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Optimal scheduling of wind-thermal power system using clustered adaptive teaching learning based optimization

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Abstract

This paper proposes an optimal scheduling/allocation of energy and spinning reserves for a wind-thermal power system. There is a considerable need for the renewable energy resources in the modern power system; therefore, in this paper, wind energy generators are used. Here, two different market clearing models are proposed. One model includes reserve offers from the conventional thermal generators, and the other includes reserve offers from both thermal generators, and demand/consumers. The stochastic behavior of wind speed and wind power generation is represented by the Weibull probability density function. The objective function considered in this paper includes cost of energy provided by conventional thermal and wind generators, cost of reserves provided by conventional thermal generators and load demands. It also includes costs due to under-estimation and over-estimation of available wind power generation. Clustered adaptive teaching learning based optimization algorithm is used to solve the proposed optimal scheduling problem for both conventional and wind-thermal power systems considering the provision for spinning reserves. To show the effectiveness and feasibility of the proposed frame work, various case studies are presented for two different test systems.

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Abbreviations

\(a_{i}\), \(b_{i}\), \(c_{i}\) :

Cost coefficients of thermal generator i

\(a_{k}^{'}\), \(b_{k}^{'}\), \(c_{k}^{'}\) :

Cost coefficients of demand-side reserve (DR) offers for kth load/demand

c, k :

Scale factor and shape factor of Weibull distribution at a given location

\(C_{\mathrm{G}i}\) :

Fuel cost function of thermal generator i

\(C_{\mathrm{w}j}\) :

Direct cost function of wind generator j

\(C_k\) :

Cost function for demand response (DR) offers of load demand k

\(C_{\mathrm{SR}i}\) :

Spinning reserve (SR) cost function of thermal generator i

\(C_{\mathrm{p,w}j}\) :

Penalty cost function for not using all available power from jth wind power generator

\(C_{\mathrm{r,w}j}\) :

Reserve cost function relating to uncertainty of wind power. This is effectively a penalty associated with over-estimation of the available wind power

\(d_{j}\) :

Direct cost coefficient of jth wind generator

\(e_i\), \(f_i\) :

Cost coefficients of the thermal generator with valve-point loading effects

\(f_p(p)\) :

Wind energy generator (WEG)/wind power probability density function (PDF)

n :

Number of buses in the system

\(N_\mathrm{G}\) :

Number of conventional thermal generators

\(N_\mathrm{L}\) :

Number of loads/demands

\(N_\mathrm{W}\) :

Number of wind generators/farms

p :

WEG power output in MWs

\(P_\mathrm{D}\) :

Total system demand in MWs

\(P_{\mathrm{G}i}\) :

Scheduled power from ith conventional thermal generator in MWs

\(P_{\mathrm{G}i}^{0}\) :

Power output of ith conventional thermal generator at previous hour in MWs

\(P_{\mathrm{G}i}^\mathrm{min}\), \(P_{\mathrm{G}i}^\mathrm{max}\) :

Minimum and maximum power limits of ith generator in MWs

\(P_{\mathrm{r}j}\) :

Rated wind power from jth wind generator

\(P_{\mathrm{SR}i}\) :

Amount of spinning reserve (SR) provided by ith conventional thermal generator in MWs

\(P_{\mathrm{SR}i}^\mathrm{max}\) :

Maximum reserve capacity of ith conventional thermal generator in MWs

\(P_{\mathrm{shd,}k}\) :

Amount of demand response (DR) provided by kth demand in MWs

\(P_{\mathrm{wf,}j}\) :

Forecasted wind power from jth wind generator in MWs

\(P_{\mathrm{w}j}\) :

Scheduled wind power from jth wind generator in MWs

\(P_{\mathrm{w}j,\mathrm{av}}\) :

Available wind power from the jth wind power generator. This is a random variable, with a range of \(0 \le P_{\mathrm{w}j,\mathrm{av}} \le P_{\mathrm{r}j}\)

\(Q_{\mathrm{G}i}\) :

Reactive power output of ith generator

\(R_{\mathrm{G}i}^\mathrm{down}\), \(R_{\mathrm{G}i}^\mathrm{up}\) :

Ramp down and ramp up limits of conventional thermal generators (MW/h)

\(T_\mathrm{F}\) :

Teaching factor

\(r_{i}\) :

Random number in the range [0, 1]

\(V_{i}\), \(V_{j}\) :

Voltage magnitudes at bus i and bus j

v :

Wind speed (m/s)

\(v_{r}\), \(v_{i}\), \(v_{o}\) :

Rated, cut-in and cut-out wind speeds (m/s)

\(x_{i}\), \(y_{i}\) :

Spinning reserve (SR) cost coefficients of thermal generator i

\(\delta _{i}\), \(\delta _{j}\) :

Voltage angles at bus i and bus j

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Authors and Affiliations

  1. Department of Railroad and Electrical Engineering, Woosong University, Daejeon, South Korea

    S. Surender Reddy

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Reddy, S.S. Optimal scheduling of wind-thermal power system using clustered adaptive teaching learning based optimization. Electr Eng 99, 535–550 (2017). https://doi.org/10.1007/s00202-016-0382-5

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  • DOI: https://doi.org/10.1007/s00202-016-0382-5

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