# IEEE 40-Units ELD Test System

I. Introduction:

$$\bullet$$ This system contains forty generating units with a load demand of 10500 MW. Some researchers could evaluate their proposed optimization algorithms with different load demands.
$$\bullet$$ The fuel-cost function of this test system is modeled using the quadratic cost function as follows:

$$C_i\left(P_i\right) = a_i + b_i P_i + c_i P^2_i$$ .......... $$(1)$$

where $$a_i$$, $$b_i$$, and $$c_i$$ are the function coefficients and tabulated in Table 1.

$$\bullet$$ If the valve-point loading effects are considered, then (1) becomes:

$$C_i\left(P_i\right) = a_i + b_i P_i + c_i P^2_i + \left|d_i \times \sin\left[e_i \times \left(P_i^{min} - P_i\right) \right]\right|$$ .......... $$(2)$$

where $$d_i$$ and $$e_i$$ are the coefficients of the valve-point loading effects. Thus, Table 1 is expanded to Table 2.

$$\bullet$$ The valve-point loading effects can be relaxed if either $$d$$ or $$e$$ of all units are set to zero.
$$\bullet$$ The network losses $$P_L$$ is neglected in this test system.
$$\bullet$$ If the emission-rates are considered, then the following equation can be used to express these rates in the optimization problem:

$$E\left(\sum_{i=1}^n P_i\right) = \sum_{i=1}^n 10^{-2} \left(\alpha_i + \beta_i P_i + \gamma_i P^2_i \right) + \xi_i e^{\delta_i P_i}$$ .......... $$(3)$$

where $$\alpha_i$$, $$\beta_i$$, $$\gamma_i$$, $$\xi_i$$, and $$\delta_i$$ are the coefficients of the $$i$$th unit emission characteristics. Thus, Table 2 is expanded to Table 3 [19].

II. Files:

$$\bullet$$ System Data (Text Format) [Download]

III. References (Some selected papers that use this test system):

[1] Z.-L. Gaing, “Particle Swarm Optimization to Solving the Economic Dispatch Considering the Generator Constraints,” Power Syst. IEEE Trans., vol. 18, no. 3, pp. 1187–1195, 2003.
[2] N. Sinha, R. Chakrabarti, and P. K. Chattopadhyay, “Evolutionary Programming Techniques for Economic Load Dispatch,” Evol. Comput. IEEE Trans., vol. 7, no. 1, pp. 83–94, Feb. 2003.
[3] A. Pereira-Neto, C. Unsihuay, and O. R. Saavedra, “Efficient Evolutionary Strategy Optimisation Procedure to Solve the Nonconvex Economic Dispatch Problem with Generator Constraints,” IEE Proc. - Gener. Transm. Distrib., vol. 152, no. 5, pp. 653–660, Sep. 2005.
[4] J.-B. Park, K. Y. K.-S. Y. K. Y. Lee, J.-R. Shin, and K. Y. K.-S. Y. K. Y. Lee, “A Particle Swarm Optimization for Economic Dispatch With Nonsmooth Cost Functions,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 34–42, Feb. 2005.
[5] C. H. Chen and S. N. Yeh, “Particle Swarm Optimization for Economic Power Dispatch with Valve-Point Effects,” in Transmission Distribution Conference and Exposition: Latin America, 2006. TDC ’06. IEEE/PES, 2006, pp. 1–5.
[6] A. I. Selvakumar and K. Thanushkodi, “A New Particle Swarm Optimization Solution to Nonconvex Economic Dispatch Problems,” Power Syst. IEEE Trans., vol. 22, no. 1, pp. 42–51, 2007.
[7] J. S. Al-Sumait, A. K. AL-Othman, and J. K. Sykulski, “Application of Pattern Search Method to Power System Valve-Point Economic Load Dispatch,” Int. J. Electr. Power Energy Syst., vol. 29, no. 10, pp. 720–730, 2007.
[8] C. C. Kuo, “A Novel Coding Scheme for Practical Economic Dispatch by Modified Particle Swarm Approach,” IEEE Trans. Power Syst., vol. 23, no. 4, pp. 1825–1835, Nov. 2008.
[9] K. T. Chaturvedi, M. Pandit, and L. Srivastava, “Self-Organizing Hierarchical Particle Swarm Optimization for Nonconvex Economic Dispatch,” Power Syst. IEEE Trans., vol. 23, no. 3, pp. 1079–1087, Aug. 2008.
[10] P. K. Roy, S. P. Ghoshal, and S. S. Thakur, “Biogeography-based Optimization for Economic Load Dispatch Problems,” Electr. Power Components Syst., vol. 38, no. 2, pp. 166–181, 2009.
[11] L. dos S. Coelho, R. C. T. Souza, and V. C. Mariani, “Improved Differential Evolution Approach Based on Cultural Algorithm and Diversity Measure Applied to Solve Economic Load Dispatch Problems,” Math. Comput. Simul., vol. 79, no. 10, pp. 3136–3147, 2009.
[12] A. I. Selvakumar and K. Thanushkodi, “Optimization Using Civilized Swarm: Solution to Economic Dispatch with Multiple Minima,” Electr. Power Syst. Res., vol. 79, no. 1, pp. 8–16, 2009.
[13] A. Bhattacharya and P. K. Chattopadhyay, “Biogeography-based optimization for different economic load dispatch problems,” IEEE Trans. Power Syst., vol. 25, no. 2, pp. 1064–1077, May 2010.
[14] A. Bhattacharya and P. K. Chattopadhyay, “Hybrid Differential Evolution with Biogeography-Based Optimization for Solution of Economic Load Dispatch,” IEEE Trans. Power Syst., vol. 25, no. 4, pp. 1955–1964, Nov. 2010.
[15] T. Niknam, “A New Fuzzy Adaptive Hybrid Particle Swarm Optimization Algorithm for Non-Linear, Non-Smooth and Non-Convex Economic Dispatch Problem,” Appl. Energy, vol. 87, no. 1, pp. 327–339, Jun. 2010.
[16] J. S. Alsumait, J. K. Sykulski, and A. K. Al-Othman, “A Hybrid GA-PS-SQP Method to Solve Power System Valve-Point Economic Dispatch Problems,” Appl. Energy, vol. 87, no. 5, pp. 1773–1781, Nov. 2010.
[17] S. Hemamalini, S. P. Simona, and S. P. Simon, “Artificial Bee Colony Algorithm for Economic Load Dispatch Problem with Non-Smooth Cost Functions,” Electr. Power Components Syst., vol. 38, no. 7, pp. 786–803, 2010.
[18] Z. Mohammed and J. Talaq, “Economic Dispatch by Biogeography Based Optimization Method,” in 2011 International Conference on Signal, Image Processing and Applications, 2011, vol. 21, pp. 161–165.
[19] S. Rajasomashekar and P. Aravindhababu, “Biogeography Based Optimization Technique for Best Compromise Solution of Economic Emission Dispatch,” Swarm Evol. Comput., vol. 7, pp. 47–57, Jun. 2012.
[20] G. Xiong, D. Shi, and X. Duan, “Multi-Strategy Ensemble Biogeography-Based Optimization for Economic Dispatch Problems,” Appl. Energy, vol. 111, pp. 801–811, Jun. 2013.
[21] T. H. Khoa, P. M. Vasant, M. S. B. Singh, and V. N. Dieu, “Solving Economic Dispatch Problem with Valve-Point Effects Using Swarm-Based Mean-Variance Mapping Optimization (MVMO),” Cogent Eng., vol. 2, no. 1, pp. 1–18, Aug. 2015.