Combined heat and power economic dispatch problem using firefly algorithm

Abstract

Cogeneration units, which produce both heat and electric power, are found in many process industries. These industries also consume heat directly in addition to electricity. The cogeneration units operate only within a feasible zone. Each point within the feasible zone consists of a specific value of heat and electric power. These units are used along with other units, which produce either heat or power exclusively. Hence, the economic dispatch problem for these plants to optimize the fuel cost is quite complex and several classical and meta-heuristic algorithms have been proposed earlier. This paper applies the firefly algorithm, which is inspired by the behavior of fireflies which attract each other based on their luminosity. The results obtained have been compared with those obtained by other methods earlier and showed a marked improvement over the earlier methods.

Keywords

combined heat and power (CHP) economic dispatch / meta-heuristic algorithm / firefly algorithm / cogeneration

Introduction

Combined heat and power (CHP) generation, also known as cogeneration systems, is an efficient production of two forms of useful energy from the same fuel resource, using the exhaust energy from one production system as the input for the other. Ordinarily, the primary energy form is thermal (steam) and the secondary form is electrical. Basically, the CHP principle could be used in any generation facility. However, it makes sense only when there is a demand for the heat. Cogeneration installations are usually sited as near as possible to the place where the heat is consumed and, ideally, are built to a size to meet the heat demand. CHP generation is an established and mature technology, which has energy efficiency and environmental advantages over other forms of energy supply.

CHP dispatch concerns the distribution of power demand and heat demand over the units that are in service, so that the fuel cost is at a minimum. The integration of cogeneration units into the power system economic dispatch is complicated because of the multiple demands (heat and power) and the heat-power capacity mutual dependencies of the cogeneration units. The mutual dependency is given in the form of a feasibility chart with heat along one axis and electric power along the other axis. The chart consists of a closed contour with the feasible points lying inside the contour. The economic dispatch problem, which includes cogeneration units, is called the CHP economic dispatch (CHPED) problem.

One of the earliest approaches to solving the CHPED problem using an iterative classical approach was proposed in Ref. [1], in which the problem proposed consisted of one power unit, two cogeneration units and one heat-only unit. Another classical method, namely the Lagrangian relaxation (LR) technique was employed in Ref. [2]. The harmony search (HS) algorithm was employed in Ref. [3], where, in addition to the problem posed in Ref. [2], another problem was suggested by including another cogeneration unit. Another version of the HS algorithm was found in Ref. [4]. Evolutionary programming was used to solve the same problem in Ref. [5]. A multi-objective particle swarm optimization was proposed in Ref. [6], where four power units, two cogeneration units and one heat-only unit were included and losses were also considered. A self-adaptive real-coded genetic algorithm was put forward in Ref. [7]. The genetic algorithm (GA), the differential evolution (DE) algorithm and a combination of GA and Tabu (GT) were employed in Refs. [8-10], respectively. The Ant Colony (ACSA) and Bee Colony approaches to the CHPED problem were suggested in Refs. [11,12]. Algorithms for the CHP problem in the deregulated environment were proposed in Ref. [13]. A mesh adaptive direct search algorithm was found in Ref. [14], and an improved genetic algorithm with multiplier updating (IGA_MU) was found in Ref. [15].

In this paper, the CHPED problem is solved using the firefly algorithm (FA). This algorithm was inspired by the behavior of fireflies, which attract their mates with their luminosity. The firefly algorithm was introduced by Yang in Ref. [16] in 2009. It was applied to solve non-convex economic dispatch problems in Ref. [17]. It was used to solve the economic emission load dispatch problem in Ref. [18], and the unit commitment problem in a deregulated environment in Ref. [19]. Other applications include image thresholding [20], reliability and redundancy [21], structural optimization [22], image compression [23] and clustering [24].

Formulation of the CHPED problem

The CHPED problem is to determine the unit power and heat production, so that the production cost of the system is minimized while the power and heat demands and other constraints are met. It can be mathematically stated as

(1)Minimize{\displaystyle \sum _{i=\mathrm{1}}^{n_p}{C}_i(P_i)}+{\displaystyle \sum _{j=\mathrm{1}}^{n_c}{C}_j(P_j,H_j)+{\displaystyle \sum _{k=\mathrm{1}}^{n_h}{C}_k(H_k)}}.

Subject to

Equality constraints

(2a){\displaystyle \sum _{i=\mathrm{1}}^{n_p}{P}_i}+{\displaystyle \sum _{j=\mathrm{1}}^{n_c}{P}_j=P_d},

(2b){\displaystyle \sum _{k=\mathrm{1}}^{n_h}{H}_k}+{\displaystyle \sum _{j=\mathrm{1}}^{n_c}{H}_j}=H_d.

Inequality constraints

(3a)P_i^{\min }\le P_i\le P_i^{\max }(i=\mathrm{1},\mathrm{2},\mathrm{...},n_p),

(3b)H_k^{\min }\le H_k^{}\le H_k^{\max }(k=\mathrm{1},\mathrm{2},\mathrm{...},n_h),

(3c)P_j^{\min }(H_j)\le P_j\le P_j^{\max }(H_j)(j=\mathrm{1},\mathrm{2},\mathrm{...},n_c),

(3d)H_j^{\min }(P_j)\le H_j\le H_j^{\max }(P_j)(j=\mathrm{1},\mathrm{2},\mathrm{....},n_c).

In addition to these constraints, the operating points of the cogeneration units have to fall within the feasibility region indicated diagrammatically.

C is the unit production cost; P is the unit power generation; H is the unit heat production; H_d and P_d are the heat and power system demands; i, j, and k are the indices of conventional power units, co-generation units and heat-only units, respectively; n_p, n_c, and n_k are the numbers of power units, cogeneration units and heat-only units; P^min and P^max are the unit power capacity limits; and H^min and H^max are the unit heat capacity limits.

The CHPED problem clearly introduces the complication of more constraints than required in the pure power economic dispatch problem. The insufficiencies and difficulties with conventional methods thus follow from the fact that CHPED is a nonlinear, highly constrained optimization problem.

Overview of the firefly algorithm

The firefly algorithm is inspired by the behavior of fireflies which flash light to attract potential mates or prey. The attraction increases with the increase in the intensity of light. The swarm of fireflies tend to converge toward the brighter fireflies. If the intensity of each firefly is equal, the fireflies move randomly. The flashing light can be associated with the objective function to be optimized. The following idealizations are made as mentioned in Ref. [16]. 1) All fireflies are unisex and attract each other regardless of their sex. 2) Attractiveness is directly proportional to the brightness and the attraction is toward the brighter fireflies. The attractiveness and brightness decrease with distance. 3) The brightness is affected or determined by the landscape of the objective function. Based on these 3 rules the firefly algorithm in pseudocode can be written as follows.

Firefly algorithm:

Objective function f(BoldItalic), BoldItalic = (x₁, x₂, ..., x_d)^T

Generate initial population of fireflies BoldItalic_i (i = 1, 2, ..., n)

Light intensity I_i at BoldItalic_i is determined by f(BoldItalic_i)

Define light absorption coefficient γ

while (t<Max generation)

for i = 1: n all n fireflies

for j = 1: i all n fireflies

if (I_j>I_i)

Move firefly i towards j in d-dimension

end if

Attractiveness varies with distance r via exp[-r]

Evaluate new solutions and update light intensity

end for j

end for i

Rank the fireflies and find the current best

end while

Post process results and visualization.

A firefly i moves toward a brighter firefly j according to the formula

(4)x_i^{t+\mathrm{1}}=x_i^t+{\beta }_{\mathrm{0}}{e}^{-\gamma r_{ij}^{\mathrm{2}}}(x_j^t-x_i^t)+\alpha {ϵ}_i^t,

where the second term({\beta }_{\mathrm{0}}{e}^{-\gamma r_{ij}^{\mathrm{2}}}(x_j^t-x_i^t)) is the attraction parameter and the third term (\alpha {ϵ}_i^t) is the random term. x_i^t is the location of the ith firefly in the tth iteration, r_{ij}=\sqrt{{\displaystyle \sum _{i\prime =1}^d{\left(x_i^{i\prime }-x_j^{i\prime }\right)}^2}}. β₀ is the attraction coefficient. γ is the absorption coefficient. α is the random parameter and ϵ is a vector of normally distributed random variables.

The firefly algorithm is an improvement over the bacterial foraging algorithm (BFA) and the particle swarm optimization (PSO). In the BFA, bacteria are attracted partly based on their fitness and partly on their distance. Whereas the firefly algorithm has adjustable visibility and is more versatile in attractiveness variations leading to greater mobility and the search space is explored more fully. The firefly algorithm improves the efficiency of the PSO algorithm by introducing the benefits of random search by appropriately varying the parameters γ and α in Eq. (4). The firefly algorithm is better than GA because fireflies aggregate more closely around the optimum instead of jumping around [16]. These features of the firefly algorithm also make it better than the algorithms which have so far been used to solve the CHPED problems as shown by the better results in the test results shown below.

Test problems and results

The test problems considered are taken from Refs. [2] and [3] and are repeated here for convenience. While generating candidate solutions randomly, the infeasible solutions are made feasible in the cogeneration units by fixing them to the nearest straight line in the contour. In the case of power only and heat-only units, infeasible candidates are moved to the nearest upper or lower limits. The equality constraints are taken care of by the use of penalty functions augmenting the objective function. The simulations were conducted in MATLAB R2008a run on a 2.40 GHz Intel (R) Core (TM) i3 processor with 1.86 GB of RAM.

Test Problem 1

A test system of four units has been taken to illustrate the performance of the proposed methods.

For the conventional power Unit-1:

C_{\mathrm{1}}=\mathrm{50}{P}_{\mathrm{1}}.

For the co-generation Units-2 and-3:

C_{\mathrm{2}}=\mathrm{2650}+\mathrm{14.5}{P}_{\mathrm{2}}+\mathrm{0.0345}{P}_{\mathrm{2}}^{\mathrm{2}}+\mathrm{4.2}{H}_{\mathrm{2}}+\mathrm{0.03}{H}_{\mathrm{2}}^{\mathrm{2}}+\mathrm{0.031}{P}_{\mathrm{2}}{H}_{\mathrm{2}},

C_{\mathrm{3}}=\mathrm{1250}+\mathrm{36}{P}_{\mathrm{3}}+\mathrm{0.0435}{P}_{\mathrm{3}}^{\mathrm{2}}+\mathrm{0.6}{H}_{\mathrm{3}}+\mathrm{0.027}{H}_{\mathrm{3}}^{\mathrm{2}}+\mathrm{0.011}{P}_{\mathrm{3}}{H}_{\mathrm{3}}.

For the heat-only Unit-4:

C_{\mathrm{4}}=\mathrm{23.4}{H}_{\mathrm{4}}.

Subject to:

\mathrm{0}\le P_{\mathrm{1}}\le \mathrm{150} \mathrm{M}\mathrm{W},

\mathrm{0}\le H_{\mathrm{4}}\le \mathrm{2695.2} \mathrm{M}\mathrm{W}\mathrm{t}\mathrm{h}.

The power and heat demand for the system are 200 MW and 115 MWth, respectively. The heat-power feasible regions for the cogeneration units are illustrated in Figs. 1 and 2.

The results obtained by the proposed method are tabulated along with other published results in Table 1. It can be seen from Table 1 that the result obtained in Ref. [2] by the classical LR technique is $9257.07. Other algorithms, which are meta-heuristic in nature, have obtained almost the same result. It can be seen that the results obtained by the proposed method is similar to the best results obtained by other methods published earlier. The parameters used in the above test problems are α = 2, β = 1.1, γ = 1.25, and the number of fireflies= 40.

Figure 3 demonstrates the convergence characteristics of the proposed FA algorithm applied to Test Problem 1, from which a rapid convergence to the optimal solution can be observed. Figure 4 displays the results obtained when the algorithm was run independently 100 times with different random initial trial solutions. It is seen that the variations have been minimal, thus showing the effectiveness of the algorithm.

Test Problem 2

The problem has one conventional power unit, three cogeneration units and a heat-only unit.

C_{\mathrm{1}}=\mathrm{254.8863}+\mathrm{7.6997}{P}_{\mathrm{1}}+\mathrm{0.00172}{P}_{\mathrm{1}}^{\mathrm{2}}+\mathrm{0.00115}{P}_{\mathrm{1}}^{\mathrm{3}}(\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}).

For cogeneration Units-2,-3 and-4:

C_{\mathrm{2}}=\mathrm{1250}+\mathrm{36}{P}_{\mathrm{2}}+\mathrm{0.0435}{P}_{\mathrm{2}}^{\mathrm{2}}+\mathrm{0.6}{H}_{\mathrm{2}}+\mathrm{0.027}{H}_{\mathrm{2}}^{\mathrm{2}}+\mathrm{0.011}{P}_{\mathrm{2}}{H}_{\mathrm{2}},

C_{\mathrm{3}}=\mathrm{2650}+\mathrm{34.5}{P}_{\mathrm{3}}+\mathrm{0.1035}{P}_{\mathrm{3}}^{\mathrm{2}}+\mathrm{2.203}{H}_{\mathrm{3}}+\mathrm{0.025}{H}_{\mathrm{3}}^{\mathrm{2}}+\mathrm{0.051}{P}_{\mathrm{3}}{H}_{\mathrm{3}},

C_{\mathrm{4}}=\mathrm{1565}+\mathrm{20}{P}_{\mathrm{4}}+\mathrm{0.072}{P}_{\mathrm{4}}^{\mathrm{2}}+\mathrm{2.3}{H}_{\mathrm{4}}+\mathrm{0.02}{H}_{\mathrm{4}}^{\mathrm{2}}+\mathrm{0.04}{P}_{\mathrm{4}}{H}_{\mathrm{4}},C_{\mathrm{5}}=\mathrm{950}+\mathrm{2.0109}{H}_{\mathrm{5}}+\mathrm{0.038}{H}_{\mathrm{5}}^{\mathrm{2}}(\mathrm{H}\mathrm{e}\mathrm{a}\mathrm{t}-\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{y}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}),

\mathrm{35}\le P_{\mathrm{1}}\le \mathrm{135}\mathrm{M}\mathrm{W},

\mathrm{0}\le H_{\mathrm{5}}\le \mathrm{60}\mathrm{M}\mathrm{W}\mathrm{t}\mathrm{h}.

The heat-power feasible operating regions of the cogeneration units are depicted in Figs. 5, 6 and 7.

The result obtained by the proposed FA is presented along with the other published results in Ref. [3] in Table 2. It can be noticed that the proposed approach has produced a cost of $13683.22 in Case 1 and $12119.86 in Case 2 compared to $13723.20 and $12284.45 as obtained in Ref. [3], respectively. It is seen that there is a reduction of $39.98 in Case 1 and $164.59 in Case 2.

Figure 8 exhibits the convergence characteristics of the proposed FA algorithm applied to Test Problem 2, Case 1. Figure 9 shows the results obtained when the algorithm was run independently 100 times. The parameters chosen for Test Problem 2, Cases 1 and 2 are the same as those used in Test Problem 1.

Figure 10 shows the convergence characteristics for Case 2. From Fig. 10, it is observed that there is a rapid convergence to the optimal solution. Figure 11 shows the results obtained when the algorithm was run independently 100 times. It is seen that the variations have been minimal, thus showing the effectiveness of the algorithm.

It can be seen from the convergence characteristics indicated in the respective figures of each problem and case that in all cases the convergence is achieved in less than 50 iterations. Since it is a heuristic algorithm, there will be variations in each run. Therefore, in each case, the costs obtained in 100 simulations have been plotted in Figs. 4, 9 and 11. It can be seen that the variation is very minimal, thus indicating that the proposed algorithm is very robust.

Conclusions

A new firefly algorithm has been successfully employed for the CHP dispatch problem. This algorithm is not only much simpler than most other meta-heuristic algorithms but also very effective in this case. It has resulted in a significant reduction in cost in Test Problem 2 while performing as well as other algorithms in Test Problem 1. It has also been seen that the algorithm is very robust, resulting in minimal random variations in as many as 100 trial runs. Thus further work can be undertaken in applying the firefly algorithm to other power system optimization problems.

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