A comparative analysis between price ‐ penalty factor method and fractional programming method for combined economic emission dispatch problem using novel hybrid CSA ‐ JAYA algorithm

Economic dispatch of power is no more a sole concern for utilities. Instead, the utilities focus on reducing toxic gases emitted to the atmosphere due to the maximum utilisation of conventional fossil ‐ fuelled generators to meet the surging demand for electricity. This can be carried out by involving renewable energy sources (RES) to generate clean power compensating the depletion in the availability of fossil fuels. This article performs combined economic emission disfpatch (CEED) on four dynamic systems with and without the involvement of RES. Two methods for solving CEED, namely the price ‐ penalty factor (ppf) method and the fractional programming (FP) method, are used to perform CEED for all the four test systems, and a comparative analysis between them is made based on the least emission of harmful and toxic gases into the atmosphere. A novel hybrid (CSA ‐ JAYA) algorithm is used as the optimisation tool for the study. Numerical results manifest that the FP method of solving CEED is economic and emits less toxic gases to the atmosphere than the ppf method. The proposed hybrid CSA ‐ JAYA outperformed a long list of algorithms from recent literature in consistently providing better and superior quality solutions.


| Overview
In the field of electricity industries, the efficacious and optimal operation and planning of electric power generating systems are of utmost importance. Problems based on cost efficient load dispatch (aka Economic Load Dispatch) are the most concerning issues in the field of control and operation of the power system. Power system optimisation problems employing ELD helps us determine the most appropriate, flawless and cost effective operation by regulating the output of various generating units supplying the load demand. The sole ambition of ELD is the reduction of the overall cost related to generation of power without violating any constraint.
On the basis of the power demand, generally referred to as load, economic dispatch problems are broadly categorised into two parts: A) Static Load Economic Dispatch: The load demand is fixed for large intervals of time, which results in the fixed generator outputs for the duration in the case of static load economic dispatch. The sole purpose is to obtain the minimum cost of generation and transmission, for every epoch of time, such that the total power generated can be exactly equal to the power required without violating any constraint. B) Dynamic Load Economic Dispatch: Unluckily, the demand of the power system is consistently varying due to which the generators need to correspondingly adapt. This means, with the increase in the load demand, the generator output needs to be increased and vice-versa. Thus in the dynamic load dispatch, the scheduling of generators committed to the grid is carried out as per the varying load at regular intervals of time with the intention of least cost of generation.
contradictory to each other, it is not possible to obtain, at the same time, the least value of both generations as well as emission. This heads to the concept of combined economic and emission dispatch (CEED). Unlike ELD where the sole target is to minimise the cost of generation, in CEED, the objective includes the concerns regarding pollution and emission along with the aim to minimise the overall cost. This calls for certain rules and regulations that need to be followed by both private and government firms so as to reduce the various toxic effluents. The present article concentrates on the evaluation of dynamic economic dispatch while allowing for valve point effects (VPE), occurring due to power transmission through the network and the contribution of renewable energy sources for multiple numbers of generating units. Considering emission and its adverse effects in mind, research has been carried out to perform multi-objective load dispatch which consists of both economic as well as emission dispatch.

| Literature Survey
Article [1] proposes load dispatch model for charging plug-in electric vehicles (PEV) to obtain the reduced cost of generation and environmental emissions. Investigations were carried out on three cases: 6-unit without PEV, 6-unit with PEV and 10-unit with PEV. 'Levenbergh Marquardt Back-Propagation Algorithm (LMBP)-based Artificial Neural Network' was used to solve Dynamic Economic Load Dispatch (DELD) problems in [2]. Tests were carried out on nine generating units considering ramp rate limit constraints (RRL). Hybridised algorithm constituted with the amalgamation of 'Artificial Algae Algorithm (AAA)' and classical 'Simplex Search method (SSM)' having dynamically tuned parameters is proposed in article [3] where AAA executes the overall optimisation, whereas SSM searches locally. The proposed algorithm was tested on various test systems which include 13 generating units, 40 generating units and 80 generating units considering the effects of valve point effects (VPE); 140 generating units considering prohibited operating zones (POZs) and VPE and 40 generating units with VPE and transmission losses. Article [4] proposes Demand side management (DSM) technique to solve optimisation problems considering time varying emission dispatch (MODEED). The DSM approach is based on day ahead load shifting and was tested on six units considering ramp rate limits, coefficients related to fuel and emission and 24 h forecasted demand considering different cases using DSM. DELD problems assuming that VPE is solved using improved PSO (IPSO), proposed in [5]. The inequality constraints are handled using the feasibility-based selection technique and power balance constraint using heuristic strategies without the use of penalty factors. Tests were performed on the 10-generator system with cases of inclusion and exclusion of transmission losses and tripled 10-unit system to obtain the 30-unit data. Authors of [6] compared genetic algorithm (GA) and dynamic programming (DP) for ELD of 26 hydro units of the Three Gorges Reservoir. Hybridised bacterial foraging (BF) algorithm with simplified swarm optimisation combined with opposition-based initialisation and new mutation operator is proposed in [7] and tested on test systems comprising different generators sets like 5, 10, 30 and 100 units considering POZs as well as VPE. Modified group search algorithm is presented by the authors of [8] for solving the problem based on the combination of economic and emission dispatch on the IEEE 30 bus system with cases of inclusion and exclusion of system loss along with the other constraints. Solution to Stochastic DELD system incorporating wind and solar-based generation systems by improved fireworks algorithm (IFWA) is presented in [9]. Fifteen-generator DELD systems is being tested with stochastic DELD systems of solar and wind having a rated capacity of 100 and 400, respectively, with consideration of POZs, ramp-rate limits and transmission losses. Authors in [10] performed economic dispatch on a renewable integrated micro-grid using neighbourhood-based differential algorithm for four different cases of load sharing. Two-stage stochastic DELD model is proposed by [11] using the stochastic decomposition algorithm with an aim of managing uncertainty and system variability influenced due to wind generation and was tested on RTS-24 and PJM-5 systems. Authors in [12] used the lambda iteration method for the optimum scheduling of six unit systems considering transmission losses. Article [13] proposed a time varying economic emission dispatch model (DEED) with the incorporation of highly penetrated wind system taking into account its intermittency and indecision, energy storage system (ESS) and DSM with weighted sum method. Modified social spider algorithm is proposed in [14] and the tests were carried out on various systems ranging from 6 to 140 units.
The major highlights of this article are as follows: a. To perform dynamic economic dispatch, emission dispatch, ppf-based CEED and FP-based CEED on 3-, 5-, 6-and 10-units systems b. To propose a novel hybrid CSA-JAYA algorithm to perform the aforementioned evaluations c. Comparative analysis between ppf-based CEED and FPbased CEED d. Comparative analysis of the proposed CSA-JAYA with an ample list of optimisation algorithms from literature e. Statistical analysis of the results obtained using the CSA-JAYA algorithm Section 2 of the article discussed the fitness functions to be evaluated while Section 3 emphasises the proposed optimisation technique in detail. The results are discussed and analysed in Section 4 and the article is concluded in Section 5.
allocate the optimal scheduling of electrical power for the dynamic demand.

| Cost function for DG units
The equation of the cost function in the case of DG units is not a linear equation. By fact, it is a quadratic equation [15][16][17] represented by equation (1A).

| Emission dispatch for DG units
The objective function of emission dispatch (ED) can be calculated as per the equation (2A) and (2B) given below, depending on the availability of the emission coefficients. When VPE is considered, the ED equation involves an exponential term as shown in equation (2B) which makes the function multimodal, where α j ; β j ; γ j ; λ j and ε j are the emission coefficients of j th DG units, while F 2 is the total emission [15][16][17].
2.3 | Combined economic emission dispatch using the ppf method ELD deals with the minimisation of the fuel costs while ED deals with the minimisation of the emission of harmful pollutants from the conventional fossil fuelled generators to the atmosphere. Hence, a compromised solution must arrive that can achieve both reduced fuel costs and release of fewer pollutants in the atmosphere. This is achieved by formulating a combined economic-emission dispatch (CEED) by combining (1) and (2) with the help of 'Price Penalty factor' (ppf ). It is a parameter which is used to get a single objective function (CEED) [15][16][17].

| Combined economic emission dispatch using the FP method
In this method, two different competing and conflicting objective functions comprising the same decision and control variables are solved as a ratio of each other. For instance, say F1 be the economic dispatch equation mathematically expressed as in (1) and F2 be the emission function as in (2). Then by the FP method, a compromised solution can be obtained by minimising the ratio F2:F1. This is mathematically expressed as in (5) [18].
DEY ET AL.

| Equality and inequality constraints
Equation (6) and (7) are the equality constraints without including RES and including RES problems, respectively. Equation (8) is the inequality constraint restricting the DERs within their limits.
where D t is the demand of the t th hour, P RES,t is the RES output in terms of power.
where UP is the utilization percentage. UP is normally used when it is an unclear and confusing attempt to represent the hourly outputs of the test systems which have larger number of DERs.

| Crow search algorithm
Crows possess the habit of observing and following other birds to determine their food storage locations and take their food in their absence. Moreover, if the crow does steal food from another bird, it becomes extra cautious and keeps shifting its own hiding place to avoid becoming a victim of robbery in future. Not only this, it also uses its own knowledge to prevents its food from the robbers. The CSA is based on these behaviours of a crow. Supposedly at iteration 'iter' crow 'h' wants to visit its hiding place m h;iter and in the same iteration, say crow 'i' plans to follow crow 'h'. At this situation, two cases may happen: Case 1: Crow 'h' is totally unaware of the fact that it is followed by crow 'i' and as a result crow 'i' will know the hiding place of crow 'h'.
Case 2: Crow 'h' knows that it is being followed by crow 'i' and hence fools crow 'i' by diverting it to a different random location within the search space.
These two cases can be mathematically represented with a set of equations as conditioned with the awareness probability (AP) of the crow 'h' [19]: where rand 1 and rand 2 are random numbers with uniform distribution between 0 and 1 and fl i is the flight length of the i th crow. If 'case 1' occurs, updating of the memory of crow 'i' will occur based on the formula below.
f(.) denotes the value of the fitness function.
The value of 'fl' decides the vicinity of search space. AP means the awareness probability of crow 'j'. Since it is a probability, its value lies between 0 and 1, both inclusive. AP maintains the exploration and exploitation of the crow search algorithm.

| JAYA algorithm
JAYA is a Sanskrit word meaning victory or success. The algorithm has only one governing equation. In every iteration, the algorithm tends to shift away from the worst solution (or failure) hence getting close to the best solution (or success/ victory) as the termination criteria is attained. The simple governing equation of the JAYA algorithm is [20].
where k and i denote dimension and particle of the population, respectively; cʹ and cʺ lie between 0 and 1, both inclusive.

| Hybrid CSA-JAYA algorithm
Jaya is a greedy search algorithm. It tends to move away from the worst value of fitness function towards the best value in every iteration. In hybrid CSA-JAYA, this property of the JAYA algorithm is utilised along with the food hunting strategy of crows and is represented by equation (13). The CSA-JAYA algorithm switches between the food hunting strategy of crows approaching the best value of fitness function of JAYA. The fitness function is calculated for every particle of the memory matrix which is updated in every iteration and the best and worst values are gathered to be utilised.where m is the memory matrix which is updated following equation (11).

| Advantages of the hybrid CSA-JAYA algorithm
Some of the crucial advantages of the CSA algorithm such as having the least amount of stages and sophisticated equations, having minimum number of random numbers, and capacity of handling large dimensional problems, are mentioned by the authors in [17]. Along with these advantages, the JAYA algorithm is modelled to strictly distance away from the worst solutions during every iteration. Therefore, the proposed hybrid CSA-JAYA algorithm yields consistently prominent and better quality solutions for both unimodal and multi-modal functions of any dimensions. One of the crucial advantage in the proposed CSA-JAYA algorithm is that AP which was a tunable parameter in CSA is formulated in the proposed CSA-JAYA so as to linearly decrease from 1 to 0 throughout the iterations. Minimum numbers of tunable parameters decrease the tediousness during execution of an algorithm.

| Implementation of CSA-JAYA for CEED problems
Step 1: Input no. of DERs (N) and DER parameters Step 2: Input population size (pop) Step 3: Input maximum of iterations Step 4: Determine the size of load and renewable output Step 5: Create population matrix as given in equation (14) Step 6: Evaluate the fitness function Step 7: Generate new generation applying equation (13) Step 8: Perform constraint violation check of the new DERs of the new position matrix Step 9: Calculate fitness function of the new position Step 10: Update the memory matrix using equation (11) Step 11: Repeat Step 7 to 10 until termination criteria are reached 4 | RESULTS AND DISCUSSION

| Overview of the test systems
Four dynamic test systems containing 3, 5, 6 and 10 fossil fuelled generating units are considered in which CEED is performed using both the ppf and FP methods. It is to be noted that the cost   Table 1-12. Table 13 shows the load demand of both the systems and Table 14 highlights the RES contribution for both the test systems. The cost of the RES was not considered for the 3-and 10-units system, whereas for the 5 unit system, the price of the wind power was 0.054 $/kW. The value T A B L E 1 Cost coefficient of the 3-unit system [15,17] Table 15. Algorithms such as pigeon inspired optimisation (PIO) [21], JAYA, CSASCA [22], TVACPSO [23], CFPSO [24,25] along with proposed CSA-JAYA were used as optimisation tools to perform CEED on the test systems. The population size was set at 80 and the maximum number of iterations was considered to be 500 for all of the optimisation techniques and each technique was executed for 20 individual trials in the MATLAB environment installed in a desktop PC of core i3 processor 4 GB RAM. emission during ppf-based CEED method. Figures 1 and 2 shows the utilisation percentage of DERS when FP and CEED were evaluated with and without RES for the 3-and 5-unit systems, respectively. For the 6-unit test system, the fitness function which is multi-modal in nature, six different optimisation algorithms are implemented along with the proposed hybrid CSA-JAYA algorithm. Table 22 lists the values of fitness function when CEED was performed based on the ppf and FP methods, respectively, utilising the seven algorithms. It can be seen in the sorted display of the table that the proposed CSA-JAYA yielded the least and best value of CEED for both the methods. Based on these values yielded by CSA-JAYA, the cost and emission were thereafter evaluated with the optimal values of DERs and listed in Table 23. Like the aforementioned 3-and 5-unit test systems, CEED based on the FP method yielded both less generation cost and pollutants emission compared with the ppf-based method for evaluating CEED. Twenty (20) individual trials were conducted for all the seven algorithms before the best values of FR-based CEED were listed in Table 22. Considering these trials, a statistical record analysis is highlighted in Table 24. Least values of standard deviation and computational time consumed by the hybrid CSA-JAYA claims the superiority of the approach. Figure 3 shows the convergence curve characteristics of the seven algorithms when the fitness function of FP-based CEED was minimised. Hybrid CSA-JAYA can be seen converging in early iterations with the best values of fitness function. All the six DERs were utilised with a different weightage when FP-based CEED and ppf-based CEED were evaluated. Figure 4 shows the utilisation percentage of the DERs when FP-and ppf-based CEED were performed using the proposed hybrid CSA-JAYA.
T A B L E 20 Cost ($) and Emission (kg) when CEED was performed with both the methods using CSA-JAYA with RES for the 3-unit system Similar to the 6-unit systems, identical and robust evaluations were performed for 10-unit dynamic system using four different algorithms viz. TVACPSO, CFPSO, JAYA, CSASCA along with the proposed hybrid CSA-JAYA. Initially, only economic dispatch was performed on the 10-unit system with wind support. The generation cost yielded by the proposed hybrid CSA-JAYA was $978,240 which is less than the cost of many algorithms mentioned in [17] and also the aforementioned four algorithms. This is shown in Figure 5. Table 25 lists the best values of fitness function after 20 trials of the mentioned algorithms when the test system was evaluated for the FP-and ppf-based CEED. Furthermore, the optimal scheduling of DERs was extracted to calculate the generation cost and emission from the best value of fitness function obtained by the hybrid CSA-JAYA for both the methods and are shown in Table 26. The aim of the study is once again fulfilled to witness that the FP-based CEED evaluation gave better generation cost and emission value compared with the ppfbased CEED. A list of statistical analysis-based data were displayed in Table 27 after 20 trials of various algorithms for FP-based CEED evaluation. Similar to the rest of the test systems, the least values of the standard deviation and T A B L E 23 Cost ($) and Emission (kg) when FP-and ppf-based CEED were performed using CSA-JAYA for the 6-unit system  Figure 6 shows the utilisation percentage of the DERs when FP-and ppf-based CEED were performed using the proposed hybrid CSA-JAYA for the wind power incorporated 10-unit system. The convergence curve characteristics observed when CEED using the ppf-method was evaluated using various optimisation algorithms are shown in Figure 7 Tables 24 and 27 were also utilised to draw the box-plot of the 6-unit and 10-unit systems when FP-based CEED was evaluated. These are displayed in Figure 9(a) and 9(b). These box-plots exhibit the distribution of quantitative data in a way that facilitates comparisons between FP among various algorithms. From these plots, it is seen that the chances of getting minimum FP is very high as the median from CSA-JAYA is nearer to the lower quartile. The least value of standard deviation claims the robustness of the proposed hybrid CSA-JAYA algorithm. A non-parametrical statistical analysis, Wilcoxon's paired sign rank test [26,27], was also conducted to analyse the superiority of the proposed hybrid approach. According to this test, two hypothetical situations H 0 and H 1 are considered. Situation H 0 states that all the algorithms considered are one and the same and there are no distinguishing differences among them. On the contrary, hypothetical situation H 1 states that the algorithms are distinctly different from each other. If the p-value is greater than the confidence level α, which is considered to be 0.5, then the hypothesis H 0 stands false. For all the paired results displayed in Table 28, it can be seen that the p-value is much lesser than 0.05. This rules out the hypothetical situation H 0 from all the pairs considered. The symbols +/−/∼ signify that the proposed CSA-JAYA is better than/less than/approximately equal to the value of the other algorithm with which it is compared. It can be seen from the T A B L E 26 Cost ($) and Emission (kg) when FP-and ppf-based CEED were performed using CSA-JAYA for the 10-unit system

| CONCLUSION
This article presented a novel comparison between two different methods for solving CEED problems. Four different test systems were considered and CEED was performed in both the methods with and without RES. The impact of RES in minimising the emission of harmful pollutants in the atmosphere was clearly highlighted from the tabulated results. Although both methods emphasised reducing the emission of toxic pollutants compromising with the active power generation cost of the system, FP proved to be more effective in solving the aforementioned purpose. The tedious and cumbersome process of evaluating the ppf of all the DERs of a system and finally sorting out the minimum from them is also avoided in FP. The proposed hybrid CSA-JAYA algorithm outperformed numerous algorithms from the literature by consistently providing better quality solutions in a minimum amount of time.
Non-parametric statistical analysis supports the superiority and robustness of the proposed approach in solving large dimensional unimodal and multi-modal optimisation problems.
As a scope of the future work, the proposed hybrid CSA-JAYA algorithms can be used in solving large dimensional problems such as optimal power flow. Energy management of smart microgrid systems consist of optimally scheduling various conventional fossil fuelled generators and other distributed generator units which emit harmful gases. The proposed FP method can be used in those cases to achieve at F I G U R E 8 Results recorded while evaluating FP-based CEED for 20 trials of (a) 6-unit system (b) 10-unit system F I G U R E 9 Box plot evaluated for FP-based CEED considering statistical data for (a) 6-unit system (b) 10-unit system