Combined heat and power economic dispatch problem using the invasive weed optimization 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 optimizing the fuel cost is quite complex and several classical and meta-heuristic algorithms have been proposed earlier. This paper applies the invasive weed optimization algorithm which is inspired by the ecological process of weed colonization and distribution. The results obtained have been compared with those obtained by other methods earlier and showed a marked improvement over earlier ones.

Keywords

combined heat and power (CHP) / economic dispatch / meta-heuristic algorithm / invasive weed optimization / cogeneration

Introduction

Combined heat and power (CHP) generation, also referred to 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 heat (steam) and the secondary form is electricity. Basically, the CHP principle could be used in any generating 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 which are in service so that the fuel cost is at the minimum. Integration of cogeneration units into the power system economic dispatch is complicated for the following reasons. First, there is a multiple demand (heat and power), and secondly, there is a heat-power capacity mutual dependency of the cogeneration units. This mutual dependency is given in the form of a feasibility chart with heat along one axis and electric power on 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 combined heat and power economic dispatch (CHPED) problem.

One of the earliest approaches to solving the CHPED problem using an iterative classical approach was proposed in Ref. [1]. 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 an additional 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 put forward in Ref. [6], where four power units, two cogeneration units and one heat only units were included and losses were also considered. A self adaptive real coded genetic algorithm was proposed in Ref. [7]. The genetic algorithm (GA), differential evolution (DE) and a combination of GA and Tabu (GT) were employed in Refs. [8–10] respectively. Ant colony search algorithm (ACSA) and bee colony approaches to the CHPED problem were found in Refs. [11] and [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 was solved by the invasive weed optimization (IWO) algorithm inspired by the behavior of weed colonization and distribution which was introduced by Mehrabian and Lucas [16]. IWO was used to solve the optimal control problem [17], optimal placement of piezo electric actuators on smart structures [18], time modulated linear array antenna synthesis [19], multi-objective optimization [20] and optimization of printed Yagi antenna [21]. It was also used in hybrid with particle swarm optimization for fast and global optimization [22] and for training feed forward neural networks [23].

Formulation of 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)\min {\displaystyle \sum _{i=\mathrm{1}}^{n_{\mathrm{p}}}{C}_i\left(P_i\right)}+{\displaystyle \sum _{j=\mathrm{1}}^{n_{\mathrm{c}}}{C}_j\left(P_j,H_j\right)+{\displaystyle \sum _{k=\mathrm{1}}^{n_{\mathrm{h}}}{C}_k\left(H_k\right)}},

subject to

equality constraints

(2a){\displaystyle \sum _{i=\mathrm{1}}^{n_{\mathrm{p}}}{P}_i}+{\displaystyle \sum _{j=\mathrm{1}}^{n_{\mathrm{c}}}{P}_j=P_{\mathrm{d}}},

and

(2b){\displaystyle \sum _{k=\mathrm{1}}^{n_{\mathrm{h}}}{H}_k}+{\displaystyle \sum _{j=\mathrm{1}}^{n_{\mathrm{c}}}{H}_j}=H_{\mathrm{d}},

and inequality constraints

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

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

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

and

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

where 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, k are the indices of conventional power units, cogeneration units and heat-only units respectively; n_p, n_c, and n_h 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. In addition to these constraints, the operating points of the cogeneration units have to fall within the feasibility region indicated diagrammatically.

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 invasive weed algorithm

The invasive weed algorithm is based on the growth of weeds within an area. At first the weeds are spread throughout an area with uniform random distribution. As the weeds grow they produce seeds in proportion to their fitness. The lowest and highest number of seeds are fixed arbitrarily with the least fit weed producing the lowest number of seeds and the most fit the highest number. The number of seeds produced by each weed varies linearly with the fitness. The seeds produce copies of parent weeds. The weeds then shift their position randomly with a normal distribution of zero mean and a standard distribution which is large at the beginning but decreases as further generations are produced. There is a maximum population size. The total number is maintained constant after this size is reached by eliminating the weaker weeds. The optimization algorithm is as follows.

Step 1Initialization: A number of candidates is generated randomly, uniformly distributed over the search space.

Step 2Reproduction: The lowest and highest number of seeds are fixed arbitrarily. The fitness of each candidate is calculated. The least fit is assigned the lowest number of the seeds f_l and the most fit the highest number f_h. Each weed is assigned a number of seeds depending linearly on its fitness. The seeds produce copies of parent.

Step 3Relocation: The weeds move randomly around their present position. This random motion follows a normal distribution with zero mean and standard deviation which decreases over the generations as given by Eq. (4).

(4)sd={\left({\displaystyle \frac{ite{r}_{\max }-iter}{ite{r}_{\max }}}\right)}^{pow}\left(s{d}_{\max }-s{d}_{\min }\right)+s{d}_{\min },

where the sd-standard deviation sd_max and sd_min are the limiting values of sd and pow is a nonlinear modulation index.

Step 4Limiting the population size: If the total number exceeds a certain maximum number NP, the excess weeds of least fitness are eliminated maintaining the population constant at this number.

Step 5Steps 1 to 4 are repeated over a fixed number of iterations (generations). After the fixed number is reached the candidate with the highest fitness is declared the optimal.

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 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 is 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 cogeneration units 2 and 3,

C_2=265\mathrm{0}+14.5P_2+\mathrm{0.0}345P_{\mathrm{2}}^{\mathrm{2}}+4.2H_2+\mathrm{0.0}3H_{\mathrm{2}}^{\mathrm{2}}+\mathrm{0.0}31P_2{H}_2,

C_3=125\mathrm{0}+36P_3+\mathrm{0.0}435P_{\mathrm{3}}^{\mathrm{2}}+\mathrm{0.}6H_3+\mathrm{0.0}27H_{\mathrm{3}}^{\mathrm{2}}+\mathrm{0.0}11P_3{H}_3.

For the heat-only unit 4,

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

subject to

0\le P_1\le 150\mathrm{M}\mathrm{W}\mathrm{a}\mathrm{n}\mathrm{d}0\le H_4\le 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 comparison of the results obtained by the proposed IWO algorithm along with other published results is presented in Table 1. From Table 1, it is observed that the IWO algorithm converged to the optimal solution of $ 9257.08. This result is comparable with those obtained by both the classical LR technique and other meta-heuristic algorithms. Thus it can be inferred that the proposed algorithm is suitably validated.

The parameters used in test problem 1 are as follows:

\mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{s}=\mathrm{15},

s{d}_{\max }=\mathrm{2},

s{d}_{\max }=\mathrm{2},

\mathrm{Maximum}\mathrm{population}\mathrm{size}\mathrm{N}\mathrm{P}=\mathrm{20},

\mathrm{N}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{mod}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}pow=\mathrm{3},

\mathrm{Max}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{m}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{s}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{s}{f}_{\mathrm{h}}=\mathrm{5},

\mathrm{Min}\mathrm{m}\mathrm{u}\mathrm{m}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{s}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{s}{f}_{\mathrm{1}}=\mathrm{1.}

Figure 3 depicts the convergence characteristics for problem 1. It can be seen from Fig. 3 that the convergence occurs after 33 iterations. Figure 4 shows the cost obtained in each of 100 simulation runs with different random initial trial solutions. The largest variation is within 0.01% of the best result. Thus it can be inferred from that the algorithm is very robust with very minor variations.

Test problem 2

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

For the conventional power unit,

C_1=254.8863+7.6997P_1+\mathrm{0.00}172P_{\mathrm{1}}^{\mathrm{2}}+\mathrm{0.00}115P_{\mathrm{1}}^{\mathrm{3}}.

For the cogeneration units 2, 3 and 4,

C_2=125\mathrm{0}+36P_2+\mathrm{0.0}435P_{\mathrm{2}}^{\mathrm{2}}+\mathrm{0.}6H_2+\mathrm{0.0}27H_{\mathrm{2}}^{\mathrm{2}}+\mathrm{0.0}11P_2{H}_2,

C_3=265\mathrm{0}+34.5P_3+\mathrm{0.}1\mathrm{0}35P_{\mathrm{3}}^{\mathrm{2}}+2.2\mathrm{0}3H_3+\mathrm{0.0}25H_{\mathrm{3}}^{\mathrm{2}}+\mathrm{0.0}51P_3{H}_3,

C_4=1565+2\mathrm{0}{P}_4+\mathrm{0.0}72P_{\mathrm{4}}^{\mathrm{2}}+2.3H_4+\mathrm{0.0}2H_{\mathrm{4}}^{\mathrm{2}}+\mathrm{0.0}4P_4{H}_4.

For the heat only unit,

C_5=95\mathrm{0}+2\mathrm{.0}1\mathrm{0}9H_5+\mathrm{0.0}38H_{\mathrm{5}}^{\mathrm{2}},

subject to

35\le P_1\le 135\mathrm{M}\mathrm{W}\mathrm{a}\mathrm{n}\mathrm{d}0\le H_5\le 60\mathrm{M}\mathrm{W}\mathrm{t}\mathrm{h}\mathrm{.}

The heat-power feasible operating regions of the cogeneration units are displayed in Figs. 2, 5 and 6.

Table 2 shows the comparison of the results obtained by the proposed IWO algorithm and HS [3]. It can be seen that the proposed approach has produced a cost of $13683.65 in case 1 and $12134.33 in case 2 compared to $13723.20 and $12284.45 obtained in Ref. [3], respectively. It is seen that there is a reduction of $39.55 in case 1 and $150.12 in case 2. This indicates that the weed algorithm shows great promise in these types of problems. Figure 7 exhibits the convergence characteristics for problem 2, case 1. From Fig. 7 it is seen that convergence occurs after approximately 55 iterations. Figure 8 shows the cost obtained at each of 100 simulations.

The parameters chosen for test problem 2 in cases 1 and 2 are the same as those used in test problem 1.

Figure 9 presents the convergence characteristics for problem 2 in case 2. As seen from Fig. 9, the convergence takes as many as 80 iterations. Figure 10 shows the cost obtained at each of 100 different simulations. Once again it is seen that the variations are extremely small (less than 0.01%), from both Fig. 8 and Fig. 10, further confirming the robustness of the algorithm.

Conclusions

A novel meta-heuristic algorithm namely the IWO algorithm was successfully employed for solving the CHPED problems. While the algorithm was found to be as good as other meta-heuristic algorithms in the first problem, it is evident from the comparison that significant improvement was observed in the second problem. This indicates that the algorithm is promising for solving power system optimization problems. Thus future work can be conducted to investigate its performance on high dimensional and more complex real time optimization problems.

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