Fuzzy Optimization and Decision Making

, Volume 15, Issue 1, pp 103–128

• Bo Wang
• Xian-zhong Zhou
Article

Abstract

In this study, we establish a bilevel electricity trading model where fuzzy set theory is applied to address future load uncertainty, system reliability as well as human imprecise knowledge. From the literature, there have been some studies focused on this bilevel problem while few of them consider future load uncertainty and unit commitment optimization which handles the collaboration of generation units. Then, our study makes the following contributions: First, the future load uncertainty is characterized by fuzzy set theory, as the various factors that affect the load forecasting are often assessed with some non-statistical uncertainties. Second, the generation costs are obtained by solving complicated unit commitment problems, rather than approximate calculations used in existing studies. Third, this model copes with the optimizations of both the generation companies and the market operator, where the unexpected load risk is particularly analyzed by using fuzzy value-at-risk as a quantitative risk measurement. Forth, a mechanism to encourage the convergence of the bilevel model is proposed based on fuzzy maxmin approach, and a bilevel particle swarm optimization algorithm is developed to solve the problem in a proper runtime. To illustrate the effectiveness of this research, we provide a test system-based numerical example and discuss about the experimental results according to the principle of social welfare maximization. Finally, we also compare the model and algorithm with conventional methods.

Keywords

Bilevel programming Fuzzy set theory Electricity trading  Unit commitment Fuzzy value-at-risk Particle swarm optimization

Constants

$$m$$

Index of generation company

$$M$$

Number of total generation companies

$$j$$

Index of each generation unit

$$t$$

Index of each scheduling period

$$T$$

Number of total scheduling periods

$$N_{m}$$

Number of units in company $$m$$

$$\textit{AP}_{t,-}^{m}$$

Empirical lower bound of $$\textit{AP}_{t}^{m}$$

$$\textit{SC}_{j_{m}}$$

Cold/hot star-up cost of unit $$j_{m}$$

$$a_{j_{m}},b_{j_{m}},c_{j_{m}}$$

Cost function coefficients of unit $$j_{m}$$

$$P_{j_{m}}^{\textit{max}}$$

Maximal generation capability of $$j_{m}$$

$$P_{j_{m}}^{\textit{min}}$$

Minimal generation constraint of $$j_{m}$$

$$P_{m}^{\textit{max}}$$

Maximal capability of company $$m$$

$$T_{j_{m},\textit{up}}$$

Minimal ‘on’ hours of unit $$j_{m}$$

$$T_{j_{m},\textit{down}}$$

Minimal ‘off’ hours of unit $$j_{m}$$

$$U_{j_{m}}$$

Number of hours unit $$j_{m}$$ is required to be on at the start of the planning period

$$D_{j_{m}}$$

Number of hours unit $$j_{m}$$ is required to be off at the start of the planning period

$$A_{j_{m}}$$

Minimal value of $$U_{j_{m}}$$ and T

$$B_{j_{m}}$$

Minimal value of $$D_{j_{m}}$$ and T

$$\textit{DR}_{j_{m}}$$

Maximal downward ramp rates of $$j_{m}$$

$$\textit{UR}_{j_{m}}$$

Maximal upward ramp rates of $$j_{m}$$

$$\widetilde{L}_{t}$$

Forecasted fuzzy load of period $$t$$

$$\textit{LB}_{t}$$

Lower bound of $$\widetilde{L}_{t}$$

$$\textit{UB}_{t}$$

Upper bound of $$\widetilde{L}_{t}$$

$$\widetilde{E}_{m}$$

Estimated fuzzy target profit of GC $$m$$

$$\widetilde{\textit{LC}}$$

Estimated fuzzy target cost of MO

$$\textit{RB}$$

Reservation budget of a MO

Functions

$$\textit{CT}_{j_{m}}(G_{t}^{j_{m}})$$

Cost function of unit $$j_{m}$$ with output $$G_{t}^{j_{m}}$$

$$F^{\prime }_{m}$$

Cost function of company $$m$$

$$F_{m}$$

Upper level objective function

$$f$$

Lower level objective function

Variables

$$\textit{AP}_{t}^{m}$$

Average bidding of company $$m$$ in $$t$$

$$\textit{HP}_{t}$$

$$G_{t}^{j_{m}}$$

Real generation of unit $$j_{m}$$ in period $$t$$

$$u_{t}^{j_{m}}$$

On/off (1/0) state of unit $$j_{m}$$ in period $$t$$

$$x_{t}^{j_{m}}$$

Startup action at time $$t$$ of generator $$j_{m}$$

$$y_{t}^{j_{m}}$$

Shutdown action at time $$t$$ of generator $$j_{m}$$

$$P_{t}^{m}$$

Generation of company $$m$$ in period $$t$$

$$P_{t}$$

Total generation of all companies in $$t$$

$$\widetilde{\textit{UL}}_{t}$$

Unexpected load of period $$t$$

$$R_{t}^{m}$$

Spinning reserve of $$m$$ in period $$t$$

$$\textit{ULC}_{t}$$

Unexpected load cost of period $$t$$

$$\textit{UMCP}_{t}$$

Unified market clearing price of period $$t$$

Notes

Acknowledgments

Supported by the Fundamental Research Funds for the Central Universities (No. 2062014286).

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