Combined Emission Economic Dispatch of Power System in Presence of Solar and Wind Using Flower Pollination Algorithm

ED (Economic Dispatch) problem is one of the vital step in operational planning. It is a nonconvex constrained optimization problem. However, it is solved as convex problem by approximation of machine input/output characteristics, thus resulting in an inaccurate result. Reliable, secure and cheapest supply of electrical energy to the consumers is the prime objective in power system operational planning. Increase in fuel cost, reduction in fossil-fuel assets and ecological concerns have forced to integrate renewable energy resources in the generation mix. However, the instability of wind and solar power output affects the power network. For solution of such solar and wind integrated economic dispatch problems, evolutionary approaches are considered potential solution methodologies. These approaches are considered as potential solution methodologies for nonconvex ED problem. This paper presents CEED (Combined Emission Economic Dispatch) of a power system comprising of multiple solar, wind and thermal units using continuous and binary FPA (Flower Pollination Algorithm). Proposed algorithm is applied on 5, 6, 15, 26 and 40 thermal generators by integrating several solar and wind plants, for both convex and nonconvex ED problems. Proposed algorithm is simulated in MATLAB 2014b. Results of simulations, when compared with other approaches, show promise of the approach.

ED [1] is a vital step in power system operational planning.
Key goal of ED problem is to allocate power to the generators, to minimize operational fuel and emission cost, subject to all related operational constraints. It is a nonconvex constrained optimization problem. However, it has been addressed comprehensively in MP (Mathematical Programming) environment by converting it into convex problem. This is due to inability of the MP approaches to tackle nonconvex problems, except dynamic programming which also suffers from the curse of dimensionality.
History of ED starts from 1920. In the start of 1930s, EICC (Equal Incremental Cost Criterion) was applied to obtain the best possible solutions. An overview of various methods, which were applied for the period of 1977-1988, for the solution of ED is presented in [2].
Electrical power system is an interrelated network and mainly depends on conventional sources of power production. Power sectors require amendments and developments for meeting the needs of large number of people. Nowadays, deployment of renewable energy resources is highly encouraged [13]. Efforts are being made, focusing on improvement in production of renewable energy while maintaining optimal scheduling of interrelated systems.
Rising trend in fuel cost, depletion of fossil energy reserves, and environmental concerns are the major factors to find alternate solutions for producing energy.
Renewable energy resources are gaining popularity due to technological advancements in the field of solar and wind power. The solar and wind energy resources generate variable power output, due to changes in climate and weather conditions. The variations in generated power output makes economic dispatch, a random optimization problem and adds complexities in it. These complexities can be solved by evolutionary algorithms, to gain benefit, by reducing both the emission and fuel cost. Strength Pareto Evolutionary Algorithm has been used in [14], in which EED (Economic and Emission Dispatch) problem has been solved using one wind plant and a solar plant.
Security constrained dynamic (EED) model has been presented in [15]. Modified Harmony Search Algorithm has been proposed in [16] for solving CEED (Combined Economic Emission Dispatch) incorporating practical constraints. CEED problem having photovoltaic generation has been solved using PSO in [2] considering power balance and power limits constraints.
FPA (Flower Pollination Algorithm) is an efficient evolutionary approach. It is a robust and reliable technique which is inspired by the fertilization mechanism of flowers.
Its meta-heuristic nature can be applied to flourish new searching algorithms. FPA has potential to solve convex as well as nonconvex problems in an efficient way. It has a single significant parameter called Sp (Switching probability). Using this key parameter, the implementation of proposed algorithm becomes simpler and faster.
Moreover, by switching between the global and local pollination, solution can be avoided from getting stuck at the local optimum.
In the recent past, researchers evaluated the efficiency of FPA and analyzed its characteristics in solving convex and nonconvex objective functions. Multi-objective optimization problem is solved by FPA in [17]. A modified FPA is given for optimization of global problems in [18]. This paper primarily focuses on a novel solution methodology that competently reduces the operational cost and emission using continuous and binary FPA. The aim of dealing with CEED in thermal generators is to maintain the optimal power sharing of a system that comprises of multiple solar, wind and thermal units. Rest of the paper is organized as follows: Section 2 contains the mathematical formulation of problem. Section 3 discusses the optimization algorithm to solve ED problem. Section 4 describes application of FPA and BFPA to solve ED problem. Section 5 describes case studies. Section 6 deals with the results and discussions. Section 7 concludes the paper.

PROBLEM FORMULATION
The ED of a power system with the integration of renewable energy has various operating constraints associated with different generating units. The objective function for CEED [19] is solved to minimize total cost of production, as given by Equation (

Modeling of Thermal Plants
Dispatch of thermal generators includes minimization of both fuel and emission cost which becomes a CEED problem [20].
where a g , b g , and c g are the fuel cost coefficients of g th generating unit.
In nonconvex fuel cost functions, valve point effect is added in the characteristic equation. Nonconvex fuel cost characteristic of thermal generator is given by Equation where ag, b g , c g , e g and f g are the fuel cost coefficients of g th generating unit.
Emissions of thermal generator are given by Equation (5).
where g ,  g ,  g ,  g and  g arethe emission coefficients of g th generating unit.
Emission costs of thermal generator are given by Equation where v g represents the emission cost coefficient of g th generating unit.

Modeling of Solar Plants
Power output produced by solar plant [20] is given by Equation (7).
where P rated is Rated power, T ref is Reference or fixed temperature of region, T amb is Ambient or variable temperature for a day, is Temperature coefficient, and S t is Incident photovoltaic radiation. Solar share of m number of solar plants in the power system is given by [20].
where P h gives the power available in h th solar plant and V h decides the working state of solar unit which is either 0 (OFF) or 1(ON).
From Equation (7), it is clear that power produced by solar plant varies with the solar radiations and temperature, but the plant was installed to work at its rated power. For maximum benefit, all solar units must be ON i.e. V h = 1 for each unit h, but since the power available at the output, changes in accordance with the temperature and solar radiations, there occurs a penalty cost for each solar unit which is called the per unit cost of each plant.
Cost of operation of h th solar plant [2] to produce solar power P h is given as: where S h represents the per unit solar cost coefficient. Solar cost coefficient S h of each h th solar unit is given in US$/MWh [2].This operation cost is associated with each unit that produces solar power P h .
For solar plants, incident solar radiations and temperature of the area are input variables along with temperature coefficient. Output power produced by solar plants is used to find the solar share in the power system depending upon it's ON or OFF condition. As the power output from solar plant and per unit cost coefficient of solar plant may vary, overall cost of production of solar power can vary.

Modeling of wind plants
Cost of wind power plant [19] associated with scheduled power P i is given by: where  i represents the wind power coefficient.
Velocity or speed of wind is a varying parameter and power produced by the wind plant does not have a linear relationship with it. Wind value is given as input data for a certain hour of the day and the power output is computed. For wind plants, power output depends on speed of wind which is a variable parameter.As a result, output power from wind plant also varies. Probability density function of wind velocity is applied to calculate the scheduled wind power P i is a function of wind velocity [22]. Wind power is calculated from wind speed and is given as: Four zones of operation of wind plant can be seen in Fig. 1. The integration of wind unit disturbs the security of the power system and affects its stability. It varies the spare capacity which accounts for spare capacity punishing cost [21]. Spare capacity is the spare or extra power which is due to difference in scheduled wind power that is produced by the plant and actual wind power that is delivered to the load. Actual power delivered to the load is less than the scheduled power due to practical losses of the system. This difference in power accounts for a penalty cost which is called spare capacity punishing cost. This cost is given by: where r i is the spare capacity coefficient, P i is the scheduled wind power generation and P i act is the actual wind power that is delivered to the load. Spare capacity punishing cost is added to the cost given in Equation (10), to give the total cost such that:

Constraints
Constraints are basically, the real time limitations for producing the power. Cost is minimized taking into account all related constraints. This is because, if these constraints are violated, solutions will not be feasible. where P loss represents losses in transmission lines and these losses can be calculated using Equation (16).
where B is the loss coefficient matrix. Thermal Power Limits: Power limit constraint must be satisfied for operation of power system. The maximum generation of power is limited by the potential of a thermal unit to produce active power and minimum generation is limited by flame instability of furnace or boiler. If the output power of a generating unit is less than a pre-defined value P g min, then this generator is not connected to the bus bar since it is not feasible to produce a small value of active power from that generator. Therefore, the power which is produced, cannot voilate the limits of boundary. The problem is to find an optimal solution within the boundary, so that the cost is minimized. Power produced by thermal generators must be within their corresponding minimum and maximum limits. Cost is minimized when maximum output is produced from the solar plant. Therefore, for solar plants, difference between the total power available by solar plant and the total solar share, must be minimum [20] such that: Wind Power Limits: Solution will not be feasible, if power limit constraint is not satisfied. Depending upon the capacity of wind plant, maximum power which is produced from the wind speed is limited. Similarly, if the wind unit violates the lower limit of power, the unit will not be connected to the system. This is because a very low value of power output is not feasible for the system. Cost is minimized by optimal scheduling of power within the generation limits. Power produced by wind plants must be in between lower and upper limits.
where UR g and DR g are the up ramp limit and down ramp limit of g th generating unit respectively while hr represents the hour or time in which the generator is in operation.
The powerP hr g produced by the g th unit in any hour may not exceed the power p hr-1 g of previous hour, by more than a particular value R g . Similarly, it should not be less than the power P hr-1 g of previous hour, by more than a defined value DR g . This is because a particular unit is not able to change its output power too rapidly due to limited thermal and mechanical stresses. Thus, taking into account this constraint, optimal power is given to each generator so that minimum cost is obtained.  Algorithm used for optimization is described as follows: (2) Self-pollination is considered as a local fertilization process of pollens.

Flower Pollination Algorithm
where i represents any pollen, y i represents a solution vector at k th iteration, and g b represents best solution among the set of feasible solutions in the present iteration.
The factor L denotes a Levy function which gives the potential of pollination i.e. step length. As pollinators travel large distances with different step lengths, Levy flight is applied to characterize their movements effectively [20]. We have L  0 expressed as: In Equation (22)

Binary Flower Pollination Algorithm
To solve binary random variables of solar and wind units, binary BFPA is applied. The stages of BFPA are identical to ordinary FPA, however, some changes are made, which are as follows: Since the initial values in BFPA are binary, the initialization of flower is needed to be done, to fit in the binary position such as: A random boolean value is assigned to every unit of the flower as given by Equation (25) (25) Solutions are updated as given by Equation (26).
where  is the random number, ranging from 0 to 1. Application of algorithm for environmental constrained ED problem is presented by following pseudo code. Encoding:  Initialize a population P g of n flowers(n=max population size).  Initialize a binary pool, 4 State switching or shifting probability Sp. 5 State max iterations M i.e. stopping criteria.
Loop(max iter:1M) Loop(P=1:n) 6 Evaluate fitness of solution by calculating Equation (1) End of loop (P) 7 Create the fittest solution in randomly generated pool. Loop i=1:n if randSp, 8 Draw L that obeys levy behavior of search. 9 Global pollination via Equation (21)  10 Draw  from uniform distribution. 11 Local pollination via Equation (23)  12 Evaluate new solutions.

end End of loop i 13
Update the population if solutions are better.

End of Loop (max iter) 14
Find the best. 15 Output the best solution found and print it. 16 Stop. and a wind power plant are taken into account. Table 1 gives the data for power demand.

CASE STUDIES
Test System-5: In test system 5, 40 thermal units having nonconvex fuel cost curves [26] are considered. A solar plant rated at 1000 MW and a wind plant are considered.
Power demand of 10000 MW is taken into account.
Termination criteria for algorithm is the total number of iterations. Best results after 30 runs have been recorded.
The graphs of each case study show the pattern of minimum cost of each hour. The power produced after satisfying all the constraints is the optimal and best possible solution at which minimum cost is obtained.

RESULTS AND DISCUSSION
Best results of power shares of all the generators give the minimum solar, wind and thermal costs. Results and discussions of each case study are presented in the following section: Case Study-1: FPA is tested on test system 1 in which 6 thermal generators with convex fuel cost curves are used.
It satisfies equality and inequality constraints. Solar unit switching status can be found in Table 3 where it can be seen that, all of these units will be off when there is zero solar radiation i.e. during night time. Table 4 gives the optimum results of power shares of generators and losses in transmission lines for 24 hours. Minimum fuel cost, emission cost, solar cost and wind cost in $/hr for 24 hours are given in Table 5. Minimum emissions of generators in kg/hr can also be found in Table 5. Total cost of each hour reported in [2], where only solar plants were integrated to thermal generators, is compared with the total cost of each hour found using proposed FPA.
This comparison proves that the integration of wind plant to the system considerably reduces the operational cost to its minimum. existing algorithms.  Fig. 3 shows that, larger renewable share results in lower fuel and emission cost. Emission cost for each hour is slightly less than fuel cost. Average cost comparison is given in Table 6. It represents that, cost over 24 hours is 31,596.95 US$/h, which is significantly less than 38,807.167 US$/h, given in [2]. For each hour, total cost calculated by using FPA is compared with the cost computed by using PSO [2]. This comparison can be seen in Fig. 4

Combined Emission Economic Dispatch of Power System in Presence of Solar and Wind Using Flower Pollination Algorithm
Hour Solar Unit Condition    less than the cost obtained by Chaotic PSO [25]. Cost comparison can be found inTable 9. Fig. 6 shows that, increase in the power output from renewable energy resources, causes the decrease in the total cost of the system. For half of the day, power output from renewable energy resources is more, with its maximum value at midday. Fig. 6 shows that the total cost is reduced to its minimum, at peak hours.   Integration of wind and solar plants to the conventional generation system shares the total load demand among all the units. As a result, total cost of the system reduces to its minimum value.

CONCLUSION
In this paper, FPA is presented for the solution of EED of a power system integrated with solar and wind plants.