# IEEE 15-Units ELD Test System

I. Introduction:

$$\bullet$$ This system contains fifteen generating units with a load demand of 2630 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, by 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:

$$\bullet$$ If the ramp-rate limits are modeled as constraints in the design function, then 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, which are tabulated in Table 2.

$$\bullet$$ Also, 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 [6] and [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.

II. Files:

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

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

[1] F. N. Lee and A. M. Breipohl, “Reserve Constrained Economic Dispatch with Prohibited Operating Zones,” IEEE Trans. Power Syst., vol. 8, no. 1, pp. 246–254, 1993.
[2] 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.
[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] 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.
[5] 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.
[6] G. Shabib, M. A. Gayed, and A. M. Rashwan, “Modified Particle Swarm Optimization for Economic Load Dispatch with Valve-Point Effects and Transmission Losses,” Curr. Dev. Artif. Intell., vol. 2, no. 1, pp. 39–49, 2011.
[7] 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.
[8] 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.
[9] 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.
[10] 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.
[11] Hardiansyah, “Modified Differential Evolution Algorithm for Economic Load Dispatch Problem with Valve-Point Effects,” Int. J. Adv. Res. Electr. Electron. Instrum. Eng., vol. 3, no. 11, pp. 13400–13409, Nov. 2014.