar energy and nonconsidering the solar power [140]. Senjyu et al. proposed a new method using genetic algorithm operated PSO to solve the thermal UCP considering wind and solar energy system. This method was able to minimize production cost and produce highquality solutions [143]. Ma et al. discussed about appliances scheduling via cooperative multiswarm PSO under photovoltaic (PV) generation and dayahead prices. This research work studied about the problem including scheduling appliances in residential system unit. The model of an appliancescheduling was established for home energy management system which was based on dayahead electricity price and PV generation [137]. Abujarad et al. discussed a review on current methods for commitment of generating unit in existence of irregular renewable energy resource [139]. Maryam Shahbazitabar and Hamdi Abdi implemented a new prioritybased stochastic unit commitment as parking lot cooperation and renewable energy sources. This paper discussed about the fastest nature of heuristic method which was established on list of priority selections to get solution for stochastic nature of the problem related to unit commitment and useful to simple 10 unit systems where the study was addition considering electrical vehicles parking allocation considering wind farm and solar farm over 24h time horizon [135]. Quan et al. proposed a comparative review on integrated renewable energy generation uncertainties which were captured by list of prediction intervals, into stochastic unit commitment considering reserve and risk [138]. Jasmin et al. implemented an optimization technique about reinforcement learning to solve unit commitment problem considering photovoltaic sources. For stochastic behavior of the associated power and solar irradiance, the arrangement of the different types of power generating sources considering solar energy turned to be an optimization problem stochastic in nature. This paper discussed about the optimization technique and reinforcement learning that can provide uncertainty of the environment of the nature which is very effective [141]. Saniya Maghsudlu and Sirus Mohammadi proposed a method to solve the problem in optimum schedule of commitment unit as appropriate control of EVs and PV uncertainty. The metaheuristic approach, cuckoo search algorithm, was developed by greatest convergence speed to attain the optimal solution and get solution of UCP. The research discussed about case study of IEEE 10 unit system which was used to examine the impact of PV and PEVs on scheduling of generating unit [134].
Problematic design
The generating power is distributed along with utilities of generator scheduling which will meet the time varying load demand for a specific time period known as unit commitment problem (UCP). The main objective of UCP is minimization of the overall cost for production considering different system constraints. The overall costs of production including sum of shutdown cost and startup cost, cost of fuel are given below:
The total cost of fuel over the scheduled time span ‘t’ is:
Here, cost for fuel \(F_{{{\text{cost}}n}} (P_{nt} )\) is stated as quadratic design that is mostly worked by researchers, also named as equation of convex function.
The cost of fuel of (n) unit at (t) hour can be mathematically represented as an equation which is given below:
where \(A_{n}\), \(B_{n}\) and \(C_{n}\) are represented as coefficients of cost that may expressed as $/h, $/MWh, and $/MWh^{2} correspondingly.
Startup cost can mathematically be represented by step function which is given below:
Usual value of the shutdown cost for standard system is denoted as zero, and this can be established as fixed cost followed by the equation number (5).
where K is represented as incremental cost for shutdown.
It is subjected through some constraints followed by (1) system constraints and (2) unit constraints.
Constraints for system
System constrains are interrelated with all generating unit existing in the systems. The systems constrains are characterized into two types like:
Power balance or load balance constraints
In power system, the constraint including power balance or load balance is more important parameter consisting of summation of whole committed generating unit at tth time span which must be larger than or equivalent to the power demand for the particular time span ‘t’
Spinning reserve (SR) constraints
Reliability of the system can be considered as facility of extra capability of power generation that is more important to deed instantly when failure occurred due to sudden change in load demand for such power generating unit which is already running. The extra capability of power generation is recognized as spinning reserve which is exactly represented as (Fig. 1):
Constraints for power generating unit
The specific constraints related to particular power generating unit existing in the systems are called generating unit constraint which are given as:
Thermal unit constraints
Thermal power units are controlled manually. This type of unit needs to undertake the change of temperature gradually, so it takes certain time span to take the generating unit accessible. Some crew members are essential to execute the maintenance and procedure of some thermal power generating units.
Minimum up time
This constraint is defined here as the minimum period of time previously the unit can be start over when the unit has already been shut down which is mathematically defined as:
where \(T_{n,t}^{\text{ON}}\) is defined as interval through which the generating unit n is constantly ON (in hours) and \(T_{n}^{\text{UP}}\) is defined as minimum up time (in hours) for the generating unit n (Fig. 1).
Minimum down time
When the power generating units will be DEcommitted, there is required least period of time for recommitment of the unit which is mathematically given as:
where \(T_{n,t}^{\text{OFF}}\) is time period for which generating unit n is constantly OFF (in hours) and \(T_{n}^{\text{DW}}\) is denoted as minimum down time (in hours) for the unit.
Adequate minimum downtime and uptime repair by heuristic mechanism accepted at those stages are stated in Fig. 2.
Maximum and minimum power generating limits
All power generating units have its individual maximum/minimum electric power generating limit, below and outside which it cannot produce, and this is known as maximum and minimum power limits, which is mathematically written as:
Initial status for operation of electrical units
For every units, the initial operating position must proceed as the day’s earlier generation schedule is taken into consideration; thus, each and all generating units can fulfill its lowest downtime/uptime (Figs. 3, 4, 5, 6, 7).
Crew constraint
When any power plant consists of more than one units, they could not turn on at the same period of time. So there need more than one crew member to attend such units in the same time while starting up.
Unit accessibility constraint
The constraint shows accessibility of power generating unit surrounded by any of the resulting various circumstances:

(A)
Accessible or Not Accessible.

(B)
Must Out or Outage.

(C)
Must Run.
Initial status of power generation unit
It signifies value of initial grade of power generating unit. Its favorable rate signifies the position of current generating unit which is already in up condition, which means that numeral time periods of the generating units are already up, and if its negative value is an index of the integer of hours, then the generating unit has been already in down condition. The position of the generating unit ± earlier the first hour through the schedule is an essential feature to define whether its latest situation interrupts the constraint of \(T_{n}^{\text{UP}}\) and \(T_{n}^{\text{DW}}\).
Methods
The mathematical formulation of the Harris Hawks optimizer has been explained in this section. The position updating mechanism of the harris hawks optimizer has been presented in Eqs. (11), (12) and (13). Presently, considering the equivalent possibility w for each adjusting system depends upon areas for additional individuals to approach sufficiency while confronting as a prey, given in Eq. (11) (Figs. 3 and 4)
where \(r_{1,} r_{2,} r_{3,} r_{4,}\) and w are random records in the middle of (0, 1); those are upgraded in every cycle, \(X(iteration + 1)\) is denoted as Rabbit’s position and N is defined as total amount of Harris hawks
Normal area for Harris Hawks is accomplished utilizing Eq. (13) (Fig. 5).
Change after the period for investigation of the period of exploitation is shown:
where E is the avoidance energy for rabbit, E_{0} the early condition for energy and \(iter\hbox{max}\) = maximum iteration
Along these lines, to find out the better solution of a soft enclose, the Hawks birds of prey are able to choose their development Y that depends upon standard that is shown in Eq. (18)
Established Lf (D) patterns are constructed which track the given instruction in Eq. (20),
where D = dimension of problems, S = dimension of random vectors with size 1 × D
where \(\mu ,\sigma\) are denoted as such kind of values randomly in between (0, 1) and \(\beta\) set to 1.5 which is a constant known as default.
The final and actual positions through this period of soft encircle can be updated using Eqs. (22) and (23) shown below:
\(X_{m} (iteration)\) can be obtained from Eq. (23).
The SCA optimization technique is mathematically written as:
Here, \(r_{4}\) is denoted as random numbers [0, 1].
This method based on the suggested process may balance exploitation as well as exploration to get favorable solutions in the area of search space and lastly meet to find global optimal solutions using Eq. (27) (Fig. 6).
Handling of spinning reserve constraints
The simple possible solution that was obtained by SCA technique is highly unsuccessful to satisfy spinning reserve necessity (Fig. 7). Also handling of minimum uptime and downtime leads to extra spinning reserve. Thus, it is compulsory to handle/adjust spinning reserve necessity heuristically. The flowchart in Fig. 8 explains whole process to repair spinning reserve necessity.
Decommitting of excess of units
It is obvious from the code given over that during fix of MDT, MUT, and spinning reserve we need to take generating unit status “ON” if these requirements are violated by putting it off. Since we do as such against the caution given by algorithm, obliviously it brings about additional save. This circumstance is exceptionally unwanted; thus, we need to recommit some of the units once again in order to archive economic operation. In the following, the heuristic methodology is received for decommitting the additional spinning reserve (Figs. 9, 10).
Results and discussion
In order to validate the efficacy of the hHHOSCA optimization technique, the outcomes of hHHOSCA algorithm have been given below. The generating units’ data are shown in Additional file 1: AnnexureA1 to A6 and its comparative analysis considering solar energy in summer and winter are shown in Table 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 2021, 22, 23, 24 and 25 (Figs. 11, 12 and 13).
5Generating Unit Test Systems: The first test system contains IEEE14 bus systems which have 24hour power demand with 10% spinning reserve. The hHHOSCA technique is considered for 100 iterations. Tables 2 and 12 show that optimal scheduling for this test system considering summer and winter, respectively, using the hHHOSCA algorithm is 8226.6 $/hour and 8572.9 $/hour. Without considering the renewable energy sources, the total generation cost is 9010.1$/hour.
6Generating Unit Test System: The second test system contains 6generating units for IEEE30 bus test systems with 24hour electrical load demand including 10% SR [145]. The hHHOSCA algorithm is assessed for 100 iterations. Tables 3 and 13 show that optimal scheduling for this test system considering summer and winter, respectively, using the hHHOSCA algorithm is 12229 $/hour and 12977 $/hour. Without considering the renewable energy sources, the total generation cost is 13489.93957 $/hour.
10Generating Unit Test System: The third system contains 10 units power generating units. This system has been verified for 24hour electric power demand outline at various spinning reserve capability. Case1 consists of spinning reserve capability of 5%, and case2 contains spinning reserve capability of 10%.
Case1: 10Generating Unit Test System (SR = 5%): The system consists of 10 power generating units with 24hour electrical load demand including 5% SR [146]. The hHHOSCA technique is evaluated for 100 iterations. Tables 4 and 14 show that optimal scheduling for this test system considering summer and winter, respectively, using the hHHOSCA algorithm is 529980 $/hour and 536200 $/hour. Without considering the renewable energy sources, the total generation cost is 557533.12$/hour.
Case2: 10Generating Unit System (SR = 10%): The system consists of 10 power generating units with 24hour electrical load demand including 10% SR [146]. The hHHOSCA technique is evaluated for 100 iterations. Tables 5 and 15 show that optimal scheduling for this test system considering summer and winter, respectively, using the hHHOSCA algorithm is 534050 $/hour and 539110 $/hour. Without considering the renewable energy sources, the total generation cost is 563937.6875$/hour.
MediumScale and LargeScale Electrical Power System (19, 20, 40, 60, 80 and 100Unit System): The data for 20 and 40 generating unit test systems and the 10unit system had been doubled and quadrupled, and electric power demand is multiplied by two and four times correspondingly [145].
19Generating Unit System: The fourth system contains 19 power generating units of IEEE118 bus test system with a 24hour electricity load demand including 10% SR [145]. The hHHOSCA technique is evaluated for 100 iterations. Tables 6 and 16 show that optimal scheduling for this test system considering summer and winter, respectively, using the hHHOSCA algorithm is 207180 $/hour and 207560 $/hour. Without considering the renewable energy sources, the total generation cost is 208510 $/hour.
20Generating Unit System: The fifth system contains 20power generating units with 24hour electricity demand including 10% SR [145]. The hHHOSCA algorithm is assessed for 100 iterations. Tables 7 and 17 show that optimal scheduling for this test system considering summer and winter, respectively, using the hHHOSCA algorithm is 1076400 $/hour and 1084100 $/hour. Without considering the renewable energy sources, the total generation cost is 1125200 $/hour.
40Generating Unit System: The sixth system contains 40power generating units having a 24hour electricity demand including 10% SR [145]. The hHHOSCA technique is evaluated for 100 iterations. Tables 8 and 18 show that optimal scheduling for this test system considering summer and winter, respectively, using the hHHOSCA algorithm is 2176900 $/hour and 2189400 $/hour. Without considering the renewable energy sources, the total generation cost is 2253700 $/hour.
60Generating Unit System: The seventh system contains 60power generating units with 24hour electricity demand including 10% SR [145]. The hHHOSCA technique is evaluated for 100 iterations. Tables 9 and 19 show that optimal scheduling for this test system considering summer and winter, respectively, using the hHHOSCA algorithm is 8226.6 $/hour and 8572.9 $/hour. Without considering the renewable energy sources, the total generation cost is 9010.1$/hour.
80Generating Unit System: The eighth system contains 80power generating units with 24hour power demand including 10% SR [145]. The hHHOSCA technique is evaluated for 100 iterations. Tables 10 and 20 show that optimal scheduling for this test system considering summer and winter, respectively, using the hHHOSCA algorithm is 5578000 $/hour and 5591700 $/hour.
100Generating Unit System: The ninth system contains 100power generating units with 24hour power demand including 10% SR [145]. The hHHOSCA technique is evaluated for 100 iterations. Tables 11 and 21 show that optimal scheduling for this test system considering summer and winter, respectively, using the hHHOSCA algorithm is 5530800 $/hour and 552990 $/hour (Tables 22, 23, 24, 25).
Conclusions
In this research work, the authors have successfully presented the fusion of Harris Hawks optimizer with SCA optimization technique and evaluated performance of the suggested hybrid optimized method for standard benchmark problem unit commitment problem, which consists of thermal generating units and along with PV generating units. The proposed research focuses on invention of hybrid variant of Harris Hawks optimizer (HHO) and sine–cosine algorithm (SCA) using memetic algorithm approach, named as intensify Harris Hawks optimizer. The efficacy of the suggested algorithm was tested for 4generating unit system, 5generating unit system, 6generating unit system, 7generating unit system, 10generating unit system, 19generating unit system, 20generating unit system, 40generating unit system and 60generating unit system. After successful experiment, it was observed that the suggested optimizer is too much effective to solve continuous, discrete and nonlinear optimization problems.
After verification, it builds up the effective outcomes of the suggested hybrid improvement optimization which are more effective to other newly defined metaheuristics, hybrid and heuristics method and advancement search calculation and suggested algorithm recommends for the efficiency of this algorithm in the search area of metaheuristics type optimization algorithms which are nature inspired. The other existing optimization techniques have good development prospect, but their research is still at initial condition and included so many problems which need to be solved or in other instance, there are several uncertainties, such as, how to adequately stay away from nearby or local optimum? What is the most effective method to consummately consolidate the upsides of distinctive enhancement calculations? How to successfully set the boundaries or parameter of a calculation? What are the compelling cycle of iteration stop conditions? etc. The most significant issue is that it comes up short on a bound together and complete hypothetical framework. So, using this novel proposed methodology, those problems are easily solved. The proposed optimization algorithm is useful to overcome those problems.
Availability of data and material
The datasets used and/or analyzed during the current research study are available from the corresponding author on reasonable request.
Abbreviations
 TFC:

Total cost of fuel
 \(F_{{{\text{cost}}\,n}} (P_{nt} )\) :

Cost of fuel for a particular generating unit nth at that particular time ‘t’ hour
 \({\text{SUC}}_{n,t}\) :

Cost of startup for nth unit within ‘t’ hours
 \({\text{SDC}}_{nt}\) :

Cost of Shutdown for nth unit within ‘t’ hours
 \(U_{nt}\) :

Unit status at time t
 \(A_{n}\) :

Coefficient of cost for nth unit
 \(B_{n}\) :

Coefficient of cost for nth unit
 \(C_{n}\) :

Coefficient of cost for nth unit
 \({\text{HSU}}_{n}\) :

hot start for nth unit
 \({\text{CSU}}_{n}\) :

cold start for nth unit
 \(P_{n,t}^{\text{MAX}}\) :

Maximum electrical power generation by unit n
 \(P_{n}^{\text{MIN}}\) :

Minimum electrical power which generation by unit n
 \(P_{n,t}\) :

Electrical power generation of unit nth at the time span ‘t’
 \({\text{PD}}_{t}\) :

load demand at ‘t’ hours
 INS_{n} :

initial status of unit n at time ‘t’
 \(T_{n,t}^{\text{OFF}}\) :

Initial OFF status for nth unit at time ‘t’
 \(T_{n,t}^{\text{ON}}\) :

Initial ON status for nth unit at time ‘t’
 \(T_{n}^{\text{UP}}\) :

UP condition for n no. of power generating unit
 \(T_{n}^{\text{DW}}\) :

DOWN condition for n no. of power generating unit
 K :

incremental cost for shut down of unit
 \({\text{PD}}_{n}\) :

Power demand for nth unit
 \({\text{SR}}_{t}\) :

spinning reserve necessity
 T ^{COLD}_{ n } :

Time span for COLD start of n no. of generating unit
 Np:

Population number
 t :

No. of hours
 NU:

No. of generators
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Acknowledgements
The corresponding author wishes to thank Dr. O.P. Malik, Professor Emeritus, Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, Alberta, CANADA for continuous support, guidance, encouragement and for providing advance research facilities for postdoctorate research at the University of Calgary, Alberta, CANADA.
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AN analyzed and interpreted the data regarding the optimal scheduling of each power generating unit during 24 h and overall cost for power generation of each unit with renewable energy sources in summer, winter, autumn and spring and also drafted the work or substantively revised it and was a major contributor in writing the manuscript. VK have made substantial contribution to the conception and design of the research work and the creation of MATLAB coding (Software) used in the work. All authors have read and approved the manuscript, and the content of the manuscript has not been published or submitted for publication elsewhere.
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Additional file 1.
Test data for standard Unit Commitment Problems.
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Nandi, A., Kamboj, V.K. A meliorated Harris Hawks optimizer for combinatorial unit commitment problem with photovoltaic applications. Journal of Electrical Systems and Inf Technol 8, 5 (2021). https://doi.org/10.1186/s43067020000263
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DOI: https://doi.org/10.1186/s43067020000263