A solution to stochastic unit commitment problem for a wind-thermal system coordination

Abstract

Unit commitment (UC) problem is one of the most important decision making problems in power system. In this paper the UC problem is solved by considering it as a real time problem by adding stochasticity in the generation side because of wind-thermal co-ordination system as well as stochasticity in the load side by incorporating the randomness of the load. The most important issue that needs to be addressed is the achievement of an economic unit commitment solution after solving UC as a real time problem. This paper proposes a hybrid approach to solve the stochastic unit commitment problem considering the volatile nature of wind and formulating the UC problem as a chance constrained problem in which the load is met with high probability over the entire time period.

Keywords

unit commitment (UC) / randomness / wind generation / univariate / chance constrained

Introduction

With increasing concern for pollution and environmental degradation, the wind power production is being increasingly implemented because of its low operation cost and great contribution to curbing emissions of green house gases. However the varying and unpredictable nature of wind power generation has brought about many difficulties to the operation of the power system. Besides, the load keeps changing constantly which is otherwise assumed to be constant each hour in a unit commitment (UC) problem. UC is a term used for the strategic choice of scheduling of generation, thereby improving the overall operational economics of the system [1]. The optimization of power generation scheduling is a complicated mathematical problem. It involves both integer and continuous variables along with a large set of constraints. For large generation systems, the UC problem (UCP) is solved using linear programming and mixed integer programming [2]. The UC problem thus determines the ON/OFF scheduling of generators in order to meet the demand while satisfying the constraints: Ramp constraints are to ensure that starting from an OFF state the generators are turned to full capacity after going through a defined sequence of intermediate generation states lasting for a specific time. Uptime and downtime constraints are used to ensure that when the generator is switched ON/OFF, it stays so for a given time duration [3]. Energy requirement constraint ensures that the system generates enough power to meet the demand throughout.

Stochasticity is the term used to account for the uncertainty in the UCP. It can be observed in two ways in a power system, load side and generation side. In UCPs, load is considered to be constant throughout. However, practically it is observed to vary during different hours of the day [2]. Hence stochasticity is to be considered at the load side. In the current scenario of increased integration of wind and other renewable generation in existing system and considering their intermittent nature, scheduling decisions need to be made in advance in order to meet the requirements of net load and spinning reserve [3]. These uncertainty factors complicate the process of adjusting the UC in order to yield an optimal schedule. A literature survey reveals the various approaches currently adopted for a UCP include dynamic programming [4], Branch and Bound method [5,6] and various evolutionary programming approaches like particle swarm optimization (PSO), shuffled frog leaping algorithm (SFLA) [7,8], bacterial foraging optimization algorithm (BFOA) [9], and evolutionary iteration particle swarm optimization (EIPSO) [10].

A few existing approaches for stochastic UCP include particle swarm optimization (PSO) based scenario generation and reduction algorithm [11], impacts of wind power on thermal generation UC and dispatch [12], and dynamic programming for commitment and dispatch of wind-thermal system [13]. A method to integrate wind generators into the UCP is proposed [14], while another method to solve the nonlinear optimal scheduling problem was suggested using EIPSO [15]. The compressed air energy storage system with wind generation is integrated into the power system. Out of the various approaches listed, the Lagrangian relaxation (LR) method is the most widely accepted one because of its computational efficiency in the case of large systems. This technique is also used in the present paper to solve the UCP. Much of the computational difficulty of the UCP arises from the linking constraints which require that the total production from all the generating units equal the electricity demand for every hour. This could lead to the infeasibility in the solution. In addition, since the variable nature of wind and load are considered, these problems need to be addressed. In view of the fact that a stochastic approach using LR was studied, a hybrid approach has been developed which integrates the heuristic method with the conventional LR method to yield a feasible solution. The economic dispatch problem is solved using PSO [16]. Additional constraints are introduced to account for the additional wind energy input. To address the problem that the hourly load is stochastic, the existing formulation of the UCP is modified by ensuring that the specified constraints are met with a high degree of probability leading to a chance constrained [17] formulation for the UCP.

Problem formulation

UC-deterministic

Objective function

(1)\min {\displaystyle \sum _{i=\mathrm{1}}^N{\displaystyle \sum _{t=\mathrm{1}}^T\left[{\mathrm{CF}}_{i,t}\left(P_{i,t}{v}_{i,t}\right)+S_{i,t}\left(v_{i,t}\right)\right]}}.

The objective function includes minimizing the total cost over the entire time period. The first term is the cost of production of each unit i at each hour t. The second term is the startup cost of each unit that is turned ON.

Subject to constraints:

Generation limit constraint

(2)v_{i,t}{P}_{i,\min }\le P_{i,t}\le v_{i,t}{P}_{i,\max }\left(i=\mathrm{1},\mathrm{2},\mathrm{...},N\mathrm{;}t=\mathrm{1},\mathrm{2},\mathrm{...},T\right),

where P_i,t is the power produced by unit i at time period t, P_i,min (P_i,max) is the minimum (maximum) amount of power unit i can produce.

Minimum up time and down time constraint

T_{i,t-\mathrm{1}}^{\mathrm{on}}\ge T_{\mathrm{u}i},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}{T}_{i,t}^{\mathrm{o}\mathrm{f}\mathrm{f}}=\mathrm{1},

(3)T_{n,t}^{\mathrm{o}\mathrm{f}\mathrm{f}}=\mathrm{O}\mathrm{f}\mathrm{f}-\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}n\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}t\mathrm{;}

T_{i,t-\mathrm{1}}^{\mathrm{off}}\ge T_{\mathrm{d}i},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}{T}_{i,t}^{\mathrm{o}\mathrm{n}}=\mathrm{1},

(4)T_{n,t}^{\mathrm{o}\mathrm{n}}=\mathrm{O}\mathrm{n}-\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}n\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}t.

Linking constraint

(5){\displaystyle \sum _{i=\mathrm{1}}^N{P}_{i,t}\ge l_t},t=\mathrm{1},\mathrm{2},\mathrm{...},T\mathrm{;}

(6)v_{i,t-\mathrm{1}}\in \left\{\mathrm{0},1\right\},i=\mathrm{1},\mathrm{2},...,N\mathrm{;}t=\mathrm{1},\mathrm{2},...,T\mathrm{.}

UC-stochastic

For considering the wind power generation (WPG) in the existing thermal model, an additional ramp constraint is included.

Maximum up and down ramp constraint of the unit

This determines the maximum value of power generation that can be increased or decreased for a unit i at hour t. In this paper, the ramp capacity of each unit is considered to be 60% of its maximum capacity [7].

(7){\mathrm{U}\mathrm{R}}_i^{\max }=d\times P_i^{\max },

(8){\mathrm{D}\mathrm{R}}_i^{\max }=d\times P_i^{\max }.

Ramping up or down capacity constraint of the unit

The ramping up or down capacity constraint of the unit is as follows:

(9){\mathrm{U}\mathrm{R}}_i\left(t\right)=\min \left\{{\mathrm{UR}}_i^{\max },P_i^{\max }-P_{i,t}\right\},

(10){\mathrm{D}\mathrm{R}}_i\left(t\right)=\min \left\{{\mathrm{DR}}_i^{\max },P_{i,t}-P_i^{\min }\right\}.

Wind generation fluctuation constraint

The variations in the wind power generation are absorbed by the system ramping capacity:

(11)P_{Wl}\left(t\right)-P_{Wl}\left(t-1\right)\le \mathrm{T}\mathrm{D}\mathrm{R}\left(t\right),\mathrm{if}{P}_{Wl}\left(t-1\right)\le P_{Wl}\left(t\right)\mathrm{;}

(12)P_{Wl}\left(t-1\right)-P_{Wl}\left(t\right)\le \mathrm{TUR}\left(t\right),\mathrm{if}{P}_{Wl}\left(t\right)\le P_{Wl}\left(t-1\right).

Load constraints

The load at hour t is considered as a random variable l_t.

(13)l_t={\mu }_t+\left(z_{\alpha }/T\right){\sigma }_t,

where μ_t is the mean of load at each hour, {\sigma }_t is the standard deviation of load, z_{\alpha }/T = 100(1 - α/T)th percentile of the standard normal distribution.

The linking constraint in Eq. (4) is replaced by a probability constraint for each hour t so that the constraint is satisfied at a given probability level at all time periods.

(14)P\left[{\displaystyle \sum _{i=\mathrm{1}}^N{P}_{i,t}}\ge l_t\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}t=1,2,\mathrm{...},T\right]\ge \mathrm{1}-\alpha .

Proposed hybrid methodology

Integrating wind generators into existing thermal system causes further difficulty in the UC problem as the variability in the wind has to be taken into account to determine the most economic schedule which might change according to the wind availability [18]. Moreover, the load is not constant. As it keeps varying each second, appropriate method has to be used to determine the best probability of the generation to meet the demand criteria. In this paper a hybrid algorithm has been developed to solve this problem. The initial UC schedule is prepared using the LR method. The LR method yields infeasible solution in some cases. This is taken care of by adding an extra heuristic method. In this paper the modified LR includes a heuristic method called the priority list method where in the units are arranged in the ascending order according to their full load average cost and turned ON. Then based on the wind availability, wind power is reduced from the thermal generation after the ramp constrained is satisfied [19]. The economic dispatch is done once again for the entire system. To consider the stochastic nature of the load, a chance constrained approach is taken where the load l_t is approximated as a univariate normal distribution (Eq. (13)) so that the linking constraint (Eq. (14)) is satisfied at a given probability level. The chance constraint technique converts the stochastic problem into a deterministic equivalent which can further be solved using normal method like modified LR in the case in this paper [20].

Proposed algorithm is as follows:

Step1: Choose an initial value of z.

Step2: Initiate the input parameters required for Lagrange function and initiate the Lagrange multiplier λ_t.

Step3: Build dynamic program having two states and T stages for each unit i and solve for dual value q*(λ_t).

Step4: Solve the economic dispatch for each hour using the UC schedule.

Step5: Calculate the up and down ramp capacity of each generator which is 60% of the maximum unit capacity. Calculate the ramping up and down capacity of the system and compare it with the available wind generation at each hour t.

Step6: If the available wind generation satisfies the ramp constraint of the system, integrate it with the total generation at each hour.

Step7: Solve the economic dispatch for the obtained wind-thermal system and calculate J*.

Step8: Calculate the relative duality gap (RG) = (J* - q*)/q*.

Step9: If RG is higher than a prescribed value, update λ_t and go to Step 2. Else go to the next step.

Step10: If the final solution obtained is not feasible, use the heuristic algorithm to achieve feasible solution. The heuristic method used is to turn ON the least costly generator with a shortage of power.

Step11: Calculate the univariate normal probability. If it differs from the specified probability level by more than a preassigned quantity e, update z and go to Step 2, otherwise STOP.

The flowchart of the proposed algorithm is shown in Fig. 1.

Numerical cases

The proposed methodology is implemented on a 10 unit 24 h system which has been studied under two cases. Case 1 deals with wind integration whose load is considered to be deterministic. Case 2 involves both variable wind and load. The input data and load demand are taken from Tables 1 and 2. Initially a standard set of load (Case 1) is considered and the required values are calculated with integration of wind. Later the load is varied with a specified standard deviation.

Each unit is assumed to have the fuel cost function of

F_c=a_i+b_i{P}_{i,t}+c_i{\left(P_{i,t}\right)}^2.

The values of the fuel cost coefficients a, b, and c are listed in Table 1.

Case 1: Test system by considering 10 thermal and 1 wind generator without consideration of load variability

In Case 1, a wind unit integrated into a 10 unit thermal system is considered. The load data is given in Table 2. The available wind power generation data is presented in Table 3. The ramp rate constraints of the thermal units are calculated as 60% (d %) of the rated capacity. The ramping up and down rates of the system are then calculated. Since the wind power generated should satisfy the up and down rates of the system, these values are then compared to incorporate wind unit into the system. This method ensures that the fluctuation in the wind does not cause any operational difficulties in scheduling of the wind-thermal system. The final UC schedule obtained is then modified using the heuristic technique to obtain a feasible solution, as shown in Table 4, and the economic dispatch is illustrated in Table 5.

The heuristic technique used is the priority list method. The full load average cost is determined and the units are arranged in ascending order of cost. The least costly unit with shortage of power is turned ON. Thus, finally the feasible solution is obtained.

The wind operating cost is assumed to be 2% of the total system operating cost of thermal generation. Therefore, the total cost of the system (in $) = 508345.4+ 0.02 × 508345.4= 518512.308.

Case 2: Test system by considering 10 thermal and 1 wind generator with consideration of variable (random) load in each hour

With the same 10 unit 24 h system considered in Case 1, a method to vary the load is introduced. The mean and standard deviation of the load in each hour (Table 6) is used to obtain the value of load in each hour as a univariate function. The load is normally distributed over mean and standard deviation values which are given in Eq. (13). The value of z is varied until the best probability condition is met in Eq. (14). The result obtained is tabulated in Tables 7 and 8. The advantage of chance constrained over other methods is that the results now obtained are feasible and there is no need to employ the heuristic technique in Case 2. Also, the iterations required for a CCO algorithm to converge for highest probability is very low and thus yields a feasible solution in lesser time. This implies that the proposed algorithm is very efficient in solving a stochastic UCP with both variable generation and load.

Wind operating cost is assumed to be 2% of the total system operating cost. Therefore, the cost of the total system (in $) = 384827.8+ 0.02 × 384827.8 = 392524.356.

Conclusions

For Case1, Tables 4 and 5 showed the results obtained with the successful integration of one unit of wind with the 10 unit 24 h thermal system. With the analysis of data, it was found that with the integration of wind power in the system, the total cost of generation in each hour was reduced as some of the costlier generators were turned off because of the optimal scheduling. The infeasibility due to Lagrange method was addressed and removed successfully by the additional ramp constraints and priority list method. Thus the modified LR method is a feasible approach in integrating wind with thermal units and thereby improving the system reliability and reducing the overall costs.

In Case 2, the results obtained were tabulated in Tables 7 and 8. This method yielded an improved solution which was more feasible due to the additional chance constraint added to the existing LR method. This method acknowledged the fact that values of hourly load consisted of random variables which were correlated and thus in the numerical case illustrated, the chance constrained programming was used to schedule the units under given amount of uncertainty. The cost for wind power generation was only 2% [4] of the system operating cost. Therefore, the addition of wind made the generation relatively cheaper in comparison to its deterministic equivalent. The full load average cost comparison shown in Table 9 validated the feasibility of the approach adopted in this paper. Even though only one wind generator was added in a group of thermal generators, the proposed algorithm was very much flexible to include more wind generators. In that case the ramp constraints of thermal generator should be taken care.

References

[1] OzturkU A, MazumdarM, NormanB A. A Solution to the stochastic unit commitment problem using chance constrained programming. IEEE Transactions on Power Systems, 2004, 19(3): 1589–1598

[2] BurnsR M, GibsonC A. Optimization of priority lists for a unit commitment program. In: IEEE/PES Summer Meeting, New York: Institute of Electrical and Electronics Engineers, 1975, Paper A 75 453–1

[3] SenjyuT, MiyagiT, SaberA Y, UrasakiN, FunabashiT.Emerging solution of large-scale unit commitment problem byStochastic Priority List. Electric Power Systems Research, 2006, 76(5): 283–292

[4] OuyangZ, ShahidehpourS M. An intelligent dynamic programming for unit commitment application. IEEE Transactions on Power Systems, 1991, 6(3): 1203–1209

[5] CohenA I, YoshimuraM. A branch-and-bound algorithm for unit commitment. IEEE Transactions on Power Apparatus and Systems, 1983, PAS-102(2): 444–451

[6] WangY, XiaQ, KangC. A novel security stochastic unit commitment for wind-thermal system operation. In: Proceedings of the 4th International Conference on Electric Utility Deregulation and Restructuring and Power Technologies (DRPT). Weihai, China, 2011, 386–393

[7] ChenC L. Optimal wind-thermal generating unit commitment. IEEE Transactions on Energy Conversion, 2008, 23(1): 273–280

[8] EbrahimiJ, HosseinianS H, GharehpetianG B. Unit commitment problem solution using shuffled frog leaping algorithm. IEEE Transactions on Power Systems, 2011, 26(2): 573–581

[9] EslamianM, HosseinianS H, VahidiB. Bacterial foraging-based solution to the unit-commitment problem. IEEE Transactions on Power Systems, 2009, 24(3): 1478–1488

[10] LeeT Y. Optimal spinning reserve for a wind-thermal power system using EIPSO. IEEE Transactions on Power Systems, 2007, 22(4): 1612–1621

[11] DaneshiH, SrivastavaA K. Security-constrained unit commitment with wind generation and compressed air energy storage. IET Generation, Transmission & Distribution, 2012, 6(2): 167–175

[12] WoodJ, WollenbergB F. Power Generation, Operation & Control. John Wiley & Sons, Inc., 1996

[13] PappalaV S, ErlichI, RohrigK, DobschinskiJ. A stochastic model for the optimal operation of a wind-thermal power system. IEEE Transactions on Power Systems, 2009, 24(2): 940–950

[14] UmmelsB C, GibescuM, PelgrumE, KlingW L, BrandA J. Impacts of wind power on thermal generation unit commitment and dispatch. IEEE Transactions on Energy Conversion, 2007, 22(1): 44–51

[15] LogenthiranT, SrinivasanD. Particle swarm optimization for unit commitment Problem. In: 2010 IEEE 11th International Conference on Probabilistic Methods Applied to Power Systems (PMAPS). Singapore, 2010, 642–647

[16] SriyanyongP, SongY H. Unit commitment using particle swarm optimization combined with Lagrange relaxation. In: 2005 IEEE Power Engineering Society General Meeting. San Francisco, USA, 2005, 2752–2759

[17] HargreavesJ J, HobbsB F. Commitment and dispatch with uncertain wind generation by dynamic programming. IEEE Transactions on sustainable energy, 2012, 3(4): 724–734

[18] SaravananB, DasS, SikriS, KothariD P. A solution to the unit commitment problem–a review. Frontiers in Energy, 2013, 7(2): 223–236

[19] SaravananB, SikriS, SwarupK S, KothariD P. Unit commitment using dynamic programming–an exhaustive working of both classical and stochastic approach. Frontiers in Energy, 2013, 7(3): 333–341

[20] SaravananB, VasudevanE R, KothariD P. A solution to unit commitment problem using invasive weed optimization algorithm. Frontiers in Energy, 2013, 7(4): 487–494

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