1. Introduction

Unit commitment (UC) in electric power systems is to optimize generating resources to supply system load while satisfying prevailing constraints, such as minimum on/off time, ramping up/down, minimum/maximum generating capacity and fuel and emission limit ^{[1, 2]}. In day-ahead electricity markets, generation companies offer their desirable hourly bid curves to participate in the market of the next day. Because of the significant impact of bid curve on the profit of each generation unit, proposed schedule of unit is very important.

A Genco is a self-interested entity which is responsible for its own risk-based maintenance outage scheduling. Such planned outage schedules will be submitted to the independent system operator (ISO) for approval. A Gencointends to minimize potential financial risks when planning its generator maintenance outage schedules ^[2]. It is very crucial for a Gencoto devise a maintenance outage schedule which guarantees its potential payoffs. The main factor determining the unit is turned off or on, is the MCP that ISO determine it.

As the market requires Gencos to bid for each unit separately or in the market with insignificant market power, the UC for each generating unit can be considered independently. Hence, for profit maximizing objective, it is profitable to operate a unit as long as the cost of producing is lower than the revenue obtained by selling that quantity produced ^[2].

Arroyo and Conejo ^[3] proposes a 0/1 mixed integer linearprogramming to maximize the unit profit from selling bothenergy and spinning reserve in spot market. Leou and Y. N. ^[4] Chang proposes a PUBC³ problem considering uncertainties of energy price at the stance of GENCOs. This problem is solved by the greedy algorithm and quadratic program using the concept of decommitment. A selective enumeration technique has been proposed in ^[5] to solve the PBUC problem using heuristic technique in coordination with dynamic programming and non-linear programming.

Ref ^[6] solve the problem for thermal and combined cycle units with pump storage solution and compare it with LR method. Finally It is expressed proposed method is beter.

All the proposed methods are, however, based on cost minimization objective, similar to the one used for the traditional markets. But, in deregulated markets, Gencos are usually entities owning generation resources and participating in the market with sole objective of maximizing the profits without concern of the system unless there is an incentive for it. Hence, if we consider generation planning for a period of, say 24 hours in advance, the Gencos based on the price forecast, generation unit characteristics, unit availability etc carryout profit based unit commitment (PBUC) and thereby determine the bidding strategy for each bidding period next day. In real time, they would have to meet generation as ordered by the ISO ^[7].

In this article, price uncertainty is considerd and supposeit has a fuzzy membership function. Also,It is assumed that the model of power market is PAB⁴ Numerical examples is presented the effectiveness of the proposed method.

2. Price Based Unit Commitment without Considering Uncertainty

Primitive and definitive solution to simplify the problem by assuming that market price is certain.

In most power markets, a company must be produced for each unit independently owned their proposed schedule to provide that in this case, the problem of PBUC for each unit is solved independently ^[5].

Since the schedule of Genco is determined by ISO, problem significance especially in terms of price uncertainty is clearer. The goal of the price-based unit commitment is to maximize the profit (i.e., revenue minus cost) of GENCOs subject to all constraints. This new price-based UC problem is different from traditional cost-minimization UC.

In the deregulated environment, market prices are unknown when GENCOs propose their bids. Therefore, this new algorithm should consider uncertainties exiting in price and the objective function is to maximize. Besides, the new UC problem doesn't consider satisfying load demand as its necessary constraint. The objective function and constraints of this model are described below. The objective function of this model is first depicted as follow:

(1)

(2)

3. Price Based Unit Commitment Considering Fuzzy Uncertainty

Figure 1. Unit benefit function and cost function

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View next figure

In actual power markets, due to the multiplicity of different Gencos and the multiplicity of bids, MCP of the market is an uncertain variable.

Unit Profit function from sales of energy and unit cost function is shown in the Figure 1. As can be seen, profit equals the difference between revenue and cost and greatest benefit may not occur in most generation.

The MCP is modeled as both, probability models ^{[4, 8, 9]} and the possibility ^{[10, 11]}. In this paper, we use possibility model and price uncertainty is modeled as fuzzy variables and membership function for MCP uncertainty is shown as in Figure 2.

Figure 2. Fuzzy membership function of MCP

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

View next figure

Figure 3 shows a membership function of possibility of price acceptance.

Figure 3. Fuzzy membership function of Acceptance price

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

View next figure

We can assume that the set P is the set of all possible price cuts of alpha (α - Cut) and fuzzy set market prices settle in relationships can be defined:

(3)

(4)

Profits and optimum production have a sensitive relation with price. And other hands, the optimal operating point will be different for different prices .Price predictions have always been a period of change and uncertainty is therefore better results can be obtained by fuzzy model, as the price of uncertainty.

(5)

: fuzzy Price prediction for hour(t)

(6)

(7)

P_min: Generation Lower Limit of Unit

P(t): Generation of Unit i for hour (t)

P_max: Generation Upper Limit of Unit

(8)

(9)

DR: Down Rate of Generation

UR: Up Rate of Generation

(10)

(11)

Given that the price variable is a fuzzy model, fuzzy variables in the form of Figure 4 will be. Sometimes, may be the unit status is changed (on or off) for different values of the fuzzy membership function. The dependence of the unit generation at different times to each other, and the constraints on the problem, lead to the generation scheduling is difficult with solving the problem by proposed method for alpha -cutting , profit and production planning will be different. benefits of each section related to alpha [13] can be expressed:

(12)

(13)

Optimal schedule of generation is calculated by equation [15]. In this regard, the combination of benefit and risk is optimized and acceptable risk level to the possible degree (n), is modeled. Parameter value of "n" in the examples and experience on the matter shall be determined by an expert.

(14)

(15)

Figure 4. Fuzzy membership function of profit

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

View next figure

4. Numerical Example

Table 1. Fuzzy membership function of profit

Download as

PowerPoint Slide

Larger image(png format)

Tables index

Veiw figure

View current table in a new window

View next table

Table 2. Trapezoidal membership function of MCP

Download as

PowerPoint Slide

Larger image(png format)

Tables index

Veiw figure

View current table in a new window

View previous table

View next table

Figure 5. Fuzzy membership function of MCP for Sample hour

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

View next figure

In this section we shall assume that the market price is certain. Since the purpose of comparison between the results of different methods, is price uncertainty, it is assumed that the values chosen for the certain MCP, is A3, corresponding point of fuzzy membership function. This price is the highest possible (with membership value equal to 1) to have comparable results.

Table 3. The hourly profits without regard to price uncertainty

Download as

PowerPoint Slide

Larger image(png format)

Tables index

Veiw figure

View current table in a new window

View previous table

View next table

Figure 6. Fuzzy membership function of profit

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

View next figure

Figure 7. Fuzzy membership function of MCP and profit

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

View next figure

In this numerical example, the effect of price uncertainty is represented in the final results, it is clear that the uncertainty can improve the unit profit. The hours that the unit is off, turn on/off cost (20$),lead to negative profit at this time.

Table 4. Fuzzy profit per hour by alpha cut method

Download as

PowerPoint Slide

Larger image(png format)

Tables index

Veiw figure

View current table in a new window

View previous table

View next table

Figure 8. Fuzzy sets (A9, 0. 5) for fuzzy variable profits and MCP

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

Table 5. Optimal selection of generation schedule by alpha-cut method

Download as

PowerPoint Slide

Larger image(png format)

Tables index

Veiw figure

View current table in a new window

View previous table

View next table

Table 6. Optimum Generation of unit with considering price uncertainty

Download as

PowerPoint Slide

Larger image(png format)

Tables index

Veiw figure

View current table in a new window

View previous table

The hours that the unit is off or on, leads to negative profits. It is possible for the generation benefit is positive, but due to the high unit cost, total profit is negative.

With respect to the previous section, in the optimum production schedule, the profit and risk of accepting is appropriate.

Conclusion

This paper proposed a novelapproach of unit scheduling under price uncertainty using PBUC. The profit obtained using PBUC considering price (MCP) uncertainty is the membership function of Trapezoidal function. Use alpha-cut method to defuzzification final profit.

The method is tested with a numerical example to analyse the effect of the price uncertainty on the expected profit. Comparisons made show that the proposed method results in higher expected profits under price uncertainty. Hence, for decision makers, this method can be a tool for improving the expected profits.

References

[1]	M. Shahidehpour, H. Yamin, and Z. Li, "Market Operations in Electric Power Systems: Forecasting, Scheduling, and Risk Management," John Wiley & Sons, Inc. 2002.
	In article		CrossRef
[2]	G.B Shrestha, B.K. Pokharel, T.T. Lie, and Fleten, S.-E.: ‘Price based unit commitment for Gencos in deregulated power market’. IEEE PES General Meeting, 2005.
	In article
[3]	J. M. Arroyo and A. J. Conejo, “Optimal response of a thermal unit toan electricity spot market,” IEEE Trans. Power Syst., 15(3): 1098-1104, 2000.
	In article		CrossRef
[4]	R. C. Leou and Y. N. Chang,” A Price-Based Unit Commitment Model Considering Uncertainties”," IEEE Trans. Power Syst., 2005.
	In article
[5]	B.K. Pokharel, G.B. Shrestha,T.T. Lie, and Fleton, S-E., “Profit Based Unit Commitment in Competitive Markets”, International Conference on Power System Technology (POWERCON), 2004 (accepted).
	In article
[6]	T. Li, and M. Shahidehpour, “Price-Based Unit Commitment: A Case of Lagrangian Relaxation Versus Mixed Integer Programming” IEEE Trans. Power Syst., 20(4): 2015-2025, 2005.
	In article		CrossRef
[7]	F. J. Nogales, J. Contreras, A. J. Conejo, and R. Espínola, “Forecasting next-day electricity prices by time series models,” IEEE Trans. Power Syst., 17(2): 342-348, 2002.
	In article		CrossRef
[8]	T. Li, M. Shahideh pour and Z. Li, “Risk-Constrained Bidding Strategy With Stochastic Unit Commitment,” IEEE Trans. Power Syst., 22(1): 449-458, 2007.
	In article		CrossRef
[9]	G. B. Shrestha, B. K. Pokharel, T.T. Lie, and S. E. Fleten, “Price-based unit commitment for bidding under price uncertainty”, IET Gener. Transm. Distrib., 1 (4): 663-669, 2007.
	In article		CrossRef
[10]	P. Attaviriyanupap, H. Kita, E. Tanaka and J. Hasegawa, “A Fuzzy-Optimization Approach to Dynamic Economic Dispatch Considering Uncertainties,” IEEE Trans. Power Syst., 19(3): 1299-1307, 2004.
	In article		CrossRef
[11]	Srivastava, A.K. ,Daneshi, A. “Application of fuzzy logic to priced-based unit commitment under price uncertainty”, Transmission and Distribution Conference and Exposition, 2010 IEEE PES, 1-5, 2010.
	In article

Notes

<sup>1

2</sup>- Generation Company

³- Market Clearing Price

⁴- Lagrangian Relaxation