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Unit Commitment in Composite
Generation and Transmission Systems Using Genetic Algorithm
K.Chandrasekaran Sishaj P Simon
Department of EEE, National Institute of Technology, Department of EEE, National Institute of Technology, Tiruchirappalli-620 015, TamilNadu, India. Tiruchirappalli-620 015, TamilNadu, India. chansekaran23@gmail.com sishajpsimon@nitt.edu
Abstract: - This paper proposes a new method for the incorporation of the generation unit and transmission line unavailability in the solution of the unit commitment problem. The above parameters are taken into account in order to assess the required spinning reserve capacity at each hour of the dispatch period, so as to maintain an acceptable reliability level. The unit commitment problem is solved by a Genetic Algorithm resulting in near-optimal unit commitment solutions. The evaluation of the required spinning reserve capacity is performed by implementing reliability constraints, based on the expected unserved energy and loss of load probability indexes. In this way, the required spinning reserve capacity is effectively scheduled according to the desired reliability level. The results are compared with LR to prove the efficiency of the proposed method.
Key words- Unit commitment (UC), composite Generation and transmission systems, Genetic Algorithm (GA), Loss of load probability (LOLP), Expected unserved energy (EUE) indexes, Spinning reserve assessment****.
I. NOMENCLATURE
FC (P(i,t))-Fuel price of unit i at t hour Ai, Bi, Ci- Generators co-efficient ST (i,t)- Status of unit i at hour t. LOADt-Demand at hour t. RSRVt -Required spinning reserve at hour t. Ton (Toff)-Minimum Up time (Minimun Down time). TOC- Total Operation Cost. SU (^) T- Total Unit Start Up Cost. SD (^) T- Total Unit Shut down Cost. PF (^) T- Penalty associated with violated reliability constraint μ- Dynamic penalty multiplier. UVL-Penalty term LOLPt-Loss of load probability at hour t. Lmax- is the maximum allowed limit of the LOLP reliability index EUEt- Expected unserved energy in hour t. EUEmax- is the maximum allowed limit of the EUE reliability index
II. INTRODUCTION Unit commitment (UC) plays a major role in the daily operation planning of power systems. System operators need to perform many UC studies, in order to economically assess the spinning reserve capacity
required to operate the system as securely as possible. The objective of the UC problem is the minimization of the total operating cost of the generating units during the scheduling horizon, subject to many system and unit constraints. The solution of the above problem is a very complicated procedure, since it implies the simultaneous solution of two sub problems: the mixed-integer nonlinear programming problem of determining the on/off state of the generating units for every hour of the dispatch period and the quadratic programming problem of dispatching the forecasted load among them. The evaluation of the system spinning reserve is usually based on deterministic criteria. According to the most common deterministic criteria, the reserve should be at least equal to the capacity of the largest unit, or to a specific percentage of the hourly system load. The basic disadvantage of the deterministic approach is that it does not reflect the stochastic nature of the system components. On the contrary, the probabilistic methods [1]–[4] can provide a more realistic evaluation of the reserve requirements by incorporating various system uncertainties, such as the availability of the generating units, the outages of the transmission system, and the load forecast uncertainty. These methods combine deterministic criteria with probabilistic indexes, in order to find a UC solution that provides an acceptable level of reliability.
Most of the existing probabilistic methods [2]---[4], are based on the priority list (PL) method in order to solve the UC problem. The PL method is very simple and fast, but it results in suboptimal solutions. Dynamic programming (DP) [1] and Lagrangian relaxation (LR) [5] have also been used for the solution of the UC problem. The main disadvantage of DP is that it suffers from the ‘‘curse of dimensionality,’’ i.e., the computational requirements grow rapidly with the system size. The Lagrangian relaxation method provides a fast solution but it may suffer from numerical convergence and solution quality problems. Aside from the above methods, there is another class of numerical techniques applied to UC problem. Specifically, there are artificial Neural Network, Simulated Annealing (SA), and Genetic Algorithms (GA’s). These methods can accommodate more complicated constraints and are claimed to have better solution quality. SA is a powerful, general-purpose stochastic optimization technique, which can theoretically converge asymptotically to a global optimum solution with probability 1 [6]. One main drawback, however, of
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SA is that it takes much CPU time to find the near-global minimum. GA’s are a general-purpose stochastic and parallel search method based on the mechanics of natural selection and natural genetics. GA’s are a search method to have potential of obtaining near-global minimum [6].
In this paper, a new probabilistic method is proposed for the incorporation of the unavailability of the generating units and transmission line [7] in the solution of the UC problem. The above parameters are taken into account in order to provide a more realistic evaluation of the system reserve requirements. The GA algorithm previously presented in [8] is used for solving the UC problem, while the evaluation of the required spinning reserve capacity is performed by using the loss of load probability (LOLP) and expected unserved energy (EUE) indexes. The inclusion of these indexes is accomplished by implementing additional constraints, which guarantee the desired level of reliability. The GA algorithm allows the direct incorporation of the above constraints in the formulation of the UC problem, using the proposed reliability constrained method, without affecting the main optimization procedure. The efficiency of the method has been tested on a system of 10 generating units for a scheduling horizon of 24 h. The proposed method is considerably fast and results in near-optimal UC solutions.
III. PROBLEM FORMULATION The objective of the UC problem is the minimization of the total operating cost of the generating units during the scheduling horizon. The total operating cost is composed of two parts: the fuel cost and the startup cost. Fuel Cost:
FC Pit = Ai Pit + BiPi , t + C i
2 ( ,) , (1)
The UC objective function is given by the minimization of the following cost function: TC=
= =
T
t
NG
i
STi tFCit Pit STit STit STRTit PF 1 1
[ , ,( ,) ,( (^1) , 1 ) , ]
Subject to
(1).Power balance constraints:
=
= ∈
NG
i
Pi t LOADt t T 1
, ,^ [^1 , ] (3)
(2).Spinning reserve requirements:
¦ STi ,^^ tP max i ≥^ LOADt + RSRVt , t ∈[^1 , T ] (4)
(3).Minimum up/down time limits:
( Ton (^) i , t − 1 − Tupi ).( STi. t − 1 − STi , t )≥ 0 i^ ∈ NG ;^ t ∈[^1 , T ] (5) ( Toff (^) , t − 1 − Tdowni ).( STi. t − STi , t − 1 )≥ 0 i^ ∈ NG ;^ t ∈[^1 , T ] (6) (4).Reliability Constraints:
The LOLP reliability constraint is given by LOLPt ≤ L max, t ∈[ 1 , T ] (7) The EUE reliability constraint is given by EUEt ≤ E max,t [1,T] (8)
IV. GENETIC ALGORITHM
Genetic algorithms (GAs) are stochastic global search and optimization methods that mimic the metaphor of natural biological evolution. GAs operates on a population of potential solutions applying the principle of survival of the fittest to produce successively better approximations to a solution. At each generation of a GA, a new set of approximations is created by the process of selecting individuals according to their level of fitness in the problem domain and reproducing them using operators borrowed from natural genetics. This process leads to the evolution of populations of individuals that are better suited to their environment than the individuals from which they were created, just as in natural adaptation.
A. Solution Encoding A solution of the UCP is determined by both the on off schedule, ST(i,t) (0-1 values), of the units at each hour and the power, P (i,t) (real values), generated by each individual unit every hour. In this approach, the values of ST(i,t) are generated by the GA, while the optimum values of P(i,t) are computed by solving a nonlinear program with 2N constraints.
Fig.1 Encoding an On and Off schedule
1 0 1 0 111…………
0 0 1 0 101…………
1 1 1 0 100…………
N 1 1 1 0 000…………
3
2
1
Units
Hours 1 2 3 4 ……………
outaged, the total capacity that remains in service, and the probability that corresponds to this state [4]. Assuming that the load of the system is constant within each hour, the LOLP for each hour can be calculated by
n LOLPt PRj LOSSj , t [ 1 , T ]
Where LOSS (^) j is given by
¯
®
<
otherwise
ifCR LOAD LOSS
j j 0 ,
1 , (14)
Similarly, the EUE for each hour can be calculated by
( ), [ 1 , ] 1
EUEt PR LOSSj LOADt CRj t T
n
j
= ¦ j − ∈
= (15) The computational time required for the formation of each COPT can be considerably reduced by rounding the outage levels to a fixed increment, e.g., 5 MW [9]. The increment must be carefully chosen in order to retain the precision of the final results. A further reduction in the time requirements of the proposed method can be achieved by omitting the outage levels for which the cumulative probabilities are less than a predefined limit, e.g., 10 -7.
B. Implementation of the LOLP Reliability Constraint
The LOLP reliability constraint is implemented in order to incorporate this index in the formulation of the UC problem. It is noted that LOLP is calculated separately for each hour of the dispatch period. Based on the above analysis, the proposed method can be described as follows.
Step 1) A new candidate solution is generated by the Genetic algorithm. Step 2) Set time counter .t=H Step 3) Calculate the LOLP (^) t:
- If LOLPtLmax, go to Step 6.
- If LOLPt >Lmax, the solution is rejected. Step 4) Update RSRV. Due to the fact that the current value of RSRVt leads to violation of the LOLP reliability Constraint, set RSRV=RSRV+1. Step 5) Generate a new trial solution, and go to Step 8. Step 6) If t<H2, increase time counter t=t+1, and go to Step 3; else, go to Step 7. Step 7) The solution is accepted. Step 8) Return to the GA algorithm.
C. Implementation of the EUE Reliability Constraint
The EUE reliability constraint is implemented in order to incorporate this index in the formulation of the UC problem. In contrast with the LOLP index, the EUE is calculated over the entire dispatch period. The proposed method is based on the implementation of a dynamic penalty function. For each feasible solution
provided by the Genetic algorithm, the corresponding EUE of the dispatch period is calculated by the procedure described in the relevant section. Then, the calculated is compared to a predefined maximum allowed limit, the determination of which is based on the desired level of reliability.
- If EUEtEmax, the trial solution provides the desired level of reliability.
- If EUEt>Emax, the quadratic penalty term given by (11) is added in the operating cost of the current solution, resulting in the augmented objective function given by (2).
Fig.3 Flowchart of the proposed reliability constrained method.
LOLPt>Lmax
Generation of new trial solution by GA
Set t-H
Calculate LOLP
EUEt>EUEma
Accept Trial solution
Calculate EUEt
UVL=EUEt-EUEmax
Update value of μ
UVL=
Calculate TC
- Reject Trial solution
- RSRV=RSRV+
- Generation of new trial solution by GA
Return to GA solution
Y
N
N
Y
The value of the penalty multiplier used in (2) is low at the early stages of the algorithm and gradually grows, until it reaches an upper bound. Each time a trial solution of the Genetic algorithm violates the EUE reliability constraint, the penalty multiplier increases by the following rule:
min( , )
0 μ (^) k = μmax inck μ (16)
Where μ k is the value of μ after k violations of
the EUE reliability constraint, μ 0 is the initial value
of μ , and inc is a constant greater than but very close to
- The initial value of the penalty multiplier is chosen small enough (close to 0) to encourage the acceptance of solutions that violate the EUE reliability constraint during the first stages of the algorithm. In this way, a more extensive exploration of the solution space is achieved. As the algorithm proceeds,^ μ^ the constantly grows, until
it reaches a very high final value. Afterward, the probability of acceptance of any solution that violates the above constraint is practically insignificant due to the great increase in the total cost of the corresponding solution. It is noted that the use of μ max is necessary in order to avoid an incalculable
increase in the penalty multiplier, which could lead to numerical problems. The implementation of the dynamic penalty function allows the solutions that violate the EUE constraint to evolve into feasible ones. Thus, the final solution of the problem provides the desired level of reliability, based on the EUE index. The flowchart of the proposed method for incorporating both LOLP and EUE reliability constraints is shown in Fig. 3
VI. N UMERICAL RESULTS
The effectiveness of the method has been tested on a system of 10 generating units, considering a dispatch period of 24 h. The system generation data and transmission line data is given in Table I and II respectively.
TABLE I. GENERATION DATA Unit.no Ai Bi Ci Pmin Pmax 1 1000 16.19 0.00048 150 455 2 970 17.26 0.00031 150 455 3 700 16.6 0.002 20 130 4 680 16.5 0.00211 20 130 5 450 19.7 0.00398 25 162 6 370 22.26 0.00712 20 80 7 480 27.74 0.00079 25 85 8 660 25.92 0.00413 10 55 9 665 27.27 0.00222 10 55 10 670 27.79 0.00173 10 55
TABLE. II LINE DATA From bus
To bus R(pu) X(pu) B(pu)
The data referring to the failure rates of the units are taken from [10]. The expected load demand at each hour of the considered period is shown in Table III.
TABLE.III EXPECTED HOURLY LOAD DEMAND (MW) Hours Load Hours Load Hours Load Hours Load 1 700 7 1150 13 1150 19 1100 2 750 8 1100 14 1100 20 1150 3 850 9 1150 15 1050 21 1100 4 950 10 1150 16 1050 22 1000 5 1000 11 1100 17 1000 23 900 6 1100 12 1130 18 1100 24 800
Two cases have been studied; Case 1 is the Unit commitment problem using LR method with reliability constraint. Case 2 is the Unit commitment Using GA with reliability constraints. The desired level of reliability depends on the predefined values of the maximum allowed limits Lmax and Emax. The simulation results for various values of these limits are shown in Table IV. It is noted that Lmax is given in percent (%) form, while Emax is expressed as a percentage of the expected energy demand of the dispatch period. It can be seen that the hourly LOLP decreases as the reliability level increases. It is noted that for all considered hours, the LOLP is less than the corresponding maximum allowed limit, confirming that the final solution of the UC problem provides the desired level of reliability. TABLE. IV RESULTS FOR VARIOUS RELIABILITY LEVEL CASE Lmax Emax Operating cost $ 1 (LR)
(GA)
The results proves the GA gives better solution when compare to LR method. The results show that the total operating cost of the system increases as the maximum
