Interruptible Load Contracts Implementation in Stochastic Security Constrained Unit Commitment Mokhtar Bozorg, Ehsan Hajipour, and Seyed Hamid Hosseini, Member, IEEE Center of Excellence in Power System Management and Control, Department of Electrical Engineering Sharif University of Technology Tehran, Iran
[email protected], ehsan.hajipour@ gmail.com,
[email protected] Accordingly, methods of load shedding and re-dispatch of generation and reserve based on reliability assessment techniques, available for considering these changes, have been improved. Optimization methods for calculating load shedding and units’ outputs corresponding to each contingency, using optimal power flow in restructured environment with pool market, based on cost minimization including generation, reserve, and load shedding costs considering network constraints has been presented in the literature [4].
Abstract— Providing more choices to customers for determining the cost of their required reliability levels is one of the most important targets of restructured power system. Centralized management of reliability in Vertically Integrated Utility (VIU) applying the same policies to different customers is replaced with decentralized management allowing customers to participate in reliability management based on their required reliability levels. Hence, the importance of implementing interruptible load contracts that specifies each customer’s reliability level has been increased in power system studies. In this paper, stochastic programming based on Monte Carlo approach has been used to implement interruptible load contracts in security constrained unit commitment. The performance of the proposed method is demonstrated by applying it to a 6-bus system.
Expected Energy Not Supplied (EENS) and Loss Of Load Expectation (LOLE) can be calculated for each bus as well as the entire system. EENS and LOLE give useful information regarding compromise between economic and reliability issues, considering uncertainties and operational constraints and resources. In some countries, assessment of reliability indices in each bus is a key component in power negotiations in open access market. Meanwhile, the same indices are used in distribution network reliability assessment and large industrial loads [5].
Keywords- interruptible load; Monte Carlo; SCUC; stochastic programming; reliability
I.
INTRODUCTION
Economical needs and the growth of liberalization process in electric markets have increased the use of stochastic optimization methods in order to handle the uncertainties in power systems. Especially, due to increasing contribution of utilities in deregulated environment, uncertainty in consumption of the electric power and the cost of fuel and electricity has been increased in the spot market [1]. On the other hand, in restructured power systems, generating units and transmission lines operate near their own limits. This leads to an increase in the probability of generating units outage that reduces system security margin [2]. The impact of electricity market on energy cost and usage and also on environmental issues has emphasized the management of limited resources of fuel and environment. Therefore, long-term SCUC is an appropriate tool to deal with interaction between the purposes of market and limited resources [3]. Traditionally, to secure the network in order to manage the problems from the outage of generators and transmission lines, spinning and nonspinning reserves based on maximum load or worst outage case have been used. This is a very conservative method that increases the system operating cost. Moreover, multiple component outages are not considered in this method. In stochastic approach, uncertainty in load estimation as well as outage of generators and transmission lines can be taken into account through different scenarios [1].
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Contract interruptible load agreement is an appropriate approach to reduce the cost of providing energy. Due to estimation and assessment of costs and revenues, details of these contracts should be carefully calculated. This calculation is performed according to current energy cost and investment rate of return using economic techniques in a life cycle. Some of the most significant parameters of these contracts based on IEEE 798 standard are [6]: • The time period between the power company request for a load reduction and the reduction. •
The magnitude of the reduction.
•
The maximum number of curtailments per year.
•
The maximum length of each curtailment.
•
The total number of curtailment hours in a year.
From power system point of view, each parameter, especially the second parameter has a consequential role in determining operating costs. On the other hand, from customer point of view, reduction in the cost of consumed energy is the most important benefit obtained from these contracts [7]. In a model presented in this paper beside EENS and LOLE as the entire system reliability indices, parameters
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corresponding to each consumer contract are considered as decision variables. With regards to these parameters, value of shed load in each bus and consequently effective load of the entire system are changed. Therefore, long-term SCUC solution will change.
B. Scenario Reduction Computational effort needed for optimization model based on scenarios is a function of the number of scenarios. Scenario reduction techniques lead to acceptable approximation of original system by means of less number of scenarios. The main approach for scenario reduction is controlling a fitness function, as probabilistic index, based on distance between random distributions. Details of the algorithms used in this paper can be found in [11].
By use of appended constraints which are introduced in this paper, the amount of load shedding in each bus can be controlled. In conventional methods, Value Of Lost Load (VOLL) is the single index that represents importance of each bus. Since VOLL is often modeled linearly, after considering network constraints, the most possible load shedding occurs in the bus with minimum VOLL. By adding new constraints, implementation of interruptible load contracts in long-term SCUC is possible. Therefore, system operator will have a proper tool to handle different types of interruptible load contracts in the market.
Many scenarios are generated using Monte Carlo simulation. After scenario reduction, S scenarios will remain. Standard deviation of LOLE corresponding to the S remained scenarios is an index to determine the value of S. In other words, scenario reduction process is continued until standard deviation of LOLE calculated using the remained scenarios would be less than a predefined value. Typically, a value in the range of 0.01-0.05 is suggested in literature [2]. Therefore,
The rest of this paper is organized as follows. Section II discusses the Monte Carlo approach in stochastic programming. Long-term SCUC is introduced in Section III. A method of implementing interruptible load constraints is presented in Section IV. In Section V, the proposed method is applied to a test system and results are discussed. Finally, conclusions are given in Section VI. II.
S
σ
1
LOLE
S σ
S
LOLE 1
σ
(1) (2)
where s is scenario index, LOLE is the average value of LOLE related to S remained scenarios and σ is the desired standard deviation of LOLE corresponding to S remained scenarios.
STOCHASTIC PROGRAMMING
In this paper, Monte Carlo approach in stochastic programming has been used for managing uncertainties in implementation of interruptible load contracts. Stochastic programming is a mathematical optimization in which some or all of the parameters of the optimization problem are presented by random variables. The source of random variables may be different depending on the essence of the problem [8]. For solving multi-period optimization problems, when the statistical data is not enough to support stochastic optimization, scenario analysis is one of the common methods [9]. In power system operation and planning, several optimization methods have used scenario analysis for uncertainty management. Each scenario corresponds to the outcome of a random variable. In other words, scenarios are realization of a multi-dimensional stochastic process.
C. Scenario Aggregation A way to aggregate all scenarios is optimizing weighted average of all objective functions (each scenario with related probability). Too much time consumption and more complicated solution rather than optimizing objective function of one scenario are weaknesses of this method. It should be noted that in this method, solution sensitivity analysis with change in probability of each scenario must be done. Wellhedged method is an appropriate scenario aggregation method which is used in this paper. In the well-hedged approach, uncertainties related to parameters and system components are modeled by means of few subproblems, corresponding to each scenario, under the master optimization problem. After solving each subproblem and analyzing its optimum solution, solution of master optimization problem is obtained using the wellhedged constraint. This constraint would connect subproblems to each other [10].
It must be noted that scenarios and their related probabilities are obtained by discrete approximation of probability density function of random data. Obviously, choosing a set of scenarios for optimization model biases the solution of optimization problem. Sampling of historical time series and using statistical models such as regression model are the most common methods of scenario generation.
III.
LONG-TERM SCUC
Since reliability indices will be important in the long-term, this section briefly explains long-term SCUC considering fuel and environment constraints. There are many different approaches to solve long-term SCUC in the literature [1]-[2], [12]-[17]. Fig. 1 depicts flowchart of a method introduced in [2] and extended in this paper to include interruptible load contracts shown by dashed lines. For reducing the solution time, network is solved using DC load flow [12]. To reduce the large dimension of the problem caused by the long-term consideration, each week is solved as a subproblem separately [13].
A. Monte Carlo Approach The main idea of Monte Carlo approach is to estimate expected value of objective function which is defined by scenarios [10]. One of the significant advantages of Monte Carlo approach is that the number of samples required to achieve a specified level of accuracy is independent of the system size. Therefore, Monte Carlo is very suitable to analyze large systems such as power systems [2].
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causes of such infeasibilities and the ways to overcome them are described. A. Islanding In scenario generation, especially for networks with weak transmission lines reliability, some scenarios are encountered in which the network is divided into two or more separate parts. Since the outage of transmission lines is handled in security check block, not in weekly unit commitment block, the load balance equation in unit commitment block is written for the whole network. So, bender’s cut that returns from security check block makes unit commitment problem infeasible. To overcome this problem, at the first step of each subproblem corresponding to a scenario, the status of transmission lines at all hours of this scenario will be reviewed. In case of islanding, load balance equation will be written in each island separately. B. Bender’s Cut Considering Load Shedding In network security check block, only generators output in bender's cut will be considered and therefore the constraint from this bender’s cut does not change the amount of load shedding. In this situation, bender’s cut resulting from network security check block, causes the master problem to enter into an infinite loop. To overcome this problem, (3) should be added to the security check block equations.
Figure 1. Long-Term Security Constrained Unit Commitment Flowchart
SCUC constraints could be categorized as follow: •
Unit commitment constraints of each separate week such as min/max output powers, min up/down times, max up/down ramp rates, and load balance equation.
•
Network security constraint such as transmission line flow limits.
•
Constraint which connect weeks such as total fuel limit, total allowable pollution and maximum allowable LOLE of the entire system.
:
(3)
where , is the load shedding in bus b at time t in network comes from the solution of security problem and value of the previous master problem iteration. The bender’s cut used in this iteration of master problem is considered as follows: 0
It should be noted that several constraints of first category such as min up/down times and max up/down ramp rates connect adjacent weeks. Since the problem will be solved consequently, initial state of each week must be obtained from solution of the previous week.
(4)
where is the generator output, and are the marginal values of previous iteration of master problem, and is the value of subproblem’s objective function. By doing this, the main problem is be modified so that it has a feasible solution, although it accepts a heavy price of load shedding.
First, for each week the unit commitment is solved. Then, the solution of this weekly unit commitment is checked for network security constraints. Bender’s cuts will be added to next iteration of weekly unit commitment until all security constraints are satisfied [14]. Then, solution of each week is transferred to the next week as initial state and next week solution will be found similarly. This process continues until the last week. Third category constraints are implemented using subgradient method. In this method, Lagrangian terms corresponding to total fuel consumption constraint, total allowable pollution, and maximum allowable LOLE of entire system are appended to the objective function. In each outer iteration, each Lagrangian multiplier is updated proportional to the deviation of related constraint. This process is continued until the maximum number of outer iterations is reached. Finally, the best solution is achieved from all obtained optimum solutions.
IV.
INTERRUPTIBLE LOAD CONTRACTS IMPLEMENTATION
In typical approaches to long-term SCUC, the amount of load shedding multiplied by VOLL is added to objective function and the details of interruptible load contracts are not considered. In this section, the implementation of some of the significant parameters of these contracts based on IEEE 798 standard is presented. 1) The magnitude of the reduction Implementing this item is relatively simple and it is enough to add a new constraint to load shedding variables in the weekly unit commitment block. (5) where is the load shedding value in bus b at time t of is the maximum amount of reduction that period p and has been mentioned in the contract of bus b.
Because of random nature of this problem, some of the created scenarios may be infeasible. In what follows, two
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is the maximum allowable number of where curtailments in bus b, is the correspondent Lagrangian multiplier at iteration k and is an appropriate coefficient.
2) Maximum length of each curtailment For each bus this constraint is similar to min up/down times of generating units. So, it is considered in weekly unit commitment block. To implement this constraint, as a new variable has been introduced. If there is a curtailment in bus b at time t of period p, this binary variable is set to 1, else set to 0. To set this variable, (6) and (7) are added to weekly unit commitment block constraints.
4) Total number of curtailment hours in a year The parameter is an appropriate index to handle this item and can be calculated by (14) and the corresponding Lagrangian multiplier is updated by (15).
(6)
,
(7)
∑
∑
(14)
where , represents demand of bus b at time t and M is a very large number. Equations (8) and (9) depict the constraint of maximum length of each curtailment. 1,
(15)
(8)
where
is maximum allowable LOLP at bus b.
5) The time period between the power company request for a load reduction and the reduction The time period between the power company request for a load reduction and the reduction is a short-term parameter that always is less than an hour. Since the minimum time period in the long-term SCUC is an hour, this item can not be modeled with this method.
(9) represents the maximum length of where curtailment that has been mentioned in the contract of bus b and is the initial state of this bus. It should be noted that (8) is also used in the next week solution.
V.
As a case study, we use the 6-bus system introduced in [1]. This system having 7 transmission lines is depicted in Fig. 2. Transmission lines data including transmission capacity and mean up/down times are presented in Table I. The entire system peak load is 300 MW that is located at buses 4, 5, and 6. The VOLL for these buses are 1000, 900, and 800$, respectively. Forecasted load in each week is assumed to be constant. Total period of optimization is considered to be 4 weeks. Table III shows the load data of each week.
3) Maximum number of curtailments per year This item and the next item of the interruptible load contracts are considered as constraints that are modeled using the subgradient method in which Lagrangian terms are added to the objective function. This approach is shown by the dashed lines in Fig. 1. The number of curtailments in each time period is obtained using and as new binary variables. is 1 at the starting time of curtailment and 0 at other times. Similarly, is 1 at the time of finishing curtailment and 0 at other times. So, to set these variables, (10) and (11) should be added to weekly unit commitment constraints.
Three generators that are located at buses 1, 2, and 3 supply the entire system load. Technical and economical data related to these generators are given in Table II. Fuel of generator 1 is coal and two other generators consume gas and oil. So, there are no fuel constraints for generators 2 and 3. Fuel constraint of generator 1 is 1,800,000 units.
(10) 1 1
CASE STUDY
TABLE I.
TRANSMISSION LINE DATA
(11)
Thus, the number of curtailments at each bus can be obtained from serial weekly optimization solution as follows: (12) is the number of hours in each period and is the where total number of periods. In each outer iteration, the Lagrangian multiplier is updated by (13). (13)
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Line Number
From Bus
To Bus
Flow Limit(MW)
1 2 3 4 5 6 7
1 2 1 2 4 5 6
2 6 4 4 5 3 3
200 100 100 100 100 100 100
Mean Up Time(h) 23.5 23.7 23.6 23.6 23.7 23.6 23.8
Mean Down Time(h) 0.5 0.3 0.4 0.4 0.3 0.4 0.2
TABLE II.
Unit
Bus No.
G1 G2 G3
1 2 3
Unit Cost Coefficient c b a (MBtu/ (MBtu/ (MBtu) MW2h) MWh) 176.95 13.51 0.00045 129.97 32.63 0.00100 137.41 17.69 0.00500
GENERATORS TECHNICAL AND ECONOMICAL DATA
Pmax (MW)
Pmin (MW)
Ini. St. (h)
Min Dn (h)
Min Up (h)
Ramp (MW/h)
Start Up (MBtu)
Fuel Price ($/MBtu)
220 100 100
100 10 10
4 2 1
4 3 1
4 2 1
55 50 20
100 200 0
1.2469 1.2461 1.2462
Mean Down Time (h) 0.4 0.3 0.2
Mean Up Time (h) 23.6 23.7 23.8
In addition to the data in Table II, the shut down cost for generator one is 30 $ and two other generators do not have shut down cost. In initial state, generators output powers are 200, 50 and 50 MW, respectively. A. Scenario Analysis Scenarios are realization of a multi-dimensional stochastic process. Each scenario corresponds to the outcome of a random variable. In this work, error in load estimation, outage of generators and transmission lines are random variables used in scenario generation process. Maximum error in load estimation is assumed to be 2% and for a given time the load is modeled by three states. Time sequence of generators and transmission lines availability status is generated using an exponential distribution based on their failure rates and repair times [1]. Initial number of generated scenarios is 100 with equal probability of 1/100 that are reduced to 8 using GAMS/SCENRED program with the fast back forward method. Maximum allowable standard deviation of LOLE obtained from remaining scenarios that were explained in section III is assumed to be 0.05. Table IV shows the probability corresponding to each final scenario.
Figure 2. One-line diagram of 6-bus test system TABLE III. Week 1 88.0%
Peak Load
B. Interruptible Load Contract Analysis 1) Case 1: In this base case, it is assumed that none of the interruptible load contracts have been applied in system operation. As seen in table V, due to lower value of lost load in bus 6 rather than buses 4 and 5, as expected, all the shed load has been occurred in this bus. So, similar to the entire system, LOLE in this bus is 0.006 and in average we have one curtailment in a month that remained 4 hours. EENS is 231.21 MWh and maximum amount of curtailment is 62.91 MW. The simulation results are given in Tables V and VI. As we will discuss in the next cases, customers in this bus can prevent this condition by signing a formal interruptible load contract.
TABLE IV. Scenario Probability
1 0.10
LOAD DATA
Week 2 90.0%
Week 3 86.2%
Week 4 84.5%
PROBABILITY OF SCENARIOS AFTER SCENARIO REDUCTION 2 0.15
3 0.11
4 0.12
5 0.18
6 0.15
7 8 0.07 0.12
Table V shows operation cost, EENS, and maximum load shedding in each bus. Table VI shows the curtailments results. 3) Case 3: As seen in the base case, maximum load shedding in bus 6 is 62.91 MW. In this case, system operator is forced to apply maximum allowable load shedding based on interruptible load contract in each bus. Maximum allowable reduction in each bus is 50 MW. As seen in Table V, load shedding is shifted from bus 6 to bus 5 which has lower VOLL than bus 4.
2) Case 2: Maximum allowable LOLE in each bus is applied in this case. Suppose bus 4 is an important bus, so maximum allowable LOLE for buses 4, 5 and 6 respectively are 0.001, 0.003 and 0.003. By applying these LOLE values in interruptible load contracts, the value of shed loads in each bus will be controlled partly. As seen in table V, although the VOLL in bus 6 is lower than other buses due to LOLE control in each bus, system operator is forced to perform curtailments in other buses. In this situation, in average, bus 6 has two onehour lasting curtailments, bus 4 has a one-hour lasting curtailment and bus 5 has a two-hour lasting curtailment.
4) Case 4: In this case, in addition to the items applied in case 3, system operator is forced to apply maximum length of each curtailment constraint which is assumed to be two hours. The simulation results are given in Tables V and VI.
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TABLE V.
Case
System LOLE
Case 1 Case2 Case3 Case4 Case5
0.006 0.007 0.006 0.006 0.006
RELIABILITY AND ECONOMICAL RESULTS
Costumer LOLE Bus 4 0 0.001 0 0.003 0.004
Bus 5 0 0.003 0.006 0.004 0.003
Costumer EENS(MWh) Bus 6 0.006 0.003 0.006 0.004 0.004
Bus 4 0 14 0 8.7 26.01
Bus 5 0 108.7 31.21 72.51 55.2 [4]
TABLE VI.
Case Case 1 Case 2 Case 3 Case 4 Case 5
CURTAILMENTS RESULTS (IN AVERAGE)
Length of Each Curtailment (hour) Bus 4 Bus 5 Bus 6 0 0 4 1 1 2 0 4 4 2 2 2 2 2 2
[5]
Number of Curtailments in a month Bus 4 Bus 5 Bus 6 0 0 1 2 1 1 0 1 1 2 1 2 2 1 2
[6] [7]
[8]
5) Case 5: In this case, maximum number of allowed curtailments is added to the items considered in case 3. Setting the maximum number of curtailments in bus 5 to 12 times a year, yields the results shown in Tables V and VI.
[9]
[10]
[11]
VI.
CONCLUSION
Contract interruptible load agreement is an appropriate approach to reduce the cost of providing energy. Due to estimation and assessment of costs and revenues, details of these contracts should be carefully calculated. In typical approaches to long-term SCUC, the amount of load shedding multiplied by VOLL is added to objective function, and the detail of interruptible load contracts is not considered. In this paper, implementation of some of the significant parameters of interruptible load contracts based on IEEE 798 standard is presented. Using additional constraints which are applied in this paper, system operator would find an appropriate tool to implement interruptible load contracts in long-term SCUC considering fuel and environmental limits.
[12]
[13]
[14]
[15]
[16]
REFERENCES [1]
[2]
[3]
L. Wu, M. Shahidhpour, T. Li, “Stochastic Security-Constrained Unit Commitment”, IEEE Transactions on Power Systems, Vol. 22, No. 2, May 2007, pp. 800-811. L. Wu, M. Shahidhpour, T. Li, “Cost of Reliability Analysis Based on Stochastic Unit Commitment”, IEEE Transactions on Power Systems, Vol. 23, No. 3, August 2008, pp. 1364-1374. Yong Fu, Mohammad Shahidehpour and Zuyi Li, “Long-Term SecurityConstrained Unit Commitment: Hybrid Dantzig–Wolfe Decomposition and Subgradient Approach”, IEEE Transactions on Power Systems, Vol. 20, No. 4, November 2005, pp. 2093-2106.
[17]
[18]
[19]
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Bus 6 231.21 122.51 200 150 150
Costumer Max. LS.(MWh) Bus 4 0 14 0 5.2 12.91
Bus 5 0 55.2 12.91 50 50
Bus 6 62.91 62.91 50 50 50
Operation Cost ($) 2949123 2974453 2952244 2958114 2959845
A. Grey and A. Sekar, “Reliability assessment of restructured power systems using optimal load shedding technique” IET Generation Transmission & Distribution, Vol. 3, July 2009, pp 628-640. A. B. R. Kumar, S. Vemuri, L. P. Ebrahimzadeh, and N. Farahbakhshian, “Fuel resource scheduling, the long-term problem,” IEEE Transaction on Power systems, vol. PWRS-1, November 1986, pp. 145–151. IEEE Standard 739-1984 "Recommended Practice for Energy Conservation and cost-effective planning in Industrial facilities”. G. J. Nolan, V. J. Puccio and C. W. Calhoun, “Standby power generation under utility curtailment contract agreements,” IEEE Transaction on Power systems, vol. 33, November-December 1997, pp. 1432–1438. Singiresu S. Rao, Engineering Optimization : Theory and Practice , Third Edition , John and Wiley, 605 Third Avenue, New York, NY 10158, 1996 , 903 pp. W. Oliveira, C. Sagastiz´abal, D. Penna, M. Maceira, J. Dam´azio, ” Optimal Scenario Tree Reduction for Stochastic Streamflows in Power Generation Planning Problems” EngOpt Conference, 1-5 June, 2008. R. T. Rockafellar and R. J.-B.Wets, “Scenarios and policy aggregation in optimization under uncertainty,” Math. Oper. Res., vol. 16, no. 1, pp.119–147, 1991. N. Growe-Kuska, H. Heitsch, W.Romisch, “Scenario reduction and scenario tree construction for power management problems, IEEE Bologna Power Tech Proceedings, Vol. 3, June 2003. Yong Fu, Mohammad Shahidehpour and Zuyi Li, “Fast SCUC for largescale power systems”, IEEE Transactions on Power Systems, Vol. 22, No. 4, November 2007, pp. 2144-2151. Yong Fu, Mohammad Shahidehpour and Zuyi Li, “Long-Term SecurityConstrained Unit Commitment: Hybrid Dantzig–Wolfe Decomposition and Subgradient Approach”, IEEE Transactions on Power Systems, Vol. 20, No. 4, November 2005, pp. 2093-2106. Mohammad Shahidehpour, Yong Fu, “Benders decomposition: applying Benders decomposition to power systems”, IEEE Power and Energy Magazine, Vol. 3, November March-April 2005, pp. 20-21. Xiaohong Guan, Sangang Guo, and Qiaozhu Zhai, “The Conditions for Obtaining Feasible Solutions to Security-Constrained Unit Commitment Problems”, IEEE Transactions on Power Systems, Vol. 20, No. 4, November 2005, pp. 1746-1756. A. Grey and A. Sekar, “Unified solution of security-constrained unit commitment problem using a linear programming methodology” IET Generation Transmission & Distribution, Vol. 2, November 2008, pp 856-867. Yong Fu, Mohammad Shahidehpour and Zuyi Li, “AC Contingency Dispatch Based on Security-Constrained Unit Commitment”, IEEE Transactions on Power Systems, Vol. 21, No. 2, May 2006, pp. 897-908. S. Takriti, J. R. Birge, and E. Long, “A stochastic model for the unit commitment problem,” IEEE Transaction on Power systems, vol. 11, August. 1996, pp. 1497–1508 GAMS/SCENRED user guide available at http://www.gams.com/
