Introduction

Power utilities have acquired a highly competitive status, particularly in generation and in the marketing of electricity. The ED aims to dispatch the committed generating units, such that, the operating cost if minimized while all the operating constraints are satisfied. In multi-area power systems the fuel cost of a pool can be decreased by importing power from areas having cheaper generating units. In such cases, the cost will depend on area-exchange agreements, characteristics of a pool, the policies adopted by utilities, types of interconnections, tie-line limits and load demands in individual areas. Transmission limits have a very significant role in deciding the cost of operation and in maintaining reliability. The traditional economic dispatch problem is normally solved without including the tie-line limits. The added tie-line constraints and area power balance requirements make the MAED problem more difficult to solve as compared to the conventional ED problem. This paper aims to formulate the ED problem with tie-line constraints and to analyze the effect of area loads and tie-line limits on the optimal operating cost for large multi-area power systems.

A complete formulation of multi-area generation was presented [1]. Desell, et al. [2] proposed an application of LP and Farmer, et al. [3] presented a probabilistic method. Hopfield neural network based approach was also proposed to solve the MAED problem [4]. MAED problem by using spatial dynamic programming with linear losses has been solved [5]. Linear programming [6] and decomposition approach by Shahidehpour [7] also addressed this problem.

These days nature inspired (NI) optimization methods are becoming very popular due to their ability to solve discontinuous and non-convex optimization problems in a very simple manner. The other advantages of NI techniques as compared to the traditional solvers are (i) non-dependence on nature of objective function, (ii) effective constraint handling, and (iii) population based powerful parallel search capability. Over the last few years many new NI methods like PSO [8, 9], differential evolution (DE) [10], bacterial foraging (BF) [11, 12], biogeography based optimization (BBO) [13], and artificial bee colony optimization (ABC) [14] have been proposed for solving complex economic load dispatch problems.

Recently and group search optimizer [15] and hybrid methods [16, 17] which combine evolutionary and swarm intelligence based techniques are also proposed. A series hybrid of PSO and DE can be found [17]. DE based on Lévy-flights has been proposed to improve convergence [18]. Multiobjective evolutionary approaches for optimizing cost and emission simultaneously are also available [19–22].

Many improved techniques based on parameter automation [8–10, 12] and hybridization of two methods [11, 16–18] are seen to enhance the performance. Iterative tuning of parameters and randomization of velocity vector [8] and introduction of additional operators [9] are used in PSO to prevent stagnation. In [10] a comparison of DE strategies for the MAED problem is carried out and some time-varying DE variants are proposed. Hybridization of BF with Nelder-Mead method [11] and an improved BF algorithm [12] with crossover and chaotic variation of step size is proposed to improve performance.

A traditional method General Algebraic Modeling System (GAMS) has been effectively used for large scale ED problems without considering practical multi-area operation [23]. For large dimensional problems the nature inspired optimization techniques may sometimes converge to near-global solutions due to saturation and premature convergence. The present paper proposes some modified PSO variants where a tuning of cognitive and social coefficient is carried out to improve global search. Static/dynamic MAED is solved for three test systems having different complexity levels. The performance is validated using NLP solver in GAMS and some recently published results from literature.

Multi-area Static/Dynamic Economic Dispatch

The objective of the economic dispatch problem is to determine the generated powers P i of units for a total load of P D so that the total fuel cost, \(F_{T}\) for the N number of generating units is minimized subject to the power balance constraint and unit upper and lower operating limits. The objective is to minimize \(\sum\nolimits_{q}^{M} {\sum\nolimits_{i}^{Nq} {F_{iq} (P_{iq} )} }\) subject to the following equality and in equality constraints given below. Here F iq is the total fuel cost for the ith generator in qth area defined by [8–10],

$$F_{iq} \left( {P_{iq} } \right) = a_{iq} P_{iq}^{2} + b_{iq} P_{iq} + c_{iq} + \left| {e_{iq} \times \sin \left( {f_{iq} \times \left( {P_{iq}^{\hbox{min} } - P_{iq} } \right)} \right)} \right|$$
(1)

where a iq , b iq , c iq , e iq and f iq are the fuel-cost coefficients.

Equality Constraints

Area-Wise Power Balance Constraint

In MAED problem the power balance constraints need to be satisfied for each area. The power balance constraints for area q can be given as [10]

$$\sum\limits_{i = 1}^{N_q} {P_{iq} - \left( {P_{Dq} + \sum\limits_{j}^{M_q} {T_{jq} - P_{qL} } } \right)} = 0,\quad {\text{such}}\;{\text{that}}\;j \ne q$$
(2)

For the qth area, P Dq is the load; P qL , the power loss; T jq , the tie-line flows from other areas; N q , the number of generating units; and M q is the count of tie-lines connected to the qth area.

Transmission Losses

The transmission losses using the B-loss coefficients is expressed as [24]

$$P_{qL} = \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{N} {P_{iq} } B_{ij} P_{jq} + \sum\limits_{i = 1}^{N} {B_{oi} } P_{iq} + B_{oo} }$$
(3)

Inequality Constraints

Unit Operating Limits Constraint

The output of the ith generating unit should lie within the minimum and maximum operating limits as given by

$$P_{iq}^{\hbox{min} } \le P_{iq} \le P_{iq}^{\hbox{max} } ;\quad i = 1,2, \ldots ,N_{q} ;\quad {\text{for all}}\;q$$
(4)

Unit Ramp-Rate Limit Constraints

When the generator ramp rate limits are considered, the operating limits are modified as follows:

$${ \hbox{max} }\left( {P_{iq}^{\hbox{min} } ,P_{iq}^{o} - DR_{iq} } \right) \le P_{iq} \le { \hbox{min} }\left( {P_{iq}^{\hbox{max} } ,P_{iq}^{o} + UR_{iq} } \right)$$
(5)

The previous operating point of ith generator in qth area is \(P_{iq}^{o}\) and DR iq and UR iq are the down and up ramp-rate limits, respectively.

Dynamic Economic Dispatch

Dynamic economic dispatch deals with sharing the system load including system losses among the available generators in such a way that all equality and inequality constraints are met and the cost of operation is minimized for each time interval ‘t’ in a time period T such that ∑t = T. In order to solve dynamic load dispatch problem, ramp-rate limit must be considered. The dynamic economic dispatch (DED) model can be described as follows:

$$\left\{ {\begin{array}{*{20}l} {\hbox{min} \,F = \sum\limits_{t}^{T} {\sum\limits_{q}^{M} {\sum\limits_{i = 1}^{N_q} {F_{iq} \left( {P_{iq} (t)} \right)} } } } \\ {\sum\limits_{i = 1}^{N_q} {P_{iq} \left( t \right) = P_{Dq} \left( t \right) + P_{Lq} \left( t \right)} } \\ {P_{iq\,\hbox{min} } \le P_{iq} \left( t \right) \le P_{iq\,\hbox{max} } } \\ { - DR_{iq} \le P_{iq} \left( t \right) - P_{iq} \left( {t - 1} \right) \le UR_{iq} } \\ \end{array} } \right.$$
(6)

PSO Variants

A number of different PSO strategies are being applied by researchers for solving the ED and other power system problems. Here, a short review of the PSO variants is presented.

Classical PSO

The PSO [25] is a population based NI method inspired by the movement of a flock of birds searching for food. It is a simple and powerful optimization tool which scatters random particles into the problem space. The particles represent the various random solutions of the optimization problem. The position and velocity vectors of the ith particle of a d-dimensional search space can be represented as

$$X_{i} = (x_{i1} ,x_{i2} , \ldots ,x_{id} );\quad V_{i} = (v_{i1} ,v_{i2} , \ldots ,v_{id} )$$
(7)

The best prior location of a particle is stored as \(p{\text{best}}_{i} = (p_{i1} ,p_{i2} , \ldots ,p_{id} )\). If the gth particle is the best among all particles in the group so far, it is represented as \(p{\text{best}}_{g} = g{\text{best}}_{{}} = (p_{g1} ,p_{g2} , \ldots ,p_{gd} )\). The updated velocity and location of each particle for fitness evaluation in the next, that is, (k + 1)th iteration are calculated using the following equations [25]:

$$\begin{aligned} v_{id}^{k + 1} & = [w \times v_{id}^{k} + c_{1} \times {\text{rand}}_{1} \times (p{\text{best}}_{id} - x_{id} ) + c_{2} \\ & \quad \times \,{\text{rand}}_{2} \times (g{\text{best}}_{gd} - x_{id} )\!\!\ ]\end{aligned}$$
(8)
$$x_{id}^{k + 1} = x_{id} + v_{id}^{k + 1}$$
(9)

The global and local search capabilities of the particle are controlled by \(w\), the inertia weight parameter, constriction factor is C, the cognitive and social coefficients are \(c_{1} ,c_{2}\), respectively, and \({\text{rand}}_{1} ,{\text{rand}}_{2}\) are random numbers between 0 and 1. The inertia weight w is modified with time as given by

$$w = \left( {w_{\hbox{max} } - w_{\hbox{min} } } \right) \times \frac{{\left( {{\text{iter}}_{\hbox{max} } - {\text{iter}}} \right)}}{{{\text{iter}}_{\hbox{max} } }} + w_{\hbox{min} }$$
(10)

where \({\text{iter}}_{\hbox{max} }\) is the maximum number of iterations. Constant c 1 pulls the particles towards local best position whereas c 2 pulls it towards the global best position.

PSO with Chaotic Inertia Weight (PSO_CIW)

The weight w (11) is changed iteratively in chaotic fashion by making use of the logistic map as given by

$$w(t) =\upmu \times w(t - 1) \times \left[ {1 - w(t - 1)} \right]$$
(11)

Here µ is a control parameter between 0–4. A very small difference in w(0) causes significant difference in its variation pattern. The system at Eq. (11) displays chaotic behavior when µ = 4 and \(w(0) \notin \left\{ {0,0.25,0.5,0.75,1.0} \right\}\).

PSO with Chaotic Acceleration Coefficients (PSO_CAC)

In the proposed PSO_CAC approach the cognitive coefficient c 1 is reduced from an initial value c 1i to a final value c 1f while the social coefficient c 2 is increased chaotically from an initial value c 2i to c 2f using the following dynamics:

$$cx_{1} (t) =\upmu \times cx_{1} (t - 1) \times [1 - cx_{1} (t - 1)]$$
(12)
$$c_{1} (t) = \left[ {\left( {c_{1f} - c_{1i} } \right)\frac{\text{iter}}{{{\text{iter}}_{ \hbox{max} } }} + c_{1i} } \right]cx_{1} (t)$$
(13)
$$cx_{2} (t) =\upmu \times cx_{2} (t - 1) \times \left[ {1 - cx_{2} (t - 1)} \right]$$
(14)
$$c_{2} (t) = \left[ {\left( {c_{2f} - c_{2i} } \right)\frac{\text{iter}}{{{\text{iter}}_{ \hbox{max} } }} + c_{2i} } \right]cx_{2} (t)$$
(15)

Time Varying PSO (PSO_TVAC)

In population-based optimization methods, the policy is to encourage exploration during initial search and exploitation as the solution approaches convergence. In PSO_TVAC the cognitive component is decreased and the social component is increased as shown in Fig. 1. The acceleration coefficients are expressed as [8, 26]:

$$c_{1} = \left( {c_{1f} - c_{1i} } \right)\frac{\text{iter}}{{{\text{iter}}_{ \hbox{max} } }} + c_{1i}$$
(16)
$$c_{2} = \left( {c_{2f} - c_{2i} } \right)\frac{\text{iter}}{{{\text{iter}}_{ \hbox{max} } }} + c_{2i}$$
(17)

where c 1i , c 1f , c 2i and c 2f are initial and final values of cognitive and social acceleration factors, respectively.

Fig. 1
figure 1

Cognitive and social coefficients in PSO_CAC for µ = 3, c 1, c 2 (t = 0) = 0.48

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Improved Search through Parameter Automation

The above PSO variants are designed to improve search by better control of the swarm as compared to classical PSO which has fixed value of w, c 1 and c 2. In PSO_CIW the inertia weight which reflects the previous position of the swarm is varied chaotically to increase population diversity. In PSO_CAC and PSO_TVAC, during the initial search, exploration of the swarm is encouraged; as solution trajectory nears convergence, exploitation is strengthened.

Static/Dynamic MAED Solution using Modified PSO Variants

The flow chart of the proposed PSO variants for solving static/dynamic MAED problem is given in Fig. 2.

Fig. 2
figure 2

Flow chart of proposed PSO variants for static/dynamic MAED solution

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Generation of the Initial Population

A population of feasible solutions is randomly generated between the lower/upper bounds.

Evaluation of Swarm Population

A fitness function is used to judge the merit of each population. This function converts the constrained problem into unconstrained problem by using penalty function method. This approach minimizes cost and achieves constraint satisfaction as shown.

$$\hbox{min} \sum\limits_{t}^{T} {\sum\limits_{q}^{M} {\sum\limits_{i = 1}^{N_q} {F_{iq} \left( {P_{iq} (t)} \right) + \sum\limits_{q = 1}^{M} {\alpha_{q} } \left[ {\sum\limits_{i = 1}^{{N_{q} }} {P_{iq} (t) - \left( {P_{Dq} (t) + \sum\limits_{j}^{{M_{q} }} {T_{jq} (t)} } \right)} } \right]^{2} } } }$$
(18)

For the static dispatch consists of solution of any one time period, that is, for fixed t.

Results and Discussion

Simulations were carried out using MATLAB 7.0.1 on a Pentium IV processor, 2.8 GHz. with 1 GB RAM.

Details of the Test Cases

The large dimensional test cases are described below. For the sake of comparison with available results, transmission losses and cost of tie-line power flow are neglected.

  1. (i)

    Test Case I: This is a large 140-unit system taken from [9] with ramp rate limits supplying a load of 49342 MW. The best cost reported is $1655685/h. The PSO variants have obtained slightly lower cost which is reported in Table 1. The results are compared with [9, 17]. For the non-convex case the PSO_TVAC obtained $1657962.7130 whereas [18] the reported cost is $1657962.7166 which are very close.

    Table 1 Performance comparison of PSO variants for Test Case I
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The optimal dispatch results for all 140-units are given in Table 2.

Table 2 Best result of PSO_TVAC method (Test Case I)
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  1. (ii)

    Test Case II: For multi-area operation the above 140-unit system is divided into two areas having 70 generators in each area. The block diagram of this system is given in Fig. 3. Optimal cost is computed for (i) different area load demands and (ii) different tie-line limits using the proposed PSO variants.

    Fig. 3
    figure 3

    Block diagram of large two-area 140 unit system

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  2. (iii)

    Test Case III: Dynamic economic dispatch is carried out for 24-h load schedule for the 140-unit single-area system, that is, Test Case I.

Parameter Setup for PSO Variants

For all PSO variants the population size was taken as 100 and number of iterations was set at 1000 for all test cases. For classical PSO both c 1 and c 2 were fixed at 2, for variants the initial and final acceleration coefficients were taken as 2.5 and 0.5, respectively. The best results are taken out of 50 trials, each with different initial populations. This is because PSO family comes under random search methods which converge to near global solutions in every run.

Figure 4 shows the final convergence for Test Case I. All there PSO variants can be seen to converge fast but the performance of PSO_TVAC was found to be the best as it produces a better solution closely followed by PSO_CAC. The performance of PSO_CIW is inferior to these two variants because the acceleration coefficients c 1 and c 2 play a more significant role in locating the new position of the swarm as compared to the inertia weigh w. Therefore effective control of these parameters gives an improved solution.

Fig. 4
figure 4

Comparison of convergence characteristics of PSO variants for Case I

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Effect of Tie-Line Limits and Load Variation on Optimal Cost in MAED

The performance of best performing variant PSO_TVAC is given in Tables 3 to 5. Traditional GAMS method also produced the same costs for the different cases. For the Test Case II, that is, two-area, 140-unit large system (total load PD = 49342 MW) three different load variation cases are taken.

Table 3 Performance of PSO_TVAC with tie-limit variation for Test Case II; load Case (i)
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Table 4 Performance of PSO_TVAC with tie-limit variation for Test Case II; load Case (ii)
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Table 5 Performance of PSO_TVAC with tie limit variation for Test Case II; load Case (iii)
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  • Case (i) PD1 = 32072 (65 %), PD2 = 17270 (35 %)

The results for this study are given in Table 3. For tie-line limit less than 4000 MW the system did not converge. Then, with increase in tie-line capacity the cost reduced as cheaper area 2 units transfer power to area 1 having costlier generators. The optimal tie-line flow was found to be 6397.023 MW. The convergence characteristics of the three PSO variants are compared in Fig. 5. The convergence behavior of PSO_TVAC is found to be superior but the other two variants also depict a stable convergence.

Fig. 5
figure 5

Comparison of convergence characteristics of PSO variants for Case II

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  • Case (ii) PD1 = 34539 (70 %), PD2 = 14803 (30 %)

Table 4 gives the results where area 1 load is increased to 70 % and tie-line limit is changed from 6000 to 9000 MW. For tie-line limit less than 6000 MW the system did not converge. For tie-line capacity 9000 MW and beyond, there is no reduction in cost as the optimal tie-line between area 1 and area 2 was found to be 8864.023 MW. The optimal cost of generation matched with the cost of operation for single area case for this tie-line limit.

  • Case (iii) PD1 = 39474 (80 %), PD2 = 9868 (20 %)

Table 5 presents the results for this case where tie-line limit is changed from 11000 to 14000 MW. The effect of variation of area load and tie-line on the optimal cost of the 140-unit multi-area system is summarized in Fig. 6.

Fig. 6
figure 6

Effect of load and tie-limit variation on optimal generation cost for multi-area system

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Dynamic Economic Dispatch

In practical economic dispatch problems the generator ramp rate limits (up limits and down limits) play a very important role in finding the optimal schedule because practical generators have to follow these constraints while increasing/decreasing their power output. The results are tabulated in Table 6 for Test Case III.

Table 6 Results of optimal dynamic dispatch for 140-unit system using PSO_TVAC (Test Case III)
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Comparison of PSO Variants

For validation, the results are compared to GAMS for convex functions. The time taken by the three PSO variants is almost comparable as shown in Table 7. However, GAMS is faster, as it is a gradient based approach. But PSO is a random search method capable of optimizing non-differentiable objective functions also, whereas GAMS is unable to solve such cases [9, 17]. Due to their non dependence on nature of objective function, the nature inspired optimization methods such as PSO have an edge over traditional solvers like GAMS which are incapable for discontinuous or non-convex objective functions.

Table 7 CPU time comparisons of PSO variants
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Conclusions

Generally the PSO algorithms experience the problem of untimely stagnation and early convergence which prohibits them in locating the global optimum solution. The proposed PSO variants employ powerful parameter automation strategies which prevent early convergence to local optimal results. The performance of these variants is tested on a large system under both static/dynamic conditions and validated using traditional solver GAMS. The test results clearly show that

  • All three proposed variants achieve significantly better results as compared to the classical PSO for a large single as well as multi-area power system.

  • The PSO variants are able to handle complex equality/inequality constraints like generation limits, area-wise power balance and ramp rate limits effectively under all static as well as dynamic test conditions.

  • The variation of optimal tie line capacity with changing load demands was also computed and analyzed. By increasing the tie-line flow limit cost can be significantly reduced.

  • The three PSO variants were capable of handling ramp rate constraints also for computing optimal dynamic dispatch solution.

  • All three variants are shown to have a stable convergence characteristic. On comparison, PSO_TVAC is found to have better performance consistently for all cases.