Research article

Multi-objective real-time integrated solar-wind-thermal power dispatch by using meta-heuristic technique

  • Received: 06 June 2022 Revised: 24 July 2022 Accepted: 29 July 2022 Published: 11 August 2022
  • The elevated demand for electrical power, expeditious expenditure of fossil fuels, and degradation of the environment because of power generation have renewed attentiveness to renewable energy resources (RER). The rapid augmentation of RER increases the convolutions in leveling the demand and generation of electrical power. In this paper, an elaborated α-constrained simplex method (ACSM) is recommended for multi-objective power dispatch problems. This methodology is devised after synthesizing the non-linear simplex method (SM) with the α-constrained method (ACM) and the evolutionary method (EM). ACSM can transfigure an optimization technique for the constrained problems by reinstating standard juxtapositions with α-level collations. The insertion of mutations and multi-simplexes can explore the periphery of the workable zone. It can also manage the fastness of convergence and therefore, the high precision solution can be obtained. A real-time multi-objective coordinated solar-wind-thermal power scheduling problem is framed. Two conflicting objectives (operating cost and emission) are satisfied. The case studies are carried out for Muppandal (Tamil Nadu), Jaisalmer (Rajasthan), and Okha (Gujarat), India. The annual solar and wind data are analyzed by using Normal Distribution and Weibull Distribution Density Factor, respectively. The presented technique is inspected on numerous archetype functions and systems. The results depict the prevalence of ACSM over particle swarm optimization (PSO), simplex method with mutations (SMM), SM, and EM.

    Citation: Sunimerjit Kaur, Yadwinder Singh Brar, Jaspreet Singh Dhillon. Multi-objective real-time integrated solar-wind-thermal power dispatch by using meta-heuristic technique[J]. AIMS Energy, 2022, 10(4): 943-971. doi: 10.3934/energy.2022043

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  • The elevated demand for electrical power, expeditious expenditure of fossil fuels, and degradation of the environment because of power generation have renewed attentiveness to renewable energy resources (RER). The rapid augmentation of RER increases the convolutions in leveling the demand and generation of electrical power. In this paper, an elaborated α-constrained simplex method (ACSM) is recommended for multi-objective power dispatch problems. This methodology is devised after synthesizing the non-linear simplex method (SM) with the α-constrained method (ACM) and the evolutionary method (EM). ACSM can transfigure an optimization technique for the constrained problems by reinstating standard juxtapositions with α-level collations. The insertion of mutations and multi-simplexes can explore the periphery of the workable zone. It can also manage the fastness of convergence and therefore, the high precision solution can be obtained. A real-time multi-objective coordinated solar-wind-thermal power scheduling problem is framed. Two conflicting objectives (operating cost and emission) are satisfied. The case studies are carried out for Muppandal (Tamil Nadu), Jaisalmer (Rajasthan), and Okha (Gujarat), India. The annual solar and wind data are analyzed by using Normal Distribution and Weibull Distribution Density Factor, respectively. The presented technique is inspected on numerous archetype functions and systems. The results depict the prevalence of ACSM over particle swarm optimization (PSO), simplex method with mutations (SMM), SM, and EM.



    Since the industrial uprising, the worldwide energy recoupment has been governed fundamentally by fossil fuels. This has crucial implications for the atmosphere. The expanded employment of RER can assist in the de-carbonization of the energy system in the future. This clean energy can help in dropping the inimical fossil fuel use and energy imports. Hence, it can portray an essential role in garnering the environment green and creating economic evolution.

    With the expansion in the share of renewable energy associated with the power grid, the effectual collaboration of the function of different energy sources has emerged as a fresh challenge to the power system scheduling. The integrated operation of the RER-based power generation system may enhance the conflicts between electrical power generation and varying power outputs.

    India is a country of geographical diversities. It has a large number of treasures of RER due to its magnificent topographical location. Its huge 'Thar Desert' of Rajasthan has high wind speeds and intense solar radiation. Its vast coasts are a great source of wind, solar, and tidal energies. It has sky-high 'Himalayan Mountain Ranges', which are the origin of thousands of water bodies and forests. Its large plains and plateaus have solar and wind energies, in abundance. To meet the power need and to turn down the role of fossil fuels in the power generation system, a large number of RER-based systems have already been implanted, in the country. Up to December 2019, the total RER-based instated capacity of India was 84 GW, with a set target of inducting 175 GW, by 2022 [1]. Therefore, the deployment of many more RER-based power generating systems is required, so that the rule of fossil fuels can be overthrown and quality power can be delivered to every needy, without much disturbing nature.

    In the past, Liaquat et al. [2] have proposed multi-update position criteria for enhancing the investigation characteristics of the traditional firely technique while incorporating the effect of the globally best result on the fluctuation of the fireflies in the exploration zone of the objective function. They have put in the dynamic search space squeezing to compress the fireflies movement inside the definite boundaries to circumvent their oscillatory movement obtained while getting on for the global best solution by finding out the best trajectory for one and all fireflies. Rahimi and co-researchers [3] have elaborated a stochastic thermal and electric load scheduling problem considering the security constraints and also uncertainties of loads, RERs (wind and solar), and market price. They have used a scenario reduction approach to model all uncertain parameters. Naverson et al. [4] have adopted the continuous-time framework to design flexible hydropower sources negotiating with thermal generators (slow-ramping) to minimize the operation cost of the system. They have demonstrated their study through a small-scale case study in which a hydropower plant is connected to a thermal power plant with a manageable high voltage direct current cable.

    Narang et al. [5] have applied the predator-prey optimization method for power scheduling of variable/fixed-head hydrothermal system. Predator assists to sustain heterogeneity in the swarm and also avert ill-timed convergence to the localized sub-optimal. They have used the variable elimination technique to control the equality constraint by abolishing variable exactness. Researchers in [6] have employed an adaptive predator-prey optimization to evaluate thermal power scheduling problems in a multi-objective framework. They have maintained the velocity of prey within limits by acknowledging the supplementary and obstruction features.

    Mondal and co-workers [7] have solved the economic load dispatch problem by considering both wind turbines and thermal generators to minimize fuel cost and NOx emission by using Gravitational Search Algorithm. They have also investigated the influence of the wind system on NOx emission. Ansari et al. [8] and Das et al. [9] have used the point estimate method to flourish the unreliability of wind and solar power systems. Das and co-workers have calculated the power generation cost with the crow search algorithm. Dasgupta and co-researchers [10] have employed the sine-cosine algorithm to minimize the cost of generation and emission pollutants. The parameters of the optimization technique have been used to balance the exploitation and exploration conditions to explore the optimal global solutions. Zhang et al. [11] have presented an enhanced borg (EBorg) algorithm to optimize a short-term dual-objective co-scheduling problem of a hydro-thermal-wind system. The EBorg framework has been comprised of ϵ dominance-based archive, crowding distance and pareto-dominance-based population upgrading mechanism, and auto-adaptive multi-operator reunification. They have worked on two objectives-generation cost and emission pollutants.

    Reddy et al. [12] have presented an optimal power dispatch problem, considering an auction market with multi-mode, and solved it by using the genetic algorithm. They have maximized the total benefit of the participants at all nodes of the system. Ready has focused on the congestion management of an optimal power flow in the deregulated electricity market by using the multiobjective grenade explosion technique [13]. He has proposed a power flow problem in a multiobjective framework and optimized it by PSO. He has used fuzzy satisfying maximization for decision-making [14]. Salkuti has worked on a novel power scheduling of a hybrid system (wind, solar, and thermal generators) considering risk level and operating cost by using a non-dominated sorting genetic algorithm-Ⅱ. He has optimized the real-time day-ahead divergence costs of the system [15]. He has considered an economic environmental dispatch problem having nonlinear features (valve point loading, ramp rate, prohibited operating zone effects, etc.) of thermal generators and optimized it with PSO [16]. He has presented an optimal feeder reconfiguration/network reconfiguration approach for minimizing operating cost and power losses of the system by using the crow search algorithm [17].

    Zhang et al. [18] have developed a robust collaborative consensus algorithm for a dispersed economic dispatch having a practical communication network. The network has consisted of switching topology, noise, and transmission delay. Researchers in [19] have proposed the decentralized collaborative control structure of an independent virtual generation tribe (VGT) for a smart grid by using a VGT-based collaborative consensus algorithm (CCA) and a VGT-based robust CCA. Zhang et al. [20] have worked on a new cyber-physical-social system with parallel learning for distributed energy management of a microgrid. They have used the correlated equilibrium-based general sum game and the novel adaptive consensus algorithms for their work. Tan and co-workers [21] have presented a new fast learning optimizer for optimal energy management. Real-time non-convex energy management has been divided into two-layer optimization to reduce difficulties during optimization.

    Biswas et al. [22] have employed success history-based adaptation method of differential evolution algorithm to solve optimal power flow incorporating uncertainty of wind and solar system with traditional thermal power generation system. They have employed lognormal probability distribution function and Weibull Distribution Functions for predicting solar and wind output power, respectively. Das and co-researchers [23] have evaluated a hydrothermal scheduling problem deploying quasi-reflected symbiotic organisms search. This algorithm has been comprised of symbiotic organisms search to refine the execution of the prescribed technique.

    He et al. [24] have used an upgraded combined binary and real number differential evolution technique based upon SHADE to present a model of coordinated power generation scheduling of hydro-thermal-wind system including spinning reserve. They have demonstrated the proposed model with an example and case study. Researchers in [25] have solved a multi-objective economic load dispatch problem with emended salp swarm algorithm. This algorithm includes the solitary and colonial phases of the reproduction cycle of life of salp. They have handled the equality constrained and prescribed functioning zone constraints.

    Li et al. [26] have taken a large-scale hydro-wind-solar field in southwestern China to design an optimal power generation scheduling problem. They have maximized the total generated power and the minimum monthly collected output for the entire scheduling interim and minimized the environmental over & short discharge. Panda and Tripathy [27] have employed a new evolutionary hybrid algorithm for environmental optimal power flow problems including wind and thermal power generation systems. They have considered operational cost, emission cost, real power loss, and installation cost of FACTS devices to maintain a stable voltage.

    Takahama and Sakai [28] have worked on the α -constrained simplex method (ACSM), to solve the constrained optimization problem of the real world. They have instated three modifications in the nonlinear simplex search method to obtain the borderline of the feasible zone, moderate the convergence speed, increase the accuracy, and enhance the overall efficiency of the system. Brar et al. [29] have suggested multi-objective fuzzy satisfying power generation scheduling by using simplex weightage pattern search. They have minimized four contradictory constraints and obtained real and reactive line flows by using generalized Z-bus distribution factors.

    In this paper, a futuristic practice described as ACSM is executed to resolve a multi-objective real-time coordinated solar-wind-thermal power scheduling problem. It is a reconditioned unification of the SM introduced by Nelder and Mead. It is an improved conversion technique for constrained optimization. In this method, the non-linear simplex method is perceived as an evolutionary method in which a specific choice, substitution approach, and exceptional variation operator are employed to get high convergence speed, accuracy, and efficiency. It has been invented after hybridizing an established SM with certain other procedures (like-EM, α-constrained method, etc.). To frame this optimization method, three changes in the ordinary SM are executed: (ⅰ) α-level comparisons, (ⅱ) the worst point's mutation, and (ⅲ) use of multi-simplexes. In this study, three places from different parts of India are sorted out, where a coordinated solar-wind-thermal power system can operate efficiently. These marked out places are Muppandal (Tamil Nadu), Jaisalmer (Rajasthan), and Okha (Gujarat). A multi-objective coordinated solar-wind-thermal scheduling problem is formulated and optimized for the contemplated sites for two test systems, by using ACSM. To reflect the ascendancy of the suggested operating procedure, the outturns are differentiated with PSO, SMM, SM, and EM.

    In the real world, the coordinated multi-objective optimization problems (CMOP) prerequisite the optimization of many contradictory constraints, concurrently. In this paper, two objectives of thermal and RER systems are discerned. These objectives are total functioning cost and emission (NOx, SO2, & CO2). Therefore, the CMOP is formulated as the minimization of two objectives subject to many equality and inequality constraints. All the objectives are evaluated discretely and then they are solved concomitantly using multi-objective configuration. Two objectives of interest (cost and emission) are of conflicting nature, especially in the case of thermal power generation system. An optimal solution to one can be attained at the cost of the other. Therefore, they are solved simultaneously to achieve the best compromise solution. These objectives can be stated as:

    The operating cost of a coordinated solar-wind-thermal power generating system depends upon the cost of fossil fuel and the functioning cost including the uncertainty cost of an RER-based power system. This objective can be conceived as the minimization of the total functioning cost of the system. The economy objective of the contemplated system can be examined as [30,31,32,33,34]:

    F1=Tgi=1(aTiTp2i+bTiTpi+cTi)+Wgj=1Ywj+SgQ=1YSQ(Rs/h) (1)

    where Tg, Wg, and Sg are the number of thermal generators, wind generators, and solar units, respectively. Tpi is the power output of the ith thermal generator in MW. aTi,bTi and cTi are the cost coefficients of the ith thermal generator. Ywj is the wind power cost of the jth wind generator. YsQ is the solar power cost of the Qth solar unit.

    Contrasting with RER-based plants the substantial environmental pollution is originated from the thermal power generation system, which is comprised mostly of nitrogen oxides (NOx), sulfur dioxide (SO2), and carbon dioxide (CO2). In this paper, NOx, SO2, & CO2 are specified as the emission determining index and treated as a single objective instead of three objectives. The economy and emission functions can be directly associated through the persistent factor known as emission rate per Mkcal for the defined grade and categorization of fossil fuel. The total thermal emission content is taken as the quadratic functions of thermal power output and can be expressed as [30,31,32,33,34]:

    F2=Tgi=1(dXiTp2i+eXiTpi+fXi)(kg/h) (2)

    where dXi,eXi & fXi are the emission coefficients of the ith thermal generator and X is the emission (NOx, SO2, & CO2).

    The goal of the multi-objective coordinated optimization problem for solar-wind-thermal power system is the acquisition of the optimal power dispatch by effectuating the minimization of incongruous objectives, simultaneously. The multi-objective power scheduling problem can be stated as [30,31,32,33,34]:

    Minimize [F1,F2]T

    Subject to:

    ⅰ. The equality constraint:

    The total generated power of the solar-wind-thermal system must be equal to the addition of power demand and transmission losses. Therefore, the load demand equality constraint of the developed problem can be defined as [30,31,32,33]:

    Tgi=1Tpi+Wgj=1Wpj+Sgq=1Spq=PD+PLoss (3)

    where Wpj is the scheduled power of the jth wind generator in MW. SpQ is the scheduled power of the Qth solar unit in MW. PD is the system power demand in MW. PLoss is the total system transmission losses in MW.

    ⅱ. Power generation limits of generating units:

    The decision variables of thermal, wind, and solar systems (Tpi, Wpj, & SpQ) must lie between the power generation limits of the respective generating unit. The lower and upper generation limits enforced on thermal, wind, and solar power generating units are [30,31,32,33]:

    TpiminTpiTpimax(i=1,2,,Tg) (4)
    0WpjWprj(j=1,2,,Wg) (5)
    0SpQSprQ(Q=1,2,,Sg) (6)

    where Tpimin and Tpimax are lower and upper limits of the power output of the ith thermal generator in MW, respectively. Wprj is the rated power output of the jth wind generator and SprQ is the rated power output of the Qth solar unit in MW.

    Electrical power generation has arisen as the principal implementation of wind energy, globally. This energy renders an accepted contemporary power generation source and a vital participant in the world's energy trade. Wind power generation exceedingly depends upon wind speeds. To procure the precise solution of wind power dispatch prognostication of wind power is decisive. In this paper, the Weibull Distribution Density Factor is used to examine irregular wind data. The wind speed variations are demonstrated by using the Probability Density Function (PDF) and it can be evaluated as [32,33,34,35,36]:

    FPDF=(kc)(vc)k1exp[(vc)k],(0v) (7)

    where is the annual average wind speed in m/sec. k is the shape factor. c is the scale factor in m/sec.

    The shape factor is a parameter that displays the span of allocation of wind speeds. It can be obtained as [32,34]:

    k=(σvm)1.086 (8)

    where vm and σ are the mean wind speed and the mode wind speed, in m/sec, respectively.

    The scale factor displays the capability of the wind power of that location. It can be defined as [32,34]:

    c=vmΓ(1+1k) (9)

    The Gamma Function has frequently applied extension of the factorial functions to the complex numbers and can be observed as [32,34]:

    Γ(x)=0ettx1dt (10)

    Wind energy approximation is decisive to guarantee grid regulation and optimal wind power dispatch. Wind velocity distribution for a specific wind power zone can be designed by using probability distribution functions (PDF). The PDF of wind power can be expressed as [32,34]:

    f(Wavj)={(kIjvinjc)((1+ρjIj)vinjc)k1exp[((1+ρjIj)vinjc)k];for0<vop<vRj1exp[(vRjc)k]+exp[(v0jc)k];forvop=0exp[(vRjc)k]exp[(v0jc)k];forvop=vRj (11)

    where vinj,vRj,&voj are the cut-in speed, the rated speed, and the cut-out speed of the jth wind generator in m/sec, respectively. vop is the operating wind speed in m/sec.

    ρj=vopvRj (12)
    Ij=(vRjvinj)vinj (13)

    The available wind power at a particular location depends upon the specifications of the jth wind generator and operating wind speeds during the considered period. The available wind power for the jth wind generator, at different wind velocities, can be calculated as [32,33,34,35]:

    Wavj={0;forvop<vinjandvop>vojWprj(vopvinj)(vRjvinj);forvinj<vop<vRjWprj;forvRjvopvoj (14)

    Electrical power systems which assimilate RER have to deal with unreliability about the accessibility of load or injected power. This causes the consideration of uncertainty costs in the representation of stochastic economic dispatch. The observation of these costs is vital for the accepted management of RER and the proper issuance of the available energy amount for the power system. The actual cost of wind power is often found more than its anticipated cost. The direct cost function of the jth wind generator can be evaluated as [32,34]:

    ydwj=yw1jWpj (15)

    where yw1j is the direct cost coefficient of the jth wind generator.

    When the actual wind power is found less than the planned wind power, the operator has to a pay penalty cost, which is called the overestimation cost. The overestimation cost function of the jth wind generator is determined as [32,34]:

    yowj=yw2jWpj0(WpjWavj)f(Wavj)d(Wavj) (16)

    where yw2j is the overestimation cost coefficient of the jth wind generator.

    On the other hand, underestimation cost is fine for not utilizing the available wind power for the certain duration. The underestimation cost function of the jth wind generator can be obtained as [32,34]:

    yuwj=yw3jWprjWpj(WavjWpj)f(Wavj)d(Wavj) (17)

    where yw3j is the underestimation cost coefficient of the jth wind generator.

    The total operating wind power cost of the jth wind generator is equal to the sum of the direct cost, the overestimation cost, and the underestimation cost, for a specific time. The total operating cost function of the jth wind generator can be stated as [32,33,34]:

    Ywj=ydwj+yowj+yuwj (18)

    Solar energy can be pivotal to the clean energy future. The sun daily radiates far more energy than the power requirements of all the human beings on earth. Solar radiations vary with the topography and climate of a certain area. In this paper, to analyze the irregular solar data, normal distribution is utilized. The Probability Density Function (PDF) of solar irradiance can be calculated as [31,34]:

    fs(It)=e(ItM)22D2D2π (19)

    where It is the solar irradiance at a given time, M is the mean of solar irradiance over the year, and D is the standard deviation of solar irradiance, in kWh/m2/day.

    The available power of the Qth solar unit can be evaluated as [31,33,34]:

    SavQ=SprQ(1+ka(ToTrQ))IrIm (20)

    where To is the operating temperature and TrQ is the reference temperature of the Qth solar unit, in ºC. ka is the temperature coefficient in /ºC. Im is the maximum value of solar radiation incident under standard conditions in kWh/m2/day.

    Solar radiation is a broad expression for the electromagnetic radiation discharged by the sun. These can be seized and converted into useful formations of energy, such as electricity and heat, employing different technologies. The solar radiations incident on an inclined plane is expressed as [31,33,34,37]:

    Ir=It[cos(A)cosδcosω+sin(A)sinδ](cosδcosωcos+sincosδ) (21)

    where Ø is geographical latitude, A is the angle of the tilt of the solar collector, δ is the sun's declination, and ω is the hour angle, in degrees. A = Ø ± 15°

    The declination angle of the sun varies seasonally because of the Earth's tilt on its rotation axis and its rotation around the sun. This angle would always be 0° if the Earth were not leaning on its rotation axis. As the Earth is sloped by 23.45° and the angle of declination depends upon this amount [37]. The angle of the sun's declination can be obtained as [31,33,34]:

    δ=23.45°sin(360(284+dn)365) (22)

    where dn is the number of the day of the year.

    Similar to the wind power system, due to the unsure conduct of the sun, the forecasted solar power may not always be equal to the scheduled solar power. The operating cost of the solar power system also depends upon the direct cost and the uncertainty cost of the solar unit. The direct cost function of the Qth solar unit can be determined as [31,34]:

    ydsQ=S1sQSpQ (23)

    where S1sQ is the direct cost coefficient of the Qth solar unit.

    The penalization for deploying another energy resource or for not supplying energy is called overestimation cost as discussed in the wind system. The overestimation cost function of the Qth solar power unit is given as [31,34]:

    yosQ=S2sQ(SpQSavQ)fs(It) (24)

    where S2sQ is the overestimation cost coefficient of the Qth solar unit.

    The castigation for not utilizing all the available power or the underestimation cost function of the Qth solar power unit can be obtained as [31,34]:

    yusQ={S3sQ(SavQSpQ)}fs(It) (25)

    where S3sQ is the underestimation cost coefficient of the Qth solar unit.

    The total operating cost of the Qth solar unit can be evaluated as [31,34]:

    YSQ=ydsQ+yosQ+yusQ (26)

    These losses directly rely upon the characteristics of the network and the operation mode. They are mostly caused by energy dissipation in the conductors, appliances used in transmission lines, etc. Regardless of how the electrical power system is modeled, these losses are unpreventable and must be fabricated before depiction can be evaluated. In this paper, the transmission losses of the coordinated solar-wind-thermal power system are obtained by making use of Kron's approximated loss formula through beta-coefficients and these can be expressed as [30,31,32,33,34]:

    PTLoss=Tgi=1(Tgj=1TpiBTiTjTpj) (27)
    PWLoss=Wgi=1(Wgj=1WpiBwiwjWpj) (28)
    PSLoss=Sgi=1(Sgj=1SpiBsisjSpj) (29)

    where PTLoss, PWLoss, and PSLoss are the transmission losses due to the thermal system, the wind system, and the solar system respectively, in MW.

    Total transmission losses of the system can be evaluated as [31,32,33,34]:

    PLoss=PTLoss+PWLoss+PSLoss (30)

    In this paper, ACSM is applied to optimize the proposed real-time constrained power scheduling problem. In this technique, SM is upgraded by inlaying α-level comparisons rather than ordinary comparisons, mutations of the worst points, and multi-simplexes as a substitute for a single simplex.

    The α -level comparisons are used to transform algorithms for unconstrained optimization problems into algorithms for constrained optimization problems. By employing these comparisons, the search points are compared based on the pre-defined satisfaction level of their constraints. It means that the points are differentiated on the grounds of their constraint infringement.

    The optimal solutions to the constrained problems are frequently found near the frontier of the feasible region. Therefore, to explore the points around the boundary of the feasible zone, the mutations to the worst points and multi-simplexes are adjoined, in the algorithm. It can also control the convergence speed.

    The nonlinear simplex method is upgraded by performing the following alterations to enhance its efficacy [28]:

    (ⅰ) α -level collations:

    To transfigure a constrained optimization problem to an unconstrained optimization problem, α -level comparisons can be applied in place of usual comparisons. Consider f(z) is an objective function and μ(z) is its satisfaction level. The α -level comparisons are the order relation on (f(z),μ(z)) set, in which the viability of z is more vital than minimizing f(z). α -level collations between any two objective functions fA & fB having satisfaction levels μA & μB respectively and can be stated as follows:

    (fA,μA)<α(fB,μB)<=>{fA<fB,;ifμA,μBαfA<fB,;ifμA=μBμA>μB;else (31)
    (fA,μA)α(fB,μB)<=>{fAfB,ifμA,μBαfAfB,ifμA=μBμA>μB,else (32)

    The value of α lies between 0 and 1. The α level collations are analogous to the usual differentiations if the value of α is zero.

    (ⅱ) Incorporation of mutations:

    In constrained optimization problems usually, the optimal solutions are found very close to the borderline of the feasible province. While solving these problems with the help of the nonlinear simplex method, when simplex reduces some search points encircling the boundary of the feasible zone are ignored, sporadically. Therefore, to evade such situations mutations are included because they are capable of producing optimal results surrounding the frontier of the feasible stretch of the surveyed points. The least desirable point is exchanged by mutations utilizing Eqs (46, 47).

    Mutations can also sway the consolidation speed of the algorithm. Therefore, the preferable value of the mutation rate should be picked out. High values of mutation rate gravitate towards a large number of calculations and therefore sometimes computation speed may decay.

    (ⅲ) Inclusion of multi-simplexes:

    In the nonlinear simplex method, simplex may overlook affine autonomy occasionally and therefore this technique can't search for the optimal solutions. To handle such situations multi-simplexes are included. The affine autonomous simplexes can work for the optimal solutions, even when some simplexes mislay affine sovereignty. In a nonlinear simplex method, for the decision variable of n dimension, initially, n+1 points are searched, whereas to formulate multi-simplexes at least n+2 points are generated.

    The count of explored points regulates the diversification of the investigation operation and also the simulation speed. If the number is very small but the convergence speed is lofty the surveyed points usually concur to a confined optimum. If the number is very large and the convergence speed is low, the explored points can't arrive at the global optimum.

    Consider f(zi) is an objective function, where zi is an n -dimensional vector of decision variable such that (i=1,2,,N). The algorithm of ACSM can be stated as [28,33,34]:

    (1) Put expansion factor γ>1, contraction factor b(0,1), mutation rate PM(01), tolerance limit Ɛ = 0.001, algorithm parameters β = 0.03, and Tα = 50.

    (2) Randomly generate the initial search points (N(>n+1)) in the extremities of the search zone. Scrutinize the ith dimension arbitrarily and now either the upper limit or the lower limit is designated to zi. The remaining variables are created inconstantly betwixt the upper limit and lower limit of each variable.

    (3) Determine the feasible solution by employing the designed casual heuristic search approach. In this paper, two distinct search proceedings are executed auspiciously to attain the feasible solution that delivers the solar-wind-thermal power generations during appeasing the equality constraints (Eq (3)). One strategy is to encounter the accessible power demand constraint of the system over the inspected interim. The second maneuver is based upon the handling of the uncertainty of the RER system using Eq (7) to Eq (26).

    (4) All the objectives and objective functions commensurate to inequality constraints of the solar-wind-thermal generation problem are collaborated to evaluate the amalgamated impact of all the objectives concurrently deploying their membership functions Eq (44). The maximum value of the clubbed membership function demonstrates the satisfaction level analogous to the non-inferior produce of the objective function Eq (45).

    (5) Obtain zl (the best point), zh (the worst point), and zs (next to the worst point), by using the following equations:

    zl=argminif(zi) (33)
    zh=argmaxif(zi) (34)
    zs=argmaxihf(zi) (35)

    (6) Create the random number R and rationalize zh as:

    (7) Skip the worst point and form the initial simplex with n+1 points. Find the centroid of the simplex from the following equation:

    z0=1n+1n+1i=1ihzi (38)

    (8) The value of the α can be restrained as stated in Eq (39). The starting value of α is α0 and it is found in the initial search. It is computed as the average of the best satisfaction level value and the mean of the total satisfaction level values. When the iteration number alters to t, the value of α is upgraded as the multiple of Tα. The value of α is disposed to 1 when the iteration number exceeds Tmax2. Its values lie between (0–1).

    α(t)={12(best(μd(zi))+1NNi=1μd(zi));t=0(1β)α(t1)+β;0<tTmax2and(tmodTα)=0α(t1);0<tTmax2and(tmodTα)01;t>Tmax2 (39)

    where t is the iteration number. Tmax is the maximum number of iterations.

    (9) Calculate the reflected point zr by reflecting the best point about the centroid, with the support of the following equation:

    zr=(1+α)z0αzh (40)

    (10) If zr is superior to the best point, ie. (f(zr),μd(zr))<α(f(zl),μd(zl)), then go to step 11, else go to step 12.

    (11) The expansion process takes place, in which with the help of reflection operation the simplex progresses towards the better zone of the search space. Determine the expansion point ze as:

    ze=γzr+(1γ)z0 (41)

    If the expansion point is more reformed than the best point, i.e., (f(ze),μd(ze))<α(f(zl),μd(zl)), then zh is reinstated by ze, else zh is replaced by zr, and go back to step 4. Figure 1 represents the flowchart of ACSM.

    Figure 1.  Flowchart of α-Constrained simplex method.

    (12) If the reflection operation drags the simplex towards the deplorable zone, i.e., the reflection point is finer than and equal to the next to the worst point of the simplex ((f(zr),μd(zr))α(f(zs),μd(zs))), then displace zh by zr and go back to step 4, else go to step 13.

    (13) If the worst point is less scholarly than the reflected point, i.e., (f(zr),μd(zr))<α(f(zh),μd(zh)), then zh is replaced by zr and go to step 4, else evaluate contraction process is supervened. The contraction point zc can be calculated as:

    zc=bzh+(1b)z0 (42)

    If (f(zc),μd(zc))<α(f(zh),μd(zh)), then zh is replaced by zc, else update zh as:

    zh=bzh+(1b)zl (43)

    and go back to step 4.

    (14) Evaluate all the objective functions.

    (15) Check the stopping criteria. If abs (flfh)ϵ then go to step 16, else go back to step 4.

    (16) Stop.

    The contraction factor controls the convergence speed of the computation process. If it is small the investigating process reaches its centroid very soon. If the convergence speed is high the search operation may omit the global optima and coincide with local optima. If the contraction factor is very large the processing speed turns out to be low therefore search may not attain global optima.

    The feasible zone can be extended by moderating the value of α and the extended feasible zone can be brought down to the primal by increasing the value of α to 1, in a constrained SM. The value of β regulates the growing speed of α and the speed of bringing down the extended feasible zone. The value of α reaches 1 moderately if the value of β is less. Here, the possibility that the explored points approach to local optimum is less. If the value of β is very small the explored points must examine a large area, therefore efficacy reduces. If the extended zone happens to be large at the half iterations, the explored points might reach to feasible zone quickly therefore, it can omit the global optima [28].

    The decision-making has an indefinite character and fuzzy targets for the objective functions. The goals comprise categories of alternatives whose limits are not distinctly defined. The fuzzy goals and fuzzy objectives can be defined accurately as fuzzy sets in the zone of substitutes. Here, a fuzzy decision can be observed as the junction of the specified targets and objectives. These fuzzy aims are adjusted by establishing their membership functions, whose values vary from 0–1. The value 0 of the membership function means irreconcilability and value 1 indicates complete complementarity. It can be defined as [30,31,32,33,34]:

    μ(Fi)={1;fifminifmaxififmaxifmini;fmini<fi<fmaxi0;fifmaxi(i=1,2,,M) (44)

    where fi is the objective function. fmaxi and fmini are the maximum and minimum values of the objective function, respectively.

    The Fuzzy Cardinal Priority Ranking (the membership function) of the non-dominated (pareto-optimal) solution to a fuzzy set can be stated as [30,33,34]:

    μKd=Mi=1μ(fKi)Kk=1Mi=1μ(fKi) (45)

    where K is the number of non-dominated solutions.

    The optimization problems of the electrical power systems (EPS) are very strenuous to solve because the EPSs are very sizeable, composite, structurally extensively distributed, and are impacted by several unpredicted circumstances. Real-time optimization techniques utilize the accessible computations in the optimization structure and are, thus, competent in conducting the appropriate self-optimizing regulation. In this paper, a real-time multi-objective coordinated solar-wind-thermal power scheduling problem is optimized for three different places in India, by using ACSM. The names of these places are:

    1. Muppandal (Tamil Nadu)

    2. Jaisalmer (Rajasthan)

    3. Okha (Gujarat)

    The same set of power generating units (PGU) and coefficients (cost and emissions) are used for all three places, to access the performance of PGU in the different geographical and environmental conditions. The functions of cost coefficients of the RER power system are also taken as the same for all three sites. This work is executed with the help of the FORTRAN-90 programming language using ACSM and results are collated with PSO, SMM, SM, and EM.

    India is the seventh-largest country in the world and it lies on the north of the equator between 8°4' north to 37°6' north latitude and 68°7' east to 97°25' east longitude [38]. It possesses large topographical and meteorological variations. Many parts of the country are rich in solar and wind energies. Muppandal, Jaisalmer, and Okha are among such places.

    Muppandal is a village in the Kanyakumari district of Tamil Nadu state of India. It is situated at the southernmost point of India. It has India's largest wind farm, with a 1500 MW installed capacity [39,40]. Jaisalmer city of Rajasthan is placed in the northwestern region of India. It is a segment of the 'Great Indian Thar Desert'. India's second-largest wind farm of 1064 MW capacity, is installed here [41]. Okha is a famous town in the Dwarka district of Gujarat state. It is a port at the west-central tip of India. It has a high solar and wind energy prospect. The solar and wind parameters of these three sites are tabulated in Table 1.

    Table 1.  Solar and wind parameters of Muppandal, Jaisalmer, & Okha.
    System variables Muppandal Jaisalmer Okha
    Geographical latitude-Ø (degrees) 8.15 26.95 22.469
    Annual average solar irradiance (kWh/m2/day) 5.68 5.79 5.86
    Average solar irradiance of June (kWh/m2/day) 5.48 5.47 4.74
    Reference temperature (℃) 32 41 33
    Mean wind speed (m/sec) 11.50 9.00 8.70
    Mode speed (m/sec) 5.50 3.42 2.90

     | Show Table
    DownLoad: CSV

    Every single locality on Earth acquires solar radiation at the minimum fragment of the year. The quantity of solar radiation that arrives at any one part on the surface of Earth differs according to geographic location, time of day, season, local weather, local landscape, etc. Due to the round shape of Earth, the sun hits the surface at dissimilar angles, varying from 0° to 90° [37]. When the rays of the sun are vertical the surface of Earth achieves all the possible energy. If the sun's rays are more tilted, they travel through the atmosphere for a longer period and therefore, become more diffused and scattered. Figure 2 displays the variation of solar radiation in three considered zones, over the year. It can be seen that all three locations have a high value of solar radiation (except in the monsoon season, from June to August). The maximum drops of radiation can be seen in Okha during this season. Therefore, it has the minimum value of available solar power, in this span of the year.

    Figure 2.  Variation of solar radiations in Muppandal, Jaisalmer, & Okha, over the year.

    The mean and mode wind speeds of the year are nearly 11.50 m/sec, 9.0 m/sec & 8.70 m/sec, and 5.50 m/sec, 3.42 m/sec & 2.90 m/sec, respectively for Muppandal, Jaisalmer, and Okha.

    In fact, because of the intermittency of wind and solar radiation, generating extra power results in increasing output fluctuations. Therefore, in this paper, an integrated scheduling model of a territorial RER-based energy system combined with a conventional thermal power generation system is established to encounter the oscillating power requirements of consumers. The considered coordinated solar-wind-thermal power system contains six generating units (two thermal generators, two solar units, and two wind farms).

    The fuel costs functions and the pollutant emission (NOx,SO2&CO2) functions of two thermal generators are given in Table 2. The fuel cost functions and pollutant emission functions are minimized over the set of permissible decision vector Tpi. The minimum and maximum generation limits of each thermal generator are taken as 10 MW and 250 MW, respectively.

    Table 2.  The characteristic fuel and emission functions of two thermal generators.
    Fuel cost (Rs/h) equations NOx emission (kg/h) equations
    F11 = 0.001345 Tp21 + 8.30154 Tp1 + 274.2241 F21= 0.006732 Tp21 − 2.39928 Tp1 + 610.2535
    F12 = 0.005963 Tp22 + 6.91559 Tp2 + 202.0258 F22= 0.006181 Tp22 − 0.39077 Tp2 + 50.3808
    SO2 emission (kg/h) equations CO2 emission (kg/h) equations
    F31 = 0.000813 Tp21 + 4.97641 Tp1 + 165.3433 F41= 0.106409 Tp21 − 12.73642 Tp1 + 1819.625
    F32 = 0.003578 Tp22 + 4.14938 Tp2 + 121.2133 F42= 0.40314 4Tp22 − 121.9812 Tp2 + 11381.07

     | Show Table
    DownLoad: CSV

    The solar system parameters for the 15th day of June of each year are enlisted in Table 3 and wind system parameters are charted in Table 4. At 1 PM the value of the hour angle is found as −15°. The values of angle of tilt of solar collector and cost coefficients of solar & wind systems are contemplated as the same for the described sectors of India.

    Table 3.  Parameters of solar units (PV).
    Solar system variables Specifications
    Hour angle (°) −15
    The angle of tilt of the solar collector (°) 20
    Temperature coefficient (/℃) −4.7 e−3
    The capacity of each solar unit (MW) 30
    Coefficient of direct cost (Rs/kWh) 4.50
    Coefficient of underestimation cost (Rs/kWh) 17.280
    Coefficient of overestimation cost (Rs/kWh) 12.280

     | Show Table
    DownLoad: CSV
    Table 4.  Parameters of wind farms.
    Wind system variables Specifications
    The capacity of each wind farm (MW) 30
    Cut in velocity vi (m/sec) 3.5
    Cut out velocity vo (m/sec) 25
    Rated speed vr (m/sec) 15
    Coefficient of direct cost (Rs/kWh) 4.00
    Coefficient of underestimation cost (Rs/kWh) 17.280
    Coefficient of overestimation cost (Rs/kWh) 12.280

     | Show Table
    DownLoad: CSV

    The fuel cost and the emissions (NOx,SO2&CO2) of the thermal generating system are obtained by using Eqs (1) and (2). The power balance Eq (3) is solved, subject to the equality constraint and power generation limits of thermal generators, wind generators, and solar units, using Eqs (4–6). The wind data is contemplated according to the Weibull distribution density function. The PDF of wind behavior is observed from Eq (7). The shape factor 'k' is evaluated from the mean and the mode wind speeds by using Eq (8), which is found as 2.229, 2.86 & 3.33 for Muppandal, Jaisalmer, and Okha, respectively. The scale factor 'c' is calculated from Eqs (9) and (10), by using the Gamma Function. These values are obtained as 12.981 m/sec, 10.099 m/sec, and 9.695 m/sec for Muppandal, Jaisalmer, and Okha, respectively.

    Figure 3 represents the variation of wind speed frequency distributions with a range of wind speeds, at 1 PM on the 15th day of June of each year, for Muppandal, Jaisalmer, and Okha. It can be seen that all three places have divergent heights and areas of their wind frequency curves because of their distinct wind distributions. The PDF of wind powers is determined from Eqs (11–13) and available wind powers for different locales are observed from Eq (14). The direct cost, overestimation cost, underestimation cost, and total operating wind power cost for all three wind power systems are evaluated from Eqs (15–18). Depending upon the regional wind distributions for the considered period of specified zones, the available wind powers are observed as 20.869565 MW, 14.3478 MW, and 13.565217 MW for Muppandal, Jaisalmer, and Okha, respectively.

    Figure 3.  Variation of wind speed frequency distributions with wind speeds of Muppandal, Jaisalmer, & Okha.

    The solar data is examined under the normal distribution of solar irradiance. The PDF of solar radiation is determined with the help of standard deviations and the mean of solar irradiance of three considered places (Figure 4), using Eq (19). The hourly beam solar irradiance incidents on an inclined plane are calculated with the help of Eq (21). The angle of declination of the sun is obtained from Eq (22). The available solar power depends upon solar radiation and the reference temperature of the examined area. Since Jaisalmer has the highest values of both of these variables for the testing interval, therefore it has the highest available solar power (27.39851 MW), which is succeeded by Muppandal (25.26320660 MW) and Okha (24.02259351 MW). The direct cost, overestimation cost, underestimation cost, and total solar power cost are obtained from Eqs (23–26). Transmission losses of the system are observed from Eqs (27–30). Fuzzy cardinal priority ranking of non-dominating solutions is employed to obtain the best compromise solution (BCS), with the help of Eqs (44–45).

    Figure 4.  Variation of PDF of solar irradiance of Muppandal, Jaisalmer, & Okha, over the year.

    This test system is comprised of power scheduling of solar-wind-thermal of three inspected places of India for 250 MW power demand. Since Muppandal has the maximum available wind power for the given spell, therefore it has the maximum scheduled wind power (40.1869 MW). It is found that Muppandal has the highest direct cost of wind power (97654.1700 Rs/h), which is followed by Jaisalmer 68422.3300 Rs/h and Okha (65499.0300 Rs/h). After optimizing the power scheduling problem, the values of fuel cost, NOx emission, SO2 emission, CO2 emission, total operating cost of wind system, total operating cost of solar system, transmission losses, and simulation time for Muppandal, Jaisalmer, and Okha are obtained as 1765.4010 Rs/h, 519.1340 kg/h, 1059.7230 kg/h, 5600.0180 kg/h, 103553.3200 Rs/h, 135482.1000 Rs/h, 3.872521 MW/h & 0.78 sec; 1846.7690 Rs/h, 513.3438 kg/h, 1108.5220 kg/h, 5444.6470 kg/h, 36270.0900 Rs/h, 147552.5000 Rs/h, 4.263066 MW/h & 0.78 sec, and 1906.8030 Rs/h, 514.9072 kg/h, 1144.5360 kg/h, 793.7719 kg/h, 66702.1700 Rs/h, 127596.0000 Rs/h, 4.554507 MW/h & 0.78 sec, respectively, with ACSM. The solution of power scheduling problem of test system-Ⅰ by using ACSM is tabulated in Table 5.

    Table 5.  Solution of power scheduling problem of test system-Ⅰ, by using ACSM.
    Output variables Muppandal Jaisalmer Okha
    Thermal power system Scheduled power (MW) Unit 1 81.456790 87.801320 89.685880
    Unit 2 81.599080 85.044940 90.563100
    Fuel cost (Rs/h) Unit 1 959.3652 1013.47900 1029.5740
    Unit 2 806.0356 833.2900 877.2296
    Total fuel cost (Rs/h) 1765.4010 1846.7690 1906.8030
    NOx emission (kg/h) Unit 1 459.4841 451.4910 449.2212
    Unit 2 59.64996 61.85275 65.6860
    Total NOx emission (kg/h) 519.1340 513.3438 514.9072
    SO2 emission (kg/h) Unit 1 576.1001 608.5461 618.1964
    Unit 2 483.6227 499.9755 526.3396
    Total SO2 emission (kg/h) 1059.7230 1108.5220 1144.5360
    CO2 emission (kg/h) Unit 1 1488.2030 1521.6650 1533.2550
    Unit 2 4111.8150 3922.9820 3640.5310
    Total CO2 emission (kg/h) 5600.0180 5444.6470 793.7719
    Wind power system Shape factor‑ k 2.229 2.86 3.33
    Scale factor- c (m/s) 12.981 10.099 9.695
    Scheduled power (MW) Unit 1 19.991970 13.930720 13.420870
    Unit 2 20.19493 14.22662 13.533470
    Direct cost (Rs/h) Unit 1 48580.4900 33851.6400 32612.7100
    Unit 2 49073.6800 34570.6900 32886.3200
    Underestimation cost (Rs/h) Unit 1 11526.6200 8358.1600 3408.3810
    Unit 2 8860.8800 2428.4010 749.6717
    Overestimation cost (Rs/h) Unit 1 −8191.3680 −5939.7110 −2422.1590
    Unit 2 −6296.9700 −1725.7400 −532.7528
    Operating cost (Rs/h) Unit 1 51915.7300 36270.0900 33598.9300
    Unit 2 51637.5900 35273.3500 33103.2400
    Total operating cost (Rs/h) 103553.3200 71543.4400 66702.1700
    Solar power system Scheduled power (MW) Unit 1 25.455080 27.021360 23.793190
    Unit 2 25.174760 26.238200 23.558000
    Direct cost (Rs/h) Unit 1 68219.6200 72417.2400 63765.7500
    Unit 2 67468.3700 70318.3700 63135.4400
    Underestimation cost (Rs/h) Unit 1 −1320.1950 4083.7060 793.7719
    Unit 2 608.5282 12563.6500 1607.5670
    Overestimation cost (Rs/h) Unit 1 938.1940 −2902.0780 −564.0925
    Unit 2 −432.4494 −8928.3350 −1142.4150
    Operating cost (Rs/h) Unit 1 67837.6200 73598.8700 63995.4300
    Unit 2 67644.4500 73953.6800 63600.5900
    Total operating cost (Rs/h) 135482.1000 147552.5000 127596.000
    Total operating cost of RER based power (Rs/h) 239035.4200 219095.9400 194298.170
    Transmission losses (MW) 3.872521 4.263066 4.554507
    Simulation time (sec) 0.78 0.78 0.78

     | Show Table
    DownLoad: CSV

    Figure 5 represents the comparison of load shared by thermal, solar, and wind powers for Muppandal, Jaisalmer, and Okha. Jaisalmer has the highest PDF of solar radiation for the considered time. Therefore, the solar power generated here is about 21% of the total power generated by the coordinated system. Load shared by solar power in Muppandal and Okha is 20% and 19%, respectively.

    Figure 5.  Load shared by thermal, solar, and wind powers in Muppandal, Jaisalmer, & Okha.

    Muppandal has the lowest value of shape factor (k = 2.229) and highest value of scale factor (c = 12.981 m/sec). The mean wind speed in Muppandal is 11.50 m/s. Therefore, the generated wind power is about 16% of the total power generated by the system, whereas this value is 11% for Jaisalmer and 10% for Okha. The overall load share of RER-based power is found 36% for Muppandal, 33% for Jaisalmer, and 29% for Okha. Therefore, the thermal power generation is the minimum at Muppandal (64%), for the examined period.

    Test system-Ⅱ comprehends the same set of six generators as used in test system-Ⅰ but the power demand is increased from 250 MW to 400 MW. Since the parameters of solar and wind systems are not changed, therefore, the available solar and wind powers are also not changed. Now, the extra load is supplied by the thermal generating system. It is therefore the value of fuel cost, NOx emission, SO2 emission, and CO2 emission also arise. These values are observed for Muppandal, Jaisalmer, and Okha as 3118.4610 Rs/h, 546.6220 kg/h, 1871.3220 kg/h & 4722.1490 kg/h; 3204.7020 Rs/h, 552.6953 kg/h, 1923.0520 kg/h & 4885.2050 kg/h; and 3277.6320 Rs/h, 560.0665 kg/h, 1966.8020 kg/h & 5030.6830 kg/h, respectively. The total operating costs of RER-based power are almost similar to the previous system. These are 237429.1900 Rs/h, 218736.6800 Rs/h, & 197487.2300 Rs/h for Muppandal, Jaisalmer, and Okha, respectively. There is a slight change in simulation time this time. It has increased from 0.78 sec to 0.79 sec. The results of test system Ⅱ using ACSM are tabulated in Table 6.

    Table 6.  Solution of power scheduling problem of test system Ⅱ, by using ACSM.
    Output variables Muppandal Jaisalmer Okha
    Thermal power system Scheduled Power (MW) Unit 1 161.02400 166.45350 170.03970
    Unit 2 161.29570 165.66960 170.32660
    Fuel cost (Rs/h) Unit 1 1645.8460 1693.3100 1724.7040
    Unit 2 1472.6160 1511.3920 1552.9280
    Total fuel cost (Rs/h) 3118.4610 3204.7020 3277.6320
    NOx emission (kg/h) Unit 1 398.4640 397.4069 396.9263
    Unit 2 148.1580 155.2884 163.1401
    Total NOx emission (kg/h) 546.6220 552.6953 560.0665
    SO2 emission (kg/h) Unit 1 987.7449 1016.2100 1035.0370
    Unit 2 883.5767 906.8428 931.76480
    Total SO2 emission (kg/h) 1871.3220 1923.0520 1966.8020
    CO2 emission (kg/h) Unit 1 2527.8060 2647.8530 2730.5850
    Unit 2 2194.3430 2237.3520 2300.0980
    Total CO2 emission (kg/h) 4722.1490 4885.2050 5030.6830
    Wind power system Shape factor‑ k 2.229 2.86 3.33
    Scale factor- c (m/s) 12.981 10.099 9.695
    Scheduled Power (MW) Unit 1 20.67054 14.03151 13.34126
    Unit 2 20.56050 14.06176 13.28956
    Direct cost (Rs/h) Unit 1 50229.4200 34096.5700 32419.2600
    Unit 2 49962.0000 34170.1000 32293.6310
    Underestimation cost (Rs/h) Unit 1 2614.0010 6338.3090 5288.0540
    Unit 2 4059.3600 5732.1317 6508.7900
    Overestimation cost (Rs/h) Unit 1 −1857.6350 −4504.3080 −3757.9460
    Unit 2 −2884.7800 −4073.5300 −4625.4580
    Operating cost (Rs/h) Unit 1 50985.7900 35930.5000 33949.3700
    Unit 2 51136.6000 35828.6800 34176.9610
    Total operating cost (Rs/h) 102122.3900 71759.1800 68126.3310
    Solar power system Scheduled Power (MW) Unit 1 25.42640 27.49806 24.43054
    Unit 2 24.94910 27.03075 23.97192
    Direct cost (Rs/h) Unit 1 68142.7700 73694.8000 65473.8400
    Unit 2 66863.5900 72442.4000 64244.7600
    Underestimation cost (Rs/h) Unit 1 −1122.8950 −1077.8910 −1411.5360
    Unit 2 2161.2340 3982.0750 175.3275
    Overestimation cost (Rs/h) Unit 1 797.9830 766.0012 1003.1060
    Unit 2 −1535.8770 −2829.8540 −124.5962
    Operating cost (Rs/h) Unit 1 67817.8500 73382.9100 65065.4100
    Unit 2 67488.9500 73594.6200 64295.4900
    Total operating cost (Rs/h) 135306.8000 146977.500 129360.900
    Total operating cost of RER based power (Rs/h) 237429.1900 218736.680 197487.230
    Transmission losses (MW) 13.926230 14.74452 15.399560
    Simulation time (sec) 0.79 0.79 0.79

     | Show Table
    DownLoad: CSV

    Test system-Ⅰ & Ⅱ are also validated with PSO, SMM, SM, & EM, and obtained results are charted in Table 7. It can be seen that ACSM possesses the highest value of Cardinal Priority Ranking and least values of fuel cost & emissions for both the test systems. It takes the least time to achieve BCS as compared to the other four techniques and also its performance is not affected by using a large number of decision variables.

    Table 7.  Comparison of results.
    Applied Technique Test System F1 (Rs/h) F2 (kg/h) YSQ (Rs/h) Ywj (Rs/h) μkd Simulation Time (sec)
    NOx emission SO2 emission CO2 emission
    MUPPANDAL
    EM 1770.6972 540.6914 1062.9021 5616.8181 135888.5460 103863.9800 0.291 4.61
    3127.8164 548.2619 1876.9360 4736.3155 135712.7200 102428.7570 0.290 4.67
    SM 1768.9318 520.1723 1061.8425 5611.2180 135753.0640 103760.4270 0.412 1.42
    3124.6979 547.7152 1875.0646 4731.5933 135577.4140 102326.6350 0.411 1.44
    SMM 1768.0491 519.9127 1061.3126 5608.4180 135685.3230 103708.6500 0.453 1.52
    3123.1387 547.4419 1874.1290 4729.2322 135509.7600 102275.5740 0.451 1.52
    PSO 1767.1664 519.6531 1060.7827 5605.6180 135617.5820 103656.8730 0.472 1.51
    3121.5795 547.1686 1873.1933 4726.8711 135442.1070 102224.5120 0.470 1.50
    ACSM 1765.4010 519.1340 1059.7230 5600.0180 135482.1000 103553.3200 0.592 0.78
    3118.4610 546.6220 1871.3220 4722.1490 135306.8000 102122.3900 0.589 0.79
    JAISALMER
    EM 1852.3093 514.8838 1111.8476 5460.9809 147995.1570 71758.0703 0.292 4.71
    3214.3161 554.3533 1928.8211 4899.8606 147418.4330 71974.4575 0.290 4.73
    SM 1850.4625 514.3705 1110.7390 5455.5362 147847.6050 71686.5269 0.412 1.42
    3211.1114 553.8007 1926.8981 4894.9754 147271.4550 71902.6984 0.413 1.43
    SMM 1849.5392 514.1138 1110.1848 5452.8140 147773.8290 71650.7552 0.453 1.51
    3209.5090 553.5243 1925.9366 4892.5328 147197.9660 71866.8188 0.451 1.52
    PSO 1848.6158 513.8571 1109.6305 5450.0917 147700.0530 71614.9834 0.472 1.51
    3207.9067 553.2480 1924.9750 4890.0902 147124.4770 71830.9392 0.471 1.50
    ACSM 1846.7690 513.3438 1108.5220 5444.6470 147552.5000 71543.4400 0.592 0.78
    3204.7020 552.6953 1923.0520 4885.2050 146977.5000 71759.1800 0.590 0.79
    OKHA
    EM 1912.5234 516.4519 1147.9696 796.1532 127978.7880 66902.2765 0.291 4.68
    3287.4649 561.7467 1972.7024 5045.7751 129748.9830 68330.7100 0.289 4.72
    SM 1910.6166 515.9370 1146.8250 795.3594 127851.1920 66835.5743 0.414 1.42
    3284.1873 561.1866 1970.7356 5040.7443 129619.6220 68262.5837 0.411 1.43
    SMM 1909.6632 515.6796 1146.2528 794.9626 127787.3940 66802.2233 0.453 1.52
    3282.5485 560.9066 1969.7522 5038.2290 129554.9410 68228.5205 0.451 1.52
    PSO 1908.7098 515.4221 1145.6805 794.5657 127723.5960 66768.8722 0.473 1.50
    3280.9096 560.6266 1968.7688 5035.7137 129490.2610 68194.4573 0.471 1.50
    ACSM 1906.8030 514.9072 1144.5360 793.7719 127596.0000 66702.1700 0.592 0.78
    3277.6320 560.0665 1966.8020 5030.6830 129360.9000 68126.3310 0.589 0.79

     | Show Table
    DownLoad: CSV

    The box plots in Figure 5 differentiate the functioning of ACSM, PSO, SMM, SM, and EM, for test system-Ⅱ. The maximum and minimum values of fuel cost using ACSM, PSO, SMM, SM, and EM are 3120.6821 Rs/h, 3123.7210 Rs/h, 3127.3576 Rs/h, 3128.7332 Rs/h, & 3137.3529 Rs/h; and 3117.3257 Rs/h, 3118.7831 Rs/h, 3120.3742 Rs/h, 3120.9759 Rs/h, & 3122.9487 Rs/h, respectively. Also, the difference between first quartile Q1 and third quartile Q3 of cost function using ACSM, PSO, SMM, SM, and EM are 0.5532 Rs/h, 1.2460 Rs/h, 2.4606 Rs/h, 3.0599 Rs/h, & 4.1751 Rs/h, respectively. All these factors delineate the superiority of ACSM over other four tested techniques.

    Figure 6.  Comparison of fuel costs using ACSM, PSO, SMM, SM, and EM, for test system-Ⅱ.

    The integration of large-scale RER is of substantial thrust to electrical energy economizing and limiting emissions. The intensive solar and wind power plant constitutes an auspicious alternative source of RER technology. It also acknowledges the amalgamation of thermal power storage for the accumulation of energy for future utilizations, but RER poses multiple obstacles for power systems because of their changeability, uncertainty, and discontinuity. The employment of RER through prudent scheduling and consigning of power can impart operative pliability into the electrical power system.

    In this paper, a multi-objective coordinated solar-wind-thermal power scheduling problem is formulated and optimized for two conflicting economic and environmental objectives. ACSM has been successfully employed in the presented non-linear optimization problem and results are contrasted with some other existing popular population-based techniques. ACSM displays significant competence to runoff from local best solutions because the priority is given to the satisfaction level over the value of the objective function, by applying α -level comparisons. The addition of mutation of the least wanted point and multi-simplexes enhances the exactness of the technique. Therefore, it lowers the probability of missing out on the points around the boundary, during the reduction of the simplex. It is an effective method for constrained optimization because ACSM remains ineffective even when parameters are reshaping. It is a very quick, lethal, and stable technique for constrained optimization problems.

    I.K. Gujral Punjab Technical University, Kapurthala, India is admiringly acknowledged.

    The authors reveal that they don't have any conflicts of interest to describe regarding this study.



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