# IEEE 6-Units ELD Test System No.1

I. Introduction:

$$\bullet$$ This system consists of six thermal units, 26 buses, and 46 transmission lines.
$$\bullet$$ The base capacity used for this test system is 100-MVA.
$$\bullet$$ The load demand is 1263 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.

$$\bullet$$ The network losses are modeled using Kron's loss formula as follows:

$$P_L = \sum_{i=1}^{n} \sum_{j=1}^{n} P_i B_{ij} P_j + \sum_{i=1}^{n} B_{0i} P_i + B_{00}$$ .......... $$(2)$$

where $$B_{ij}$$, $$B_{0i}$$, and $$B_{00}$$ are called loss coefficients (or just B-coefficients) and listed below:

$$B_{ij}=\begin{bmatrix} 0.0017 & 0.0012 & 0.0007 & -0.0001 & -0.0005 & -0.0002\\ 0.0012 & 0.0014 & 0.0009 & 0.0001 & -0.0006 & -0.0001\\ 0.0007 & 0.0009 & 0.0031 & 0.0000 & -0.0010 & -0.0006\\ -0.0001 & 0.0001 & 0.0000 & 0.0024 & -0.0006 & -0.0008\\ -0.0005 & -0.0006 & -0.0010 & -0.0006 & 0.0129 & -0.0002\\ -0.0002 & -0.0001 & -0.0006 & -0.0008 & -0.0002 & 0.0150 \end{bmatrix}$$

$$B_{0i} = 10^{-3} \times \left[-0.3908,-0.1297,0.7047,0.0591,0.2161,-0.6635\right]$$

$$B_{00} = 0.056$$

$$\bullet$$ The ramp-rate limits are modeled as constraints in the design function. Thus, the feasible search space of each unit is determined through the following equation:

$$\text{max}\left(P_i^{min},P_i^{now}-R_i^{down}\right) \leqslant P_i^{new} \leqslant \text{min}\left(P_i^{max},P_i^{now}+R_i^{up}\right)$$ .......... $$(3)$$

where $$P_i^{now}$$ and $$P_i^{new}$$ are respectively the existing and new power output of the $$i$$th generator. $$R_i^{down}$$ and $$R_i^{up}$$ are respectively the downward and upward ramp-rate limits. Therefore, Table 1 should be modified to Table 2.

$$\bullet$$ If the prohibited operating zone phenomenon is considered in the design function of the ELD problem, then the fuel-cost curves will have some discontinuities. This constraint can be modeled as follows:

\begin{align} P_i^{min}  & \leqslant P_i \leqslant P_{i,j}^L \\ P_{i,j}^U & \leqslant P_i \leqslant P_{i,j+1}^L \text{ .......... $$(4)$$}\\ P_{i,\varkappa_i}^U & \leqslant P_i \leqslant P_i^{max}\end{align}

where $$P_{i,j}^L$$ and $$P_{i,j}^U$$ are respectively the lower and upper bounds of the $$j$$th prohibited operating zone on the fuel-cost curve of the $$i$$th unit. $$\varkappa_i$$ means the total number of the prohibited operating zones exist within the $i$th unit. Based on that, Table 2 is updated to Table 3.

$$\bullet$$ In [11], the valve-point loading effects are considered in the cost function, so (1) becomes:

$$C_i\left(P_i\right) = a_i + b_i P_i + c_i P^2_i + \left|d_i \times \sin\left[e_i \times \left(P_i^{min} - P_i\right) \right]\right|$$ .......... $$(5)$$

where $$d_i$$ and $$e_i$$ are the coefficients of the valve-point loading effects. Thus, Table 3 is expanded to Table 4.

$$\bullet$$ The valve-point loading effects can be relaxed if either $$d$$ or $$e$$ of all units are set to zero.
$$\bullet$$ The researcher should carefully select the proper expression to have a fair comparison with other studies reported in the literature.

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,” IEEE Trans. Power Syst., vol. 18, no. 3, pp. 1187–1195, Aug. 2003.
[2] A. I. Selvakumar and K. Thanushkodi, “A New Particle Swarm Optimization Solution to Nonconvex Economic Dispatch Problems,” IEEE Trans. Power Syst., vol. 22, no. 1, pp. 42–51, Feb. 2007.
[3] L. dos S. Coelho and V. C. Mariani, “Improved Differential Evolution Algorithms for Handling Economic Dispatch Optimization with Generator Constraints,” Energy Convers. Manag., vol. 48, no. 5, pp. 1631–1639, Jan. 2007.
[4] 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.
[5] K. T. Chaturvedi, M. Pandit, and L. Srivastava, “Self-Organizing Hierarchical Particle Swarm Optimization for Nonconvex Economic Dispatch,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1079–1087, Aug. 2008.
[6] 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.
[7] 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.
[8] 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.
[9] N. A. Rahmat and I. Musirin, “Differential Evolution Ant Colony Optimization (DEACO) Technique in Solving Economic Load Dispatch Problem,” in 2012 IEEE Inter. on Power Engineering and Optimization Conference (PEOCO2012), 2012, pp. 263–268.
[10] A. Nazari and A. Hadidi, “Biogeography Based Optimization Algorithm for Economic Load Dispatch of Power System,” Am. J. Adv. Sci. Res., vol. 1, no. 3, pp. 99–105, Sep. 2012.
[11] M. I. Abouheaf, S. Haesaert, W. Lee, and F. L. Lewis, “Q-Learning with Eligibility Traces to Solve Non-Convex Economic Dispatch Problems,” Int. J. Electr. Robot. Electron. Commun. Eng., vol. 6, no. 7, pp. 41–48, 2012.
[12] N. Singh and Y. Kumar, “Economic Load Dispatch with Valve Point Loading Effect and Generator Ramp Rate Limits Constraint Using MRPSO,” Int. J. Adv. Res. Comput. Eng. Technol., vol. 2, no. 4, pp. 1472–1477, Apr. 2013.
[13] S. Aravind and R. Scaria, “Biogeography-Based Optimization for the Solution of the Combined Heat and Power Economic Dispatch Problem,” Int. J. Eng. Innov. Technol., vol. 3, no. 1, pp. 429–432, Jul. 2013.
[14] K. P. S. Parmar and B. Khokhar, “Oppositional Biogeography-Based Optimization for Solving Economic Dispatch Problems: An Efficient Method,” in International Conference on Advances in Computer Engineering & Applications (ICACEA-2014) at IMSEC,GZB, 2014, pp. 53–58.
[15] K. Zare and T. G. Bolandi, “Modified Iteration Particle Swarm Optimization Procedure for Economic Dispatch Solving with Non-Smooth and Non-Convex Fuel Cost Function,” in 3rd IET International Conference on Clean Energy and Technology (CEAT) 2014, 2014, pp. 1–6.