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Benders decomposition applied to security constrained unit commitment: Initialization of the algorithm
Electrical Power and Energy Systems 66 (2015) 53–66
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Benders decomposition applied to security constrained unit commitment: Initialization of the algorithm J. Alemany a,⇑, F. Magnago a,b a b
Department of Electrical and Electronic Engineering, Universidad Nacional de Río Cuarto, Argentina Nexant Inc., AZ, USA
a r t i c l e
i n f o
Article history: Received 5 December 2013 Received in revised form 3 August 2014 Accepted 12 October 2014
Keywords: Benders decomposition Initialization methodology Mixed integer linear programming Security constrained unit commitment
a b s t r a c t Benders decomposition has been broadly used for security constrained unit commitment problems, despite the fact that it may present convergence difficulties due to instabilities and to the mixed integer nature of the unit commitment problem. The initialization of Benders decomposition has been recognized as a prominent feature for the algorithm enhancement. In this work, a new Benders decomposition initialization methodology is proposed. The objective of the initialization is to include inexpensive network signals that can be added during the initial unit commitment master problem. Numerical simulations using the IEEE-118 and RTS-96 systems are performed to illustrate the benefits of the proposed initialization methodology. Results suggest that the initialization of Benders decomposition applied to security constrained unit commitment problems improves the overall convergence of the algorithm. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction The unit commitment (UC) calculation is extensively used in daily power system operation. The UC is an exercise of large-scale, time-varying, non-convex, mixed-integer modeling and optimization. Security constrained unit commitment (SCUC) is an extension of conventional UC with the inclusion of system network constraints, in both normal and contingency operation states. The main objectives of the SCUC are to ensure not only the economic operation but also the security of the system [1,2]. These two different objectives can be separately solved as a two-level optimization problem. It is a common practice to define a master level, which includes the UC calculation, and the sub-problem level, which checks the network security constraints. Benders Decomposition (BD) is an algorithm that has been broadly used for large-scale optimization problems, particularly for power systems [3,4]. BD has three main advantages: modularity, flexibility and robustness. With regard to modularity, master and sub-problems can be separately solved by specialized algorithms, thus providing speed and efficiency on the overall performance of the global optimization process. Additionally, Benders flexibility is mainly supported by the different existing power system applications. For example, it is possible to find its application in areas like security constrained economic dispatch ⇑ Corresponding author. E-mail address: [email protected] (J. Alemany). http://dx.doi.org/10.1016/j.ijepes.2014.10.044 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
[5], generation–transmission planning [6,7], hydrothermal coordination [8,9], and optimal power flow [10,11]. Finally, in relation to robustness, despite the different nature of the master and the sub-problems in SCUC applications, both are essentially solved using Linear Programming (LP) algorithms. This is an important feature because LP algorithms are considered to be among the most mature methodologies in optimization techniques [12]. Nevertheless, since BD is a cutting-plane method [13], it may present instabilities which are translated into delays of the algorithm convergence [14,8]. In addition, since the master level is formulated as a mixed integer linear problem (MILP), the convergence time is strongly affected by the high computational burden of the master problem [15]. Therefore, there are many research efforts regarding BD improvements [16]. Among the different suggested possibilities, having a better initialization of BD is recognized by several authors as one of the most important enhancements, concluding that it could have a significant effect on BD performance. Several initialization methodologies have been developed. In [17] an initial set of cuts is constructed from routes of an aircraft routing model. The author concluded that the initial cuts are advisable to be used with any network, since there is a reduction in the total computation time. Furthermore, in [18] a strategy to initialize BD with the addition of a series of valid inequalities is developed. The initialization procedure is applied to a fixed charge network problem, and different variants of refinery systems are studied.
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Nomenclature T G B N L b t g l
scheduling horizon number of generation units number of piece-wise blocks number of buses number of branches power block index hour index unit index branch index contingency scenario loop iteration out of service lapse initial hour production cost startup cost piece-wise power block active power relaxation variable load supplying capability MW bus injection state binary variable power block binary variable start-up binary variable shut-down binary variable generation vector
m k
s Cpgt Cagt dbgt pgt slack
q pbt ugt jbgt sgt hgt p
f puc h Kg s cg Fbg Trbg Dt MUTg and MDTg off Ton g and Tg Pg and Pg RULg and RDLg k peak alb k cs cT D D f and f P and P
c S
The aim of this paper is to propose a new BD initialization methodology applied to SCUC problems. The initialization methodology is based on the combination of three main concepts: a fast UC resolution, a redundant constraint elimination, and the obtainment of upgraded cuts within a range of load bounds. The remaining sections are outlined as follows. Section ‘Benders decomposition based SCUC’ gives a general background on BD. Section ‘Outline of the initialization methodology’ provides a detailed description of the proposed initialization strategy. This section includes a detailed description of the model and concepts in which the Benders initialization is based. In Section ‘Computational results’, numerical results are presented and discussed. Finally, Section ‘Conclusion’ presents the most relevant conclusions.
branch flow vector UC generation level vector bus voltage angle vector start-up cost for step s fix cost power block slope min–max power block limits hourly system demand min on/off service times initial on/off hours of service max–min power limits up–down fixed ramp limits iterative demand linear sensitivity factors load participation factors vector infeasibility cost vector dispatch costs vector demand vector allowed MW deviation vector max–min flow limit vectors generation bound vectors branch susceptances matrix branch-bus incidence matrix (0 is transpose)
8bgt
ðTrbg Trb1;g Þjbgt 6 dbgt
ð6Þ
8bgt
dbgt 6 ðTrbg Trb1;g Þjb1;gt
ð7Þ
dBgt P 0 8gt
ð8Þ
8gt
dBgt 6 ðPg TrB1;g ÞjB1;gt Cagt P Kgs ugt
s X
ð9Þ
!
8gt
ug;tn
ð10Þ
n¼1
Cagt P 0 8gt s:t: :
G X pgt ¼ Dt
ð11Þ
8t
ð12Þ
g¼1 t X
8g 8t 2 MUTg þ 1; T
sgi 6 ugt
ð13Þ
i¼tMUTg þ1
Benders decomposition based SCUC In this section BD methodology applied to SCUC problem is formulated [1]. Additional information about BD can be found in the Appendix. Master problem – the decision stage
t X
hgt 6 1 ugt
8g 8t 2 MDTg þ 1; T
ð14Þ
i¼tMDTg þ1 i6Ton g
X
1 ugi ¼ 0 8g 8t ¼ 0
ð15Þ
ugi ¼ 0 8g 8t ¼ 0
ð16Þ
i¼0 off
The master problem is the single-bus UC without security constraints. The thermal UC model can be formulated as follows:
min z ¼
T X G X
Cpgt þ Cagt
ð1Þ
t¼1 g¼1
Cpgt ¼ ugt cg þ
B X Fbg dbgt
8gt
ð2Þ
b¼1
pgt ¼ ugt Pg þ
B X dbgt
i6Tg
X i¼0
ugt Pg 6 pgt 6 ugt Pg
8gt
ð17Þ
pgt pg;t1 6 RULg
8g 8t P 0
ð18Þ
pg;t1 pgt 6 RDLg
8g 8t P 0
ð19Þ
8gt
ð20Þ
sgt hgt ¼ ugt ug;t1
8gt
ð3Þ
b¼1
pgt P 0 8gt ðTr1g Pg Þj1gt 6 d1gt 6 ðTr1g Pg Þugt
ð4Þ
8gt
ð5Þ
sgt þ hgt 6 1 8gt
ð21Þ
jbgt ; ugt ; sgt ; hgt 2 f0; 1g 8bgt
ð22Þ
Benders cutstm
8t m
ð23Þ
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J. Alemany, F. Magnago / Electrical Power and Energy Systems 66 (2015) 53–66
min w ¼
N X cs slack
ð24Þ
s:t: : Sf slack 6 puc D
pd1
Sf slack 6 puc þ D
ð25Þ
pd2
ð26Þ
0
f cS h ¼ 0
ð27Þ
f 6 f pf 1 f 6 f
ð28Þ ð29Þ
pf 2
0 6 slack 6 D
ps
ð30Þ
The associated costs cs are usually set to one. After solving the problem represented by Eqs. (24)–(30), relaxing the balance constraints (slack), one Benders cut is created for each hour of a network scenario as follows:
w þ
pd1 pd2 puc puc 6 0
ð31Þ
In the next section an initialization methodology applied to BD based SCUC is presented.
Outline of the initialization methodology The main motivation to develop an initialization methodology applied to BD based SCUC is to improve the convergence rate of the Benders decomposition. The core of the initialization methodology is to provide a set of inexpensive cuts to be added in the initial master problem, in order to provide early signals of the network to the UC calculation. The proposed initialization methodology comprises six different steps:
Fig. 1. MIL algorithm.
Fig. 2. Relaxed horizon scheme.
Eq. (1) is the objective function that accounts for the total operational day-ahead costs. Eq. (2) represents the production costs. Eqs. (3)–(9) are the piece-wise linearization of production costs. Eqs. (10) and (11) represents the step-wise start-up costs. Eq. (12) is the total energy balance. Eqs. (13)–(16) represent min up–down times. Eq. (17) represent the generator operational limits. Eqs. (18) and (19) are the ramp limits. Eqs. (20)–(22) represent the logic among binary variables. The last term stands for the eventual Benders cuts. Sub-problem – the feasibility stage The coupling between the master and the sub-problems is done through the variable puc , which is the solution of problem represented by Eqs. (1)–(22). The sub-problem can be formulated as:
1. Calculation of the Load Supplying Capability (LSC) [19]. The LSC is an upper bound of network infeasibility that allows forming the cuts. 2. Calculation of the Minimal Infeasible Load (MIL). The MIL is a lower bound of network infeasibility that allows forming the cuts. 3. Calculation of a Relaxed Horizon single-bus UC (RHUC) [20] for the LSC demand peak. The calculation is carried out in order to get the MWs dispatch levels. 4. Pre-processing of redundant network constraints [21]. Redundant security constraints are eliminated with a network preprocessing method. 5. Calculation of the Linear Load Flow with Rescheduling (LLFR) [22] for the given MWs UC levels and the LSC demand peak. The calculation is carried out to check network infeasibility in order to form a cut. 6. Upgrading of Benders cuts [23]. The cut in Step 5 is upgraded for every demand level within the range delimited by the LSC and MIL bounds. In the following sections, detailed information about each step is provided.
Table 1 Two generation units system. Item
Value
Variable cost Start up cost Fix demand Max power Min power B var R var
1, 2 3, 2 30, 31, 40 50, 20 10, 5 ugt ; sgt ; hgt pgt
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Table 2 Variable pattern.
Fig. 4. Inactive constraints set.
Calculation of the load supplying capability
Table 3 Relaxed horizon UC results. Iter problem 0 RMILP
1 MILP 2 MILP
3 MILP 4 MILP
The LSC of a given system is the maximum load that can be supplied with all branches and generators operating, and considering the participation factors of each load bus with respect to the total load. The concept of the LSC is to stress the network to a maximum, that is, to calculate the maximum load that can be supplied without system overloads. The LSC is the upper bound of the range within which it is possible to form a set of initial cuts. Mathematically it can be formulated as the solution of the following LP problem:
Model item
Stats.
z value
Rows Cols. Non 0 Z cols. Relaxed cols. Fixed cols.
16 25 53 0 18 0
103.4
Z cols. Relaxed cols.
6 12
111
Z cols. Relaxed cols. Fixed cols.
6 6 6
109
Z cols. Fixed cols.
6 12
109
where q is a scalar that represents the total system demand.
Z cols.
18
104
Calculation of the minimal infeasible load
max LSC ¼ q cS0 h þ p ¼ k q jShj 6 f =c
ð32Þ
06p6P
ð35Þ
ð33Þ ð34Þ
The MIL represents the minimum load that can be economically supplied without network overload. The MIL is the lower bound of the range within which a set of initial cuts can be formed. Mathematically it can be formulated as the solution of the following LP problems:
min x ¼ cT p
ð36Þ
G X k p peak P 0
ð37Þ
06p6P
ð38Þ
min y ¼ slack
ð39Þ
G X cS0 h þ p þ slack ¼ k peakk
ð40Þ
jShj 6 f=c slack P 0 Fig. 3. 3 Buses system.
ð41Þ ð42Þ
The above LP problems need to be iteratively solved. The algorithm is described next:
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Input: LSC k
Initialization: ub = LSC, lb = 0, peak = ub, k = 0 loop k Solve (36)–(38) then (39)–(41) if y > 0 then k
ub = peak
peak = ub- ub—lb 2 else if y 6 0 then k
k
lb = peak
k peak = ub- ub—lb 2 end if k¼kþ1 if ub–lb 6 then Stop MIL=ub end if end loop Output: MIL
Fig. 5. LSC-MIL range.
Table 4 First UC master results.
Calculation of a relaxed horizon unit commitment
Fixed variable sub-period (Dark Grey). Active variable sub-period (White). Relaxed variable sub-period (Light Grey). The binary variables within the fixed variable sub-period are fixed using the previous iteration values. The binary variables within the active variable sub-period are treated as integer variables. Finally, the binary variables within the relaxed variable sub-period are treated as continuous variables. Thus, as the iterations progress, the fixed variable sub-period increases while the relaxed variable sub-period decreases. At the final iteration the whole scheduling horizon is treated as a complete MILP. In this final stage, after two rounds without significant improvement of the objective value the algorithm stops. A simple example is presented to illustrate this strategy. Let’s consider a small system with two identical generators. Table 1 shows the data used for the simulations. Without loosing generality, from the full set of operational constraints, only balance, binary logic and power limit constraints are included into the UC formulation. The relaxed horizon unit commitment is formulated as follows:
8t
ugt ug;t1 6 sgt hgt
ð43Þ
ð45Þ
8gt
30 31 40
p2 [MW] 0 0 0
ð46Þ
104
Table 2 shows the relaxed horizon pattern used for the simulations. Table 3 shows the results after the simulations. This relaxation strategy does not affect the solution space because at the last iterations a full MILP problem is solved. The objective of this relaxation strategy is to provide a fast feasible solution that can be used to check network infeasibility. Nevertheless, some parameters of the algorithm need to be tuned before good performance is obtained. For general purpose application, default parameter values are recommended only for large scale UC problems. Pre-processing of redundant network constraints Power systems are not expected to operate out of limits [21], therefore, in general it is necessary to deal with only a small subset of security constraints at any stage. Based on this observation, several redundant network constraints can be eliminated prior to the LLF calculations. The aim of the strategy is to reduce the size of the LLF problem in the base case, provided the solution space is not affected because the eliminated constraints are redundant to the feasible region. This is demonstrated using a simple example next. Let’s consider again the 2 units system of Table 1 into the 3-bus network described in Fig. 3 serving 30 MW load. Fig. 4 represents the feasible region corresponding to the system of Fig. 3. The gray zone is the feasible region which is limited by restrictions P2 6 14.9, P2 P 5, the balance equation P1 + P2 = 30, and the lower limit P1 P 10. It can be seen that only 1 out of 6 security Table 5 LLFR results for LSC demand. Hour
p
1 2 3
30 31 40
p1 [MW]
ð44Þ
0 6 pgt 6 ugt P 8gt
1 2 3 Objective value
The strategy is based on sequentially fixing, relaxing and activating different subsets of the binary variables. At each iteration only a portion of the simulation horizon is considered as a MILP problem, while the remaining part is either fixed using values from a previous iteration, or relaxed in its integrality at the current iteration. Fig. 2 illustrates a general scheme for this methodology. For each iteration, the scheduling horizon is further divided into three sub-periods:
G X pgt ¼ Dt
puc p1 [MW]
The algorithm is basically based on the bisection search method [13]. Fig. 1 outlines the proposed MIL approach.
T X G X min z ¼ ðCpgt pgt þ Cagt sgt Þ
Hour
w [MW] p2 [MW] 0 2 20
0 2 20
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J. Alemany, F. Magnago / Electrical Power and Energy Systems 66 (2015) 53–66
Table 6 LLFR sensitivities. Sens
Hour 1
2
3
0 1
0 1
0 1
pr
p1 p2
pg
p1
0
0
0
p2
1
1
1
b2 b3
0 0
1 2
1 2
pd
Table 7 Final UC master results. Hour
puc p1 [MW]
1 2 3
30 26 20
Objective value
p2 [MW]
Fig. 8. Demand 3 network status.
0 5 20 131
Fig. 9. Demand 2 network with cut.
Fig. 6. Demand 1 network status.
Fig. 10. Demand 3 network with cut.
Fig. 7. Demand 2 network status.
constraints are non-redundant. The redundant security constraints have no influence on the feasible region of the problem and can thus be eliminated from the SCUC problem. Relaxation techniques are very important when only few of a large set of inequality constraints become binding [24]. The redundant constraints can be obtained considering the following LP problem:
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zlt ¼ max
N X
alb pbt
8lt
ð47Þ
b¼1 N X
N X alb pbt slack 6 f l þ alb Dbt
b¼1 N X
8l
ð48Þ
b¼1
pbt ¼ Dt
8t
ð49Þ
Table 9 118-bus Line set [%]. Hour
Non-redundant lines
Total lines [%]
1, 3, 7, 9,
16 15 14 17
8.6 8.1 7.5 9.1
2, 8, 15, 17, 18, 19, 21, 24 4, 5, 6, 22 23 10, 11, 12, 13, 14, 16, 20
b¼1
0 6 pbt 6 Pb
8t 8g 2 b
ð50Þ
slack P 0
ð51Þ
where zlt measures the power flow of each branch. Eq. (47) denotes the maximum possible power flow over a transmission branch provided that power balance, Eq. (49), and unit restrictions, Eq. (50), are satisfied. Eq. (48), monitors which branches are redundant or not. Whenever a branch overload occurs, slack assumes some positive value and the security constraint is labeled as non-redundant. Generators are dispatched to supply the system load; the more sensitive the branch is with respect to the unit, the more weight it will have on the objective function. This process, therefore, allows the monitoring of branches in order to prevent overloading during the given period. For this step, a series of LP problems (l t) need to be solved in order to detect which network constraints are redundant in the feasible region; nevertheless, this is a pre-processing step.
Table 10 RTS-96 network after pre-processing. Hour
Line 2 13 77 78
3 107
6 20 43 109 69
14 24 26 30 44 60 71 85
32 103
36 37
75 76 82
1 11 12 29 31 41 42 47 52 65 68 70 80 90 106 108 118 119 120
0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1
11111001 11111001 11111001 11111101 11111001 11111001 01111011 01101011 01101011 01101011 01111011 01101011 00000010 00001010 01001010 01101010 01101011 01111011 01111001
1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 1 1
0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Calculation of linear load flow with rescheduling Given the single-bus UC pattern, puc , calculated in Section ‘Calculation of a relaxed horizon unit commitment’, a LLFR is obtained. The objective of the LLFR, provided that network infeasibility exists, is to obtain the cut corresponding to the LSC demand level. In order to form this cut, it is necessary to minimize the system infeasibility, w, taking into account the coupling between the UC pattern, puc , and the network constrained dispatch p. Mathematically, the LLFR is formulated as follows:
Table 8 118-bus network after pre-processing. Hour
1, 2 3, 4 5, 6 7 8 9 10, 11 12, 13 14 15 16 17, 18 19 20 21 22 23 24
Line 14 39
44 50 112 115 119 129 136 162 163 165 168
63 106
120 123 127 133 166
10 10 10 10 1 01 01 01 01 01 01 01 01 01 01 1 10 10
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
00011 00001 00001 00011 00111 1 1 1 1 01111 1 11110 11110 1 01111 00011 00011 00011
1 2 3 4, 5, 6, 7 8 9 10 11 12 13 14, 15, 16 17 18 19 20 21 22 23 24
Table 11 RTS-96 line set [%]. Hour
Non-redundant lines
Total lines [%]
1 2 3 4, 5, 6, 7 8 9 10, 23, 14, 15, 16 11, 13 12, 17, 22 18 19 20 21 24
34 37 43 44 39 32 27 26 24 20 21 22 23 31
28.3 30.8 35.8 36.7 32.5 26.7 22.5 21.7 20.0 16.7 17.5 18.3 19.2 25.8
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Table 12 Initial UC comparison. Load pattern
LSC [%]
Winter
60 80
Summer
60 80
min w ¼
N X
IEEE-118 time [s]
RTS-96 time [s]
UC RHUC UC RHUC
1.20 1.60 2.34 1.87
146 29 27 14
UC RHUC UC RHUC
1.18 1.72 1.90 1.75
235 56 64 34
slack
ð52Þ
b¼1
cS0 h þ p ¼ D pd jShj 6 f =c pf 0 6 p 6 P pg
p p þ slack 6 D uc
ð53Þ
Fig. 11. Summer: IEEE-118 LSC-MIL range.
ð54Þ ð55Þ
pr
ð56Þ
Here the variable slack is a vector of slack variables which correspond to the gap between UC and LLF generation levels. The resolution of the LLFR provides the sensitivities pd ; pg ; pf , and pr . (Simplex multipliers associated to the Eqs. (52)–(56).) These sensitivities are necessary not only to form the LSC cut but also to upgrade the subsequent cuts within the LSC-MIL range. This step is explained in the next section. Upgrading of Benders cuts The feasibility cut resulting from Eqs. (52)–(56), is described by:
w þ pr puc puc 6 0
ð57Þ
Fig. 12. Winter: IEEE-118 LSC-MIL range.
This cut can be upgraded to accommodate deviations between the forecasted and actual demand, as mentioned in [4]. As a consequence, the cut could also be valid for other time steps. Before the cut can be upgraded, the feasibility cut should also accommodate changes in system configuration due to changes in the UC. Then, the upgraded cuts within the LSC-MIL range are:
w þ pr puc puc þ pg p puc þ pd ðDt Dlsc Þ 6 0
ð58Þ
where w is the optimal solution value of Eqs. (52)–(56) for the LSC demand level, Dlsc is indeed the LSC demand level, and Dt correspond to the demand levels within the LSC-MIL range. The third term of Eq. (58) represents UC changes and the fourth term represents demand changes. Fig. 5 illustrates a generic demand curve and the LSC-MIL range. The upgraded cuts represented by Eq. (58) and the LSC cut given by Eq. (57), are added to the initial master UC in order to initialize the BD algorithm. Again, a simple example is presented to illustrate this strategy. Let’s consider the small system of Table 1 and Fig. 3. Solving problems of Sections ‘Calculation of the load supplying capability and Calculation of the minimal infeasible load’ the LSC and MIL of this system are 40 and 30 MW respectively. Table 4 shows the results of the first UC master. Pre-processing of network is not applied.
Fig. 13. Summer: RTS-96 LSC-MIL range.
Table 5 shows the results after the problem resolution of Section ‘Calculation of linear load flow with rescheduling’ for the LSC demand level and Table 6 shows the sensitivities of that problem. Therefore, the LSC cut and the upgraded cuts are as follows:
Upgraded cut for t ¼ 1 puc 2 P 0
Table 13 Calculation of LSC-MIL bounds. Bound
IEEE-118 time [s]
RTS-96 time [s]
LSC MIL
0.562 3.790
0.436 1.284
ð59Þ
Upgraded cut for t ¼ 2 puc 2 P 2
ð60Þ
LSC cut for t ¼ 3 puc 2 P 20
ð61Þ
Adding the cuts (59)–(61) to the master and resolving the UC, Table 7 is obtained.
J. Alemany, F. Magnago / Electrical Power and Energy Systems 66 (2015) 53–66
Fig. 14. Winter: RTS-96 LSC-MIL range.
Table 14 Calculation of LLFR for LSC demand level. System
Load pattern
LSC [%]
Peak [h]
Infeasibility [MW]
Time [s]
IEEE118
Winter
60
18
112
0.49
Summer
80 60 80
12
365 112 363
0.54 0.51 0.50
117 197 140 275
0.43 0.44 0.42 0.44
RTS-96
Winter Summer
60 80 60 80
12 10
Fig. 15. IEEE118 – summer and winter: 80% LSC.
Table 15 IEEE-118: 80% LSC – summer and winter load patterns. Iter
Time [s]
Ini master cost
Summer Classic BD Initialized BD Improvement (%)
8 6 33
73.95 48.39 54
1,172,407 1,175,594
Winter Classic BD Initialized BD Improvement (%)
8 6 33
70.73 51.20 37
1,172,289 1,184,209
Table 16 IEEE-118: 60% LSC – summer and winter load patterns. Iter
Time [s]
Ini master cost
Summer Classic BD Initialized BD Improvement (%)
6 5 20
48.02 37.85 30
837,328 837,909
Winter Classic BD Initialized BD Improvement (%)
6 5 20
49.87 39.34 28
838,272 840,665
Finally, Figs. 6–8 show the network for the first master solution and Figs. 9 and 10 show the network after the cut signals in the master solution. The next section will describe numerically the proposed methodology.
Fig. 16. IEEE118 – summer and winter: 60% LSC.
61
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J. Alemany, F. Magnago / Electrical Power and Energy Systems 66 (2015) 53–66
Table 17 RTS-96: 80% LSC – summer and winter load patterns. Iter
Time [s]
Ini master cost
Summer Classic BD Initialized BD Improvement (%)
15 13 15
294 239 23
5,546,510 5,563,205
Winter Classic BD Initialized BD Improvement (%)
14 12 17
271 226 20
5,368,409 5,364,254
Table 18 RTS-96: 60% LSC – summer and winter load patterns. Iter
Time [s]
Ini master cost
Summer Classic BD Initialized BD Improvement (%)
9 8 13
211 173 22
3,228,775 3,236,432
Winter Classic BD Initialized BD Improvement (%)
9 7 29
215 172 25
3,175,159 3,179,984
Fig. 18. RTS96 – summer and winter: 60% LSC.
in [25,26]. Two hourly load patterns are used, summer and winter, corresponding to the week-day patterns presented in [26]. Two different load peaks are tested, 80% and 60% of LSC, respectively. The IEEE 118-bus system has 54 units and 186 branches. The RTS-96 system has 96 units and 120 branches. The UC gaps for the initial master and for the subsequent masters are set to 5% and 1%, respectively. The network infeasibility tolerance is set to 1 MW. The Benders gaps illustrated in Figs. 15– 18 are defined as:
Benders gap ¼ Upper bound Lower bound Upper bound ¼ Master UC solution þ 1T
T X Subproblem LLF solutions
Lower bound ¼ Master UC solution It is important to remark that both upper and lower bounds are costs, and that is why vector 1T is necessary. In addition, the solution refers to the optimal value of the objective function, the total cost for the UC and the hourly network infeasibility for the subproblems.
Pre-processing results
Fig. 17. RTS96 – summer and winter: 80% LSC.
Computational results Numerical examples are presented in this section to show the benefits of BD initialization for SCUC calculation. For the sake of simplicity, only base scenario is tested, no contingencies are considered. All the numerical tests are implemented in GAMS, the problem is solved with CPLEX 11.0 solver on a PC with an Intel i5-2310 CPU at 2.90 GHz and 4.00 GB RAM memory. The IEEE 118-bus and the RTS-96 systems are studied, the data are given
Table 8 summarizes the results after the pre-processing step (Section ‘Pre-processing of redundant network constraints’) on the IEEE 118-bus network. This table illustrates the pattern of non-redundant lines in relation to the total set of lines. The total number of network constraints are 4464 (186 lines 24 h), only 383 are non-zero elements which represents 8.6% of the total number of network constraints. Table 9 illustrates the percentage of non-redundant lines at each hour, in relation to the total set of 186 lines. The non-redundant subset spans from 14 to 17 lines, representing at the most 9.1% of the total set of lines. Table 10 summarizes the pre-processing on the RTS-96 network. The total number of network constraints are 2880 (120 lines 24 h), only 737 are non-zero elements which represents 25.6% of the total number of network constraints. Table 11 illustrates the percentage of non-redundant lines at each hour, in relation to the total set of 120 lines.
63
J. Alemany, F. Magnago / Electrical Power and Energy Systems 66 (2015) 53–66
RHUC results Table 12 compares the results between the relaxed horizon UC resolution (Section ‘Calculation of a relaxed horizon unit commitment’) and the standard UC resolution that conforms the initial master problem on both the IEEE-118 and RTS-96 systems. In this work, a 1/4 of the scheduling horizon to each sub-period is used. It is important to note that the relaxed UC resolution performance is not as effective as expected in small scale generation systems. The practical experience with this technique allows to conclude that their biggest impact it is performed in large scale or complex generation systems. The results reported for the RTS-96 confirm that the relaxed horizon UC strategy is more convenient for complex generation systems. LSC-MIL bounds results For the base case, the LSC of the IEEE 118-bus system is 5416 MW and the MIL is 1458 MW. The LSC of the RTS-96 system is 9825 MW and the MIL is 1228 MW. Table 13 presents the computing times for the calculation of the LSC-MIL bounds of the systems. In Figs. 11 and 12 the IEEE-118 LSC-MIL range in relation to summer and winter week-day load patterns is showed. In Figs. 13 and 14 the RTS-96 LSC-MIL range is illustrated. LLFR and Benders cuts upgrading results Table 14 presents computing times for the calculation of LLFR for the LSC demand levels for both IEEE-118 and RTS-96 systems. Benders cuts upgrading consists in the building of cuts by Ec. (58). This process do not posses a significant computing impact in relation to the other steps of BD initialization. Therefore, these computing times are not presented.
the number of iterations. Finally, the overall convergence patterns illustrated in Figs. 15–18 show the faster entrance into the pre-specified gap tolerance. Tables 19 and 20 illustrate how each initialization step impacts the global performance of BD based SCUC calculation, excluding the network pre-processing. The results presented here are consistent with previous work done in other study areas. [16,27] reported the improved general performance of BD with network problems. Additionally, [28] reported similar experience with facility location problems. Furthermore, [17] reported the reduction in the total computation time with aircraft networks. Finally, [18] reported the decrease in the number of iterations with fixed charge network problems. To summarize, it is important to remark that all the simulation instances did improve the convergence of BD algorithm. Although the results are preliminary, they seem to be very promissory and may support the development of more sophisticated initialization strategies applied to SCUC based BD.
Table 19 Initialization steps impact on global performance for IEEE-118. Step calculation
Load pattern
LSC [%]
Time [s]
Relative to initialized BD [%]
IEEE-118 LSC
Winter
60 80 60 80
0.56
1.43 1.10 1.48 1.16
60 80 60 80
3.79
9.25 7.12 9.73 7.55
60 80 60 80
1.60 1.87 1.72 1.75
4.07 3.65 4.54 3.62
60 80 60 80
0.49 0.54 0.51 0.50
1.25 1.05 1.35 1.03
Summer MIL
Winter Summer
Global results RHUC
Tables 15 and 16 summarize the results for summer and winter load patterns at 80% and 60% of LSC for the IEEE-118. Figs. 15 and 16 illustrate the convergence of Benders algorithm on the IEEE-118 for both summer and winter load patterns at 80% and 60% of LSC respectively. Tables 17 and 18 summarize the results for summer and winter load patterns at 80% and 60% of LSC for the RTS-96. Figs. 17 and 18 illustrate the convergence of Benders algorithm on the RTS-96 for both summer and winter load patterns at 80% and 60% of LSC. Computing times labeled as Initialized BD reported in Tables 15–18 include the time spent by the initialization procedure. Computing times, both Classic and Initialized BD, do not include the time spent by the pre-processing of redundant network constraints step, due to this is an external step. In addition, the network preprocessing and LSC-MIL bounds are applied to both, classic and initialized BD. Improvements in Tables 15–18 are calculated as the relation between Classic and Initialized BD, for both parameters, number of iterations and total time. Results of Benders based SCUC were validated against results calculated using PSSÒE. The comparisons are shown in Tables B.21 and B.22 in Appendix B. Also, Tables C.24 and C.25 of Appendix C correspond to iterative results of SCUC patterns. The results suggest that the initialization of Benders decomposition applied to SCUC problems has the possibility to improve the overall convergence of the algorithm. The percentage improvements reported in Tables 15–18 reveal the impact of the initialization upon the performance of BD algorithm. Additionally, the computing times reported in Tables 15–18 show the reduction in the total CPU solution times. Furthermore, the iterations reported in Tables 15–18 illustrate the decrease in
Winter Summer
LLFR
Winter Summer
Table 20 Initialization steps impact on global performance for RTS-96. Step calculation
Load pattern
LSC [%]
Time [s]
Relative to Initialized BD [%]
RTS-96 LSC
Winter
60 80 60 80
0.44
0.26 0.19 0.25 0.18
60 80 60 80
1.88
1.09 0.83 1.09 0.79
60 80 60 80
29 14 56 34
16.9 6.19 32.4 14.2
60 80 60 80
0.43 0.44 0.42 0.44
0.25 0.19 0.24 0.18
Summer MIL
Winter Summer
RHUC
Winter Summer
LLFR
Winter Summer
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J. Alemany, F. Magnago / Electrical Power and Energy Systems 66 (2015) 53–66 Table B.21 Branch loading checking: initial master.
Conclusion In this work, a new BD initialization methodology applied to SCUC problems has been proposed. The initialization was based on the addition of inexpensive cuts to the initial UC master problem. The initial cuts were obtained after the application of the following steps: calculation of LSC, calculation of MIL, calculation of relaxed horizon UC, elimination of redundant network constraints, calculation of LLF with rescheduling, and upgrading of Benders cuts. The results obtained by the application of the initialization methodology to the IEEE-118 and the RTS-96 systems have been presented. Based on these results, it is confirmed the possibility of improving the overall convergence of Benders algorithm. From a computational standpoint, it can be concluded that the procedure offers the potential for reducing the number of iterations, as well as the computing time required to reach a SCUC solution.
Load pattern
LSC [%]
Bus from
To
Loading [MW]
Rating [MW]
Perc. [%]
Classic BD Winter
60
5 17 26 5 17 23 26 37
8 30 30 8 30 25 30 38
232.0 197.9 220.1 271.2 247.2 160.4 252.4 230.4
175 175 175 175 175 140 175 175
132.5 113.1 125.8 154.9 141.2 114.6 144.2 131.7
5 17 26 5 30 23 26 37
8 30 30 8 247.2 25 30 38
231.9 198.7 212.6 271.2 175 160.4 252.4 230.4
175 175 175 175 141.2 140 175 175
132.5 113.5 121.5 154.9
5 17 26 5 17 23 26 37
8 30 30 8 30 25 30 38
221.3 196.4 210.6 268.4 235.7 158.5 239.0 214.2
175 175 175 175 175 140 175 175
126.4 112.2 120.3 153.4 134.7 113.2 136.6 122.4
5 17 26 5 17 23 26 37
8 30 30 8 30 25 30 38
221.3 196.4 210.6 268.4 235.7 158.3 238.7 214.2
175 175 175 175 175 140 175 175
126.4 112.2 120.3 153.4 134.7 113.1 136.4 122.4
80
Summer
60
80 17
Acknowledgements This work was funded by CONICET and SECyT-UNRC, to whom grateful acknowledgment is due.
Initialized BD Winter 60
Benders decomposition
80
In this appendix BD methodology is reviewed. Additional information about BD can be found in [3]. A general two-level optimization problem can be formulated mathematically as follows: Summer
min cT1 x1 þ cT2 x2
60
x1 ;x2
A1 x1 ¼ b1 B1 x1 þ A2 x2 ¼ b2
ðA:1Þ
where x1 is the vector of integer variables considered at the first level, and x2 is the vector of continuous variables considered at the second level. A1 ; A2 , and B1 are real matrices, c1 ; c2 ; b1 , and b2 are real vectors. An example of this kind of problem formulation is the SCUC, which is a large-scale optimization problem that must provide optimal decisions subject to unit, system and network constraints. Decomposition methods were proposed to solve this kind of optimization problems in an efficient way. These methods take advantage of the special structure of the problem solving iteratively smaller problems. Although they are general methods, the application of them depends mainly on the type of problem; if the problem is the variables or the restrictions. BD is normally applied when the problem is the restrictions. Hence, for SCUC problems BD method is the most used one. The BD method decomposes the two-level SCUC problem into a master problem and a sub-problem. The unit commitment master problem is represented in the first optimization level, and the network signals, known as cuts, derive from the sub-problems in the second optimization level. There are two main applications of Benders: when variables x1 make the problem too complex, and/or when the master and the sub-problems are different in nature. Mathematically, Eq. (A.1) can be reformulated as follows:
min x1 ;h2
cT1 x1
80
114.6 144.2 131.7
Table B.22 Branch loading checking: final master. Load pattern Classic BD Winter
Summer
LSC [%]
Bus from
To
Loading [MW]
Rating [MW]
Perc. [%]
60 80
17 8 17 30 37
30 30 30 38 38
175.1 175.5 179.9 187.0 180.2
175 175 175 175 175
100.1 100.3 102.8 106.9 103.0
60
17 26 5
30 30 8
175.4 176.7 175.2
175 175 175
100.2 101.0 100.1
60 80
17 17 26 37
30 30 30 38
175.1 175.6 177.0 178.9
175 175 175 175
100.1 100.3 101.2 102.3
60
5 17 5 37
8 30 8 38
175.2 175.3 176.4 175.8
175 175 175 175
100.1 100.2 100.8 100.4
80 Initialized BD Winter
Summer
80
þ h2 ðx1 Þ ðA:2Þ
A1 x1 ¼ b1 The function h2 ðx1 Þ represents the objective function of the following sub-problem:
h2 ðx1 Þ ¼ mincT2 x2 x2
A 2 x2 ¼ b 2 B 1 x1 : p 2
ðA:3Þ
Table C.23 Mapping between units and buses. Bus Unit
10 4
25 10
27 12
59 24
61 25
80 36
92 43
99 44
100 45
113 53
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J. Alemany, F. Magnago / Electrical Power and Energy Systems 66 (2015) 53–66 Table C.24 Initialized Benders status. Gen
Hr 10
it1 4 10 12 24 25 36 43 44 45 53 it2 4 10 12 24 25 36 43 44 45 53 it3 4 10 12 24 25 36 43 44 45 53 it4 4 10 12 24 25 36 43 44 45 53 it5 4 10 12 24 25 36 43 44 45 53
Table C.25 Classic Benders status.
1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
Gen 11 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
12 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
13 1 0 0 1 1 1 1 1 1 0
14 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
15 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
16 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
17 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
18 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
19 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
20 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
21 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
22 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
where p2 are the constraints dual variables (known as shadow prices). Then, the original two-level problem represented by Eq. (A.1) can be reformulated as follows:
min cT1 x1 þ h2 x1 ;h2
A1 x1 ¼ b1 T
h2 P ðb2 B1 x1 Þ : p12 ... T
h2 P ðb2 B1 x1 Þ : pm2
ðA:4Þ
Hr 10
11
12
13
14
15
16
17
18
19
20
21
22
it1 4 5 10 24 25 36 43 44 45 53
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
it2 4 5 10 24 25 36 43 44 45 53
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
it3 4 5 10 24 25 36 43 44 45 53
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
it4 4 5 10 24 25 36 43 44 45 53
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
it5 4 5 10 24 25 36 43 44 45 53
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
it6 4 5 10 24 25 36 43 44 45 53
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
where m is the maximum number of cuts. Eqs. (A.4) represent the complete master problem since contains all possible cuts. Instead of including all cuts at the same time in the master problem, the BD algorithm permits including one cut per iteration.
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J. Alemany, F. Magnago / Electrical Power and Energy Systems 66 (2015) 53–66
Results validation Tables B.21 and B.22 correspond to results of Benders based SCUC calculated using PSSÒE. SCUC patterns Table C.23 shows the mapping between buses and units. Tables C.24 and C.25 correspond to results of Benders based SCUC patterns for the summer 60% LSC. Patterns fully 0 or 1 are not presented. Hour 10–22, where the load peak is, are presented. References [1] Pinto H et al. Security constrained unit commitment: network modeling and solution issues. In: Power systems conference and exposition. IEEE PES; 2006. p. 1759–66. [2] Samiee M et al. Security constrained unit commitment of power systems by a new combinatorial solution strategy composed of enhanced harmony search algorithm and numerical optimization. Int J Electr Power Energy Syst 2013;44(1):471–81. [3] Conejo A et al. Decomposition techniques in mathematical programming: engineering and science applications. Berlin: Springer-Verlag; 2006. [4] Granville S. Mathematical decomposition techniques for power system expansion planning: decomposition methods and uses. In: Electric Power Research Institute and Stanford University, Systems Optimization Laboratory. The Institute Publisher; 1988. [5] Parastegari M et al. AC constrained hydro-thermal generation scheduling problem: application of Benders decomposition method improved by BFPSO. Int J Electr Power Energy Syst 2013;49:199–212. [6] Pereira M et al. A decomposition approach to automated generation/ transmission expansion planning. IEEE Trans Power Ap Syst 1985;PAS104(11):3074–83. [7] Asadamongkol S. Transmission expansion planning with AC model based on generalized Benders decomposition. Int J Electr Power Energy Syst 2013;47:402–7. [8] Sifuentes W et al. Hydrothermal scheduling using Benders decomposition: accelerating techniques. IEEE Trans Power Syst 2007;22(3):1351–9. [9] Sifuentes W et al. Short-term hydrothermal coordination considering an AC network modeling. Int J Electr Power Energy Syst 2007;29(6):488–96. [10] Alguacil N et al. Multiperiod optimal power flow using Benders decomposition. IEEE Trans Power Syst 2000;15(1):196–201.
[11] Kim M et al. An optimal pricing scheme in electricity markets by parallelizing security constrained optimal power flow based market-clearing model. Int J Electr Power Energy Syst 2013;48:161–71. [12] Bixby R et al. MIP: theory and practice – closing the gap. In: System modelling and optimization: methods, theory and applications. Kluwer Academic Publishers; 2000. [13] Bonnans F et al. Numerical optimization: theoretical and practical aspects. 2nd ed. Springer; 2006. [14] Pinto H, et al. Dual maximization methods for Lagrangian relaxation-based SCUC. In: Bulk power system dynamics and control conference in, VIII IREP; August 2010. [15] Alemany J et al. Symmetry issues in mixed integer programming based unit commitment. Int J Electr Power Energy Syst 2014;54:86–90. [16] Magnanti T. Accelerating Benders decomposition: algorithmic enhancement and model selection criteria. Oper Res 1981;29(3):464–84. [17] Richardson R. An optimization approach to routing aircraft. Transport Sci 1976;10(1):52–71. [18] Saharidis G et al. Initialization of the Benders master problem using valid inequalities applied to fixed-charge network problems. Expert Syst Appl 2011;38(6):6627–36. [19] Pereira M et al. Application of sensitivity analysis of load supplying capability to interactive transmission expansion planning. IEEE Trans Power Ap Syst 1985;PAS-104(2):381–9. [20] Le K et al. Rolling horizon method: a new optimization technique for generation expansion studies. IEEE Trans Power Ap Syst 1982;PAS101(9):3112–6. [21] Stott B et al. Linear programming for power system network security applications. IEEE Trans Power Ap Syst 1979;PAS-98(3):837–48. [22] Monticelli A et al. Security-constrained optimal power flow with postcontingency corrective rescheduling. IEEE Trans Power Syst 1987;2(1):175–80. [23] Granville S, et al. Mathematical decomposition techniques for power system expansion planning: security-constrained optimal power flow with postcontingency corrective rescheduling. In: EPRI EL-5299, Project 2473-6, Final Report, vol. 4; February 1988. [24] Stott B, et al. Review of linear programming applied to power system rescheduling. In: Power industry computer applications conference, PICA-79, IEEE conference proceedings; May 1979. p. 142–54. [25] motor.ece.iit.edu/data/JEAS_IEEE118.doc. [26] http://www.ee.washington.edu/research/pstca/rts/pg_tcarts.htm. [27] Mevert P. 1977. Fixed charge network flow problems: applications and methods of solution. In: Large scale and hierarchical systems workshop. Brussels; May 1977. [28] Geoffrion A et al. Multicommodity distribution system design by Benders decomposition. Manage Sci 1974;5:822–44.
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Benders decomposition applied to security constrained unit commitment: Initialization of the algorithm J. Alemany a,⇑, F. Magnago a,b a b
Department of Electrical and Electronic Engineering, Universidad Nacional de Río Cuarto, Argentina Nexant Inc., AZ, USA
a r t i c l e
i n f o
Article history: Received 5 December 2013 Received in revised form 3 August 2014 Accepted 12 October 2014
Keywords: Benders decomposition Initialization methodology Mixed integer linear programming Security constrained unit commitment
a b s t r a c t Benders decomposition has been broadly used for security constrained unit commitment problems, despite the fact that it may present convergence difficulties due to instabilities and to the mixed integer nature of the unit commitment problem. The initialization of Benders decomposition has been recognized as a prominent feature for the algorithm enhancement. In this work, a new Benders decomposition initialization methodology is proposed. The objective of the initialization is to include inexpensive network signals that can be added during the initial unit commitment master problem. Numerical simulations using the IEEE-118 and RTS-96 systems are performed to illustrate the benefits of the proposed initialization methodology. Results suggest that the initialization of Benders decomposition applied to security constrained unit commitment problems improves the overall convergence of the algorithm. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction The unit commitment (UC) calculation is extensively used in daily power system operation. The UC is an exercise of large-scale, time-varying, non-convex, mixed-integer modeling and optimization. Security constrained unit commitment (SCUC) is an extension of conventional UC with the inclusion of system network constraints, in both normal and contingency operation states. The main objectives of the SCUC are to ensure not only the economic operation but also the security of the system [1,2]. These two different objectives can be separately solved as a two-level optimization problem. It is a common practice to define a master level, which includes the UC calculation, and the sub-problem level, which checks the network security constraints. Benders Decomposition (BD) is an algorithm that has been broadly used for large-scale optimization problems, particularly for power systems [3,4]. BD has three main advantages: modularity, flexibility and robustness. With regard to modularity, master and sub-problems can be separately solved by specialized algorithms, thus providing speed and efficiency on the overall performance of the global optimization process. Additionally, Benders flexibility is mainly supported by the different existing power system applications. For example, it is possible to find its application in areas like security constrained economic dispatch ⇑ Corresponding author. E-mail address: [email protected] (J. Alemany). http://dx.doi.org/10.1016/j.ijepes.2014.10.044 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
[5], generation–transmission planning [6,7], hydrothermal coordination [8,9], and optimal power flow [10,11]. Finally, in relation to robustness, despite the different nature of the master and the sub-problems in SCUC applications, both are essentially solved using Linear Programming (LP) algorithms. This is an important feature because LP algorithms are considered to be among the most mature methodologies in optimization techniques [12]. Nevertheless, since BD is a cutting-plane method [13], it may present instabilities which are translated into delays of the algorithm convergence [14,8]. In addition, since the master level is formulated as a mixed integer linear problem (MILP), the convergence time is strongly affected by the high computational burden of the master problem [15]. Therefore, there are many research efforts regarding BD improvements [16]. Among the different suggested possibilities, having a better initialization of BD is recognized by several authors as one of the most important enhancements, concluding that it could have a significant effect on BD performance. Several initialization methodologies have been developed. In [17] an initial set of cuts is constructed from routes of an aircraft routing model. The author concluded that the initial cuts are advisable to be used with any network, since there is a reduction in the total computation time. Furthermore, in [18] a strategy to initialize BD with the addition of a series of valid inequalities is developed. The initialization procedure is applied to a fixed charge network problem, and different variants of refinery systems are studied.
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J. Alemany, F. Magnago / Electrical Power and Energy Systems 66 (2015) 53–66
Nomenclature T G B N L b t g l
scheduling horizon number of generation units number of piece-wise blocks number of buses number of branches power block index hour index unit index branch index contingency scenario loop iteration out of service lapse initial hour production cost startup cost piece-wise power block active power relaxation variable load supplying capability MW bus injection state binary variable power block binary variable start-up binary variable shut-down binary variable generation vector
m k
s Cpgt Cagt dbgt pgt slack
q pbt ugt jbgt sgt hgt p
f puc h Kg s cg Fbg Trbg Dt MUTg and MDTg off Ton g and Tg Pg and Pg RULg and RDLg k peak alb k cs cT D D f and f P and P
c S
The aim of this paper is to propose a new BD initialization methodology applied to SCUC problems. The initialization methodology is based on the combination of three main concepts: a fast UC resolution, a redundant constraint elimination, and the obtainment of upgraded cuts within a range of load bounds. The remaining sections are outlined as follows. Section ‘Benders decomposition based SCUC’ gives a general background on BD. Section ‘Outline of the initialization methodology’ provides a detailed description of the proposed initialization strategy. This section includes a detailed description of the model and concepts in which the Benders initialization is based. In Section ‘Computational results’, numerical results are presented and discussed. Finally, Section ‘Conclusion’ presents the most relevant conclusions.
branch flow vector UC generation level vector bus voltage angle vector start-up cost for step s fix cost power block slope min–max power block limits hourly system demand min on/off service times initial on/off hours of service max–min power limits up–down fixed ramp limits iterative demand linear sensitivity factors load participation factors vector infeasibility cost vector dispatch costs vector demand vector allowed MW deviation vector max–min flow limit vectors generation bound vectors branch susceptances matrix branch-bus incidence matrix (0 is transpose)
8bgt
ðTrbg Trb1;g Þjbgt 6 dbgt
ð6Þ
8bgt
dbgt 6 ðTrbg Trb1;g Þjb1;gt
ð7Þ
dBgt P 0 8gt
ð8Þ
8gt
dBgt 6 ðPg TrB1;g ÞjB1;gt Cagt P Kgs ugt
s X
ð9Þ
!
8gt
ug;tn
ð10Þ
n¼1
Cagt P 0 8gt s:t: :
G X pgt ¼ Dt
ð11Þ
8t
ð12Þ
g¼1 t X
8g 8t 2 MUTg þ 1; T
sgi 6 ugt
ð13Þ
i¼tMUTg þ1
Benders decomposition based SCUC In this section BD methodology applied to SCUC problem is formulated [1]. Additional information about BD can be found in the Appendix. Master problem – the decision stage
t X
hgt 6 1 ugt
8g 8t 2 MDTg þ 1; T
ð14Þ
i¼tMDTg þ1 i6Ton g
X
1 ugi ¼ 0 8g 8t ¼ 0
ð15Þ
ugi ¼ 0 8g 8t ¼ 0
ð16Þ
i¼0 off
The master problem is the single-bus UC without security constraints. The thermal UC model can be formulated as follows:
min z ¼
T X G X
Cpgt þ Cagt
ð1Þ
t¼1 g¼1
Cpgt ¼ ugt cg þ
B X Fbg dbgt
8gt
ð2Þ
b¼1
pgt ¼ ugt Pg þ
B X dbgt
i6Tg
X i¼0
ugt Pg 6 pgt 6 ugt Pg
8gt
ð17Þ
pgt pg;t1 6 RULg
8g 8t P 0
ð18Þ
pg;t1 pgt 6 RDLg
8g 8t P 0
ð19Þ
8gt
ð20Þ
sgt hgt ¼ ugt ug;t1
8gt
ð3Þ
b¼1
pgt P 0 8gt ðTr1g Pg Þj1gt 6 d1gt 6 ðTr1g Pg Þugt
ð4Þ
8gt
ð5Þ
sgt þ hgt 6 1 8gt
ð21Þ
jbgt ; ugt ; sgt ; hgt 2 f0; 1g 8bgt
ð22Þ
Benders cutstm
8t m
ð23Þ
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min w ¼
N X cs slack
ð24Þ
s:t: : Sf slack 6 puc D
pd1
Sf slack 6 puc þ D
ð25Þ
pd2
ð26Þ
0
f cS h ¼ 0
ð27Þ
f 6 f pf 1 f 6 f
ð28Þ ð29Þ
pf 2
0 6 slack 6 D
ps
ð30Þ
The associated costs cs are usually set to one. After solving the problem represented by Eqs. (24)–(30), relaxing the balance constraints (slack), one Benders cut is created for each hour of a network scenario as follows:
w þ
pd1 pd2 puc puc 6 0
ð31Þ
In the next section an initialization methodology applied to BD based SCUC is presented.
Outline of the initialization methodology The main motivation to develop an initialization methodology applied to BD based SCUC is to improve the convergence rate of the Benders decomposition. The core of the initialization methodology is to provide a set of inexpensive cuts to be added in the initial master problem, in order to provide early signals of the network to the UC calculation. The proposed initialization methodology comprises six different steps:
Fig. 1. MIL algorithm.
Fig. 2. Relaxed horizon scheme.
Eq. (1) is the objective function that accounts for the total operational day-ahead costs. Eq. (2) represents the production costs. Eqs. (3)–(9) are the piece-wise linearization of production costs. Eqs. (10) and (11) represents the step-wise start-up costs. Eq. (12) is the total energy balance. Eqs. (13)–(16) represent min up–down times. Eq. (17) represent the generator operational limits. Eqs. (18) and (19) are the ramp limits. Eqs. (20)–(22) represent the logic among binary variables. The last term stands for the eventual Benders cuts. Sub-problem – the feasibility stage The coupling between the master and the sub-problems is done through the variable puc , which is the solution of problem represented by Eqs. (1)–(22). The sub-problem can be formulated as:
1. Calculation of the Load Supplying Capability (LSC) [19]. The LSC is an upper bound of network infeasibility that allows forming the cuts. 2. Calculation of the Minimal Infeasible Load (MIL). The MIL is a lower bound of network infeasibility that allows forming the cuts. 3. Calculation of a Relaxed Horizon single-bus UC (RHUC) [20] for the LSC demand peak. The calculation is carried out in order to get the MWs dispatch levels. 4. Pre-processing of redundant network constraints [21]. Redundant security constraints are eliminated with a network preprocessing method. 5. Calculation of the Linear Load Flow with Rescheduling (LLFR) [22] for the given MWs UC levels and the LSC demand peak. The calculation is carried out to check network infeasibility in order to form a cut. 6. Upgrading of Benders cuts [23]. The cut in Step 5 is upgraded for every demand level within the range delimited by the LSC and MIL bounds. In the following sections, detailed information about each step is provided.
Table 1 Two generation units system. Item
Value
Variable cost Start up cost Fix demand Max power Min power B var R var
1, 2 3, 2 30, 31, 40 50, 20 10, 5 ugt ; sgt ; hgt pgt
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Table 2 Variable pattern.
Fig. 4. Inactive constraints set.
Calculation of the load supplying capability
Table 3 Relaxed horizon UC results. Iter problem 0 RMILP
1 MILP 2 MILP
3 MILP 4 MILP
The LSC of a given system is the maximum load that can be supplied with all branches and generators operating, and considering the participation factors of each load bus with respect to the total load. The concept of the LSC is to stress the network to a maximum, that is, to calculate the maximum load that can be supplied without system overloads. The LSC is the upper bound of the range within which it is possible to form a set of initial cuts. Mathematically it can be formulated as the solution of the following LP problem:
Model item
Stats.
z value
Rows Cols. Non 0 Z cols. Relaxed cols. Fixed cols.
16 25 53 0 18 0
103.4
Z cols. Relaxed cols.
6 12
111
Z cols. Relaxed cols. Fixed cols.
6 6 6
109
Z cols. Fixed cols.
6 12
109
where q is a scalar that represents the total system demand.
Z cols.
18
104
Calculation of the minimal infeasible load
max LSC ¼ q cS0 h þ p ¼ k q jShj 6 f =c
ð32Þ
06p6P
ð35Þ
ð33Þ ð34Þ
The MIL represents the minimum load that can be economically supplied without network overload. The MIL is the lower bound of the range within which a set of initial cuts can be formed. Mathematically it can be formulated as the solution of the following LP problems:
min x ¼ cT p
ð36Þ
G X k p peak P 0
ð37Þ
06p6P
ð38Þ
min y ¼ slack
ð39Þ
G X cS0 h þ p þ slack ¼ k peakk
ð40Þ
jShj 6 f=c slack P 0 Fig. 3. 3 Buses system.
ð41Þ ð42Þ
The above LP problems need to be iteratively solved. The algorithm is described next:
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Input: LSC k
Initialization: ub = LSC, lb = 0, peak = ub, k = 0 loop k Solve (36)–(38) then (39)–(41) if y > 0 then k
ub = peak
peak = ub- ub—lb 2 else if y 6 0 then k
k
lb = peak
k peak = ub- ub—lb 2 end if k¼kþ1 if ub–lb 6 then Stop MIL=ub end if end loop Output: MIL
Fig. 5. LSC-MIL range.
Table 4 First UC master results.
Calculation of a relaxed horizon unit commitment
Fixed variable sub-period (Dark Grey). Active variable sub-period (White). Relaxed variable sub-period (Light Grey). The binary variables within the fixed variable sub-period are fixed using the previous iteration values. The binary variables within the active variable sub-period are treated as integer variables. Finally, the binary variables within the relaxed variable sub-period are treated as continuous variables. Thus, as the iterations progress, the fixed variable sub-period increases while the relaxed variable sub-period decreases. At the final iteration the whole scheduling horizon is treated as a complete MILP. In this final stage, after two rounds without significant improvement of the objective value the algorithm stops. A simple example is presented to illustrate this strategy. Let’s consider a small system with two identical generators. Table 1 shows the data used for the simulations. Without loosing generality, from the full set of operational constraints, only balance, binary logic and power limit constraints are included into the UC formulation. The relaxed horizon unit commitment is formulated as follows:
8t
ugt ug;t1 6 sgt hgt
ð43Þ
ð45Þ
8gt
30 31 40
p2 [MW] 0 0 0
ð46Þ
104
Table 2 shows the relaxed horizon pattern used for the simulations. Table 3 shows the results after the simulations. This relaxation strategy does not affect the solution space because at the last iterations a full MILP problem is solved. The objective of this relaxation strategy is to provide a fast feasible solution that can be used to check network infeasibility. Nevertheless, some parameters of the algorithm need to be tuned before good performance is obtained. For general purpose application, default parameter values are recommended only for large scale UC problems. Pre-processing of redundant network constraints Power systems are not expected to operate out of limits [21], therefore, in general it is necessary to deal with only a small subset of security constraints at any stage. Based on this observation, several redundant network constraints can be eliminated prior to the LLF calculations. The aim of the strategy is to reduce the size of the LLF problem in the base case, provided the solution space is not affected because the eliminated constraints are redundant to the feasible region. This is demonstrated using a simple example next. Let’s consider again the 2 units system of Table 1 into the 3-bus network described in Fig. 3 serving 30 MW load. Fig. 4 represents the feasible region corresponding to the system of Fig. 3. The gray zone is the feasible region which is limited by restrictions P2 6 14.9, P2 P 5, the balance equation P1 + P2 = 30, and the lower limit P1 P 10. It can be seen that only 1 out of 6 security Table 5 LLFR results for LSC demand. Hour
p
1 2 3
30 31 40
p1 [MW]
ð44Þ
0 6 pgt 6 ugt P 8gt
1 2 3 Objective value
The strategy is based on sequentially fixing, relaxing and activating different subsets of the binary variables. At each iteration only a portion of the simulation horizon is considered as a MILP problem, while the remaining part is either fixed using values from a previous iteration, or relaxed in its integrality at the current iteration. Fig. 2 illustrates a general scheme for this methodology. For each iteration, the scheduling horizon is further divided into three sub-periods:
G X pgt ¼ Dt
puc p1 [MW]
The algorithm is basically based on the bisection search method [13]. Fig. 1 outlines the proposed MIL approach.
T X G X min z ¼ ðCpgt pgt þ Cagt sgt Þ
Hour
w [MW] p2 [MW] 0 2 20
0 2 20
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Table 6 LLFR sensitivities. Sens
Hour 1
2
3
0 1
0 1
0 1
pr
p1 p2
pg
p1
0
0
0
p2
1
1
1
b2 b3
0 0
1 2
1 2
pd
Table 7 Final UC master results. Hour
puc p1 [MW]
1 2 3
30 26 20
Objective value
p2 [MW]
Fig. 8. Demand 3 network status.
0 5 20 131
Fig. 9. Demand 2 network with cut.
Fig. 6. Demand 1 network status.
Fig. 10. Demand 3 network with cut.
Fig. 7. Demand 2 network status.
constraints are non-redundant. The redundant security constraints have no influence on the feasible region of the problem and can thus be eliminated from the SCUC problem. Relaxation techniques are very important when only few of a large set of inequality constraints become binding [24]. The redundant constraints can be obtained considering the following LP problem:
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zlt ¼ max
N X
alb pbt
8lt
ð47Þ
b¼1 N X
N X alb pbt slack 6 f l þ alb Dbt
b¼1 N X
8l
ð48Þ
b¼1
pbt ¼ Dt
8t
ð49Þ
Table 9 118-bus Line set [%]. Hour
Non-redundant lines
Total lines [%]
1, 3, 7, 9,
16 15 14 17
8.6 8.1 7.5 9.1
2, 8, 15, 17, 18, 19, 21, 24 4, 5, 6, 22 23 10, 11, 12, 13, 14, 16, 20
b¼1
0 6 pbt 6 Pb
8t 8g 2 b
ð50Þ
slack P 0
ð51Þ
where zlt measures the power flow of each branch. Eq. (47) denotes the maximum possible power flow over a transmission branch provided that power balance, Eq. (49), and unit restrictions, Eq. (50), are satisfied. Eq. (48), monitors which branches are redundant or not. Whenever a branch overload occurs, slack assumes some positive value and the security constraint is labeled as non-redundant. Generators are dispatched to supply the system load; the more sensitive the branch is with respect to the unit, the more weight it will have on the objective function. This process, therefore, allows the monitoring of branches in order to prevent overloading during the given period. For this step, a series of LP problems (l t) need to be solved in order to detect which network constraints are redundant in the feasible region; nevertheless, this is a pre-processing step.
Table 10 RTS-96 network after pre-processing. Hour
Line 2 13 77 78
3 107
6 20 43 109 69
14 24 26 30 44 60 71 85
32 103
36 37
75 76 82
1 11 12 29 31 41 42 47 52 65 68 70 80 90 106 108 118 119 120
0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1
11111001 11111001 11111001 11111101 11111001 11111001 01111011 01101011 01101011 01101011 01111011 01101011 00000010 00001010 01001010 01101010 01101011 01111011 01111001
1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 1 1
0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Calculation of linear load flow with rescheduling Given the single-bus UC pattern, puc , calculated in Section ‘Calculation of a relaxed horizon unit commitment’, a LLFR is obtained. The objective of the LLFR, provided that network infeasibility exists, is to obtain the cut corresponding to the LSC demand level. In order to form this cut, it is necessary to minimize the system infeasibility, w, taking into account the coupling between the UC pattern, puc , and the network constrained dispatch p. Mathematically, the LLFR is formulated as follows:
Table 8 118-bus network after pre-processing. Hour
1, 2 3, 4 5, 6 7 8 9 10, 11 12, 13 14 15 16 17, 18 19 20 21 22 23 24
Line 14 39
44 50 112 115 119 129 136 162 163 165 168
63 106
120 123 127 133 166
10 10 10 10 1 01 01 01 01 01 01 01 01 01 01 1 10 10
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
00011 00001 00001 00011 00111 1 1 1 1 01111 1 11110 11110 1 01111 00011 00011 00011
1 2 3 4, 5, 6, 7 8 9 10 11 12 13 14, 15, 16 17 18 19 20 21 22 23 24
Table 11 RTS-96 line set [%]. Hour
Non-redundant lines
Total lines [%]
1 2 3 4, 5, 6, 7 8 9 10, 23, 14, 15, 16 11, 13 12, 17, 22 18 19 20 21 24
34 37 43 44 39 32 27 26 24 20 21 22 23 31
28.3 30.8 35.8 36.7 32.5 26.7 22.5 21.7 20.0 16.7 17.5 18.3 19.2 25.8
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Table 12 Initial UC comparison. Load pattern
LSC [%]
Winter
60 80
Summer
60 80
min w ¼
N X
IEEE-118 time [s]
RTS-96 time [s]
UC RHUC UC RHUC
1.20 1.60 2.34 1.87
146 29 27 14
UC RHUC UC RHUC
1.18 1.72 1.90 1.75
235 56 64 34
slack
ð52Þ
b¼1
cS0 h þ p ¼ D pd jShj 6 f =c pf 0 6 p 6 P pg
p p þ slack 6 D uc
ð53Þ
Fig. 11. Summer: IEEE-118 LSC-MIL range.
ð54Þ ð55Þ
pr
ð56Þ
Here the variable slack is a vector of slack variables which correspond to the gap between UC and LLF generation levels. The resolution of the LLFR provides the sensitivities pd ; pg ; pf , and pr . (Simplex multipliers associated to the Eqs. (52)–(56).) These sensitivities are necessary not only to form the LSC cut but also to upgrade the subsequent cuts within the LSC-MIL range. This step is explained in the next section. Upgrading of Benders cuts The feasibility cut resulting from Eqs. (52)–(56), is described by:
w þ pr puc puc 6 0
ð57Þ
Fig. 12. Winter: IEEE-118 LSC-MIL range.
This cut can be upgraded to accommodate deviations between the forecasted and actual demand, as mentioned in [4]. As a consequence, the cut could also be valid for other time steps. Before the cut can be upgraded, the feasibility cut should also accommodate changes in system configuration due to changes in the UC. Then, the upgraded cuts within the LSC-MIL range are:
w þ pr puc puc þ pg p puc þ pd ðDt Dlsc Þ 6 0
ð58Þ
where w is the optimal solution value of Eqs. (52)–(56) for the LSC demand level, Dlsc is indeed the LSC demand level, and Dt correspond to the demand levels within the LSC-MIL range. The third term of Eq. (58) represents UC changes and the fourth term represents demand changes. Fig. 5 illustrates a generic demand curve and the LSC-MIL range. The upgraded cuts represented by Eq. (58) and the LSC cut given by Eq. (57), are added to the initial master UC in order to initialize the BD algorithm. Again, a simple example is presented to illustrate this strategy. Let’s consider the small system of Table 1 and Fig. 3. Solving problems of Sections ‘Calculation of the load supplying capability and Calculation of the minimal infeasible load’ the LSC and MIL of this system are 40 and 30 MW respectively. Table 4 shows the results of the first UC master. Pre-processing of network is not applied.
Fig. 13. Summer: RTS-96 LSC-MIL range.
Table 5 shows the results after the problem resolution of Section ‘Calculation of linear load flow with rescheduling’ for the LSC demand level and Table 6 shows the sensitivities of that problem. Therefore, the LSC cut and the upgraded cuts are as follows:
Upgraded cut for t ¼ 1 puc 2 P 0
Table 13 Calculation of LSC-MIL bounds. Bound
IEEE-118 time [s]
RTS-96 time [s]
LSC MIL
0.562 3.790
0.436 1.284
ð59Þ
Upgraded cut for t ¼ 2 puc 2 P 2
ð60Þ
LSC cut for t ¼ 3 puc 2 P 20
ð61Þ
Adding the cuts (59)–(61) to the master and resolving the UC, Table 7 is obtained.
J. Alemany, F. Magnago / Electrical Power and Energy Systems 66 (2015) 53–66
Fig. 14. Winter: RTS-96 LSC-MIL range.
Table 14 Calculation of LLFR for LSC demand level. System
Load pattern
LSC [%]
Peak [h]
Infeasibility [MW]
Time [s]
IEEE118
Winter
60
18
112
0.49
Summer
80 60 80
12
365 112 363
0.54 0.51 0.50
117 197 140 275
0.43 0.44 0.42 0.44
RTS-96
Winter Summer
60 80 60 80
12 10
Fig. 15. IEEE118 – summer and winter: 80% LSC.
Table 15 IEEE-118: 80% LSC – summer and winter load patterns. Iter
Time [s]
Ini master cost
Summer Classic BD Initialized BD Improvement (%)
8 6 33
73.95 48.39 54
1,172,407 1,175,594
Winter Classic BD Initialized BD Improvement (%)
8 6 33
70.73 51.20 37
1,172,289 1,184,209
Table 16 IEEE-118: 60% LSC – summer and winter load patterns. Iter
Time [s]
Ini master cost
Summer Classic BD Initialized BD Improvement (%)
6 5 20
48.02 37.85 30
837,328 837,909
Winter Classic BD Initialized BD Improvement (%)
6 5 20
49.87 39.34 28
838,272 840,665
Finally, Figs. 6–8 show the network for the first master solution and Figs. 9 and 10 show the network after the cut signals in the master solution. The next section will describe numerically the proposed methodology.
Fig. 16. IEEE118 – summer and winter: 60% LSC.
61
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Table 17 RTS-96: 80% LSC – summer and winter load patterns. Iter
Time [s]
Ini master cost
Summer Classic BD Initialized BD Improvement (%)
15 13 15
294 239 23
5,546,510 5,563,205
Winter Classic BD Initialized BD Improvement (%)
14 12 17
271 226 20
5,368,409 5,364,254
Table 18 RTS-96: 60% LSC – summer and winter load patterns. Iter
Time [s]
Ini master cost
Summer Classic BD Initialized BD Improvement (%)
9 8 13
211 173 22
3,228,775 3,236,432
Winter Classic BD Initialized BD Improvement (%)
9 7 29
215 172 25
3,175,159 3,179,984
Fig. 18. RTS96 – summer and winter: 60% LSC.
in [25,26]. Two hourly load patterns are used, summer and winter, corresponding to the week-day patterns presented in [26]. Two different load peaks are tested, 80% and 60% of LSC, respectively. The IEEE 118-bus system has 54 units and 186 branches. The RTS-96 system has 96 units and 120 branches. The UC gaps for the initial master and for the subsequent masters are set to 5% and 1%, respectively. The network infeasibility tolerance is set to 1 MW. The Benders gaps illustrated in Figs. 15– 18 are defined as:
Benders gap ¼ Upper bound Lower bound Upper bound ¼ Master UC solution þ 1T
T X Subproblem LLF solutions
Lower bound ¼ Master UC solution It is important to remark that both upper and lower bounds are costs, and that is why vector 1T is necessary. In addition, the solution refers to the optimal value of the objective function, the total cost for the UC and the hourly network infeasibility for the subproblems.
Pre-processing results
Fig. 17. RTS96 – summer and winter: 80% LSC.
Computational results Numerical examples are presented in this section to show the benefits of BD initialization for SCUC calculation. For the sake of simplicity, only base scenario is tested, no contingencies are considered. All the numerical tests are implemented in GAMS, the problem is solved with CPLEX 11.0 solver on a PC with an Intel i5-2310 CPU at 2.90 GHz and 4.00 GB RAM memory. The IEEE 118-bus and the RTS-96 systems are studied, the data are given
Table 8 summarizes the results after the pre-processing step (Section ‘Pre-processing of redundant network constraints’) on the IEEE 118-bus network. This table illustrates the pattern of non-redundant lines in relation to the total set of lines. The total number of network constraints are 4464 (186 lines 24 h), only 383 are non-zero elements which represents 8.6% of the total number of network constraints. Table 9 illustrates the percentage of non-redundant lines at each hour, in relation to the total set of 186 lines. The non-redundant subset spans from 14 to 17 lines, representing at the most 9.1% of the total set of lines. Table 10 summarizes the pre-processing on the RTS-96 network. The total number of network constraints are 2880 (120 lines 24 h), only 737 are non-zero elements which represents 25.6% of the total number of network constraints. Table 11 illustrates the percentage of non-redundant lines at each hour, in relation to the total set of 120 lines.
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RHUC results Table 12 compares the results between the relaxed horizon UC resolution (Section ‘Calculation of a relaxed horizon unit commitment’) and the standard UC resolution that conforms the initial master problem on both the IEEE-118 and RTS-96 systems. In this work, a 1/4 of the scheduling horizon to each sub-period is used. It is important to note that the relaxed UC resolution performance is not as effective as expected in small scale generation systems. The practical experience with this technique allows to conclude that their biggest impact it is performed in large scale or complex generation systems. The results reported for the RTS-96 confirm that the relaxed horizon UC strategy is more convenient for complex generation systems. LSC-MIL bounds results For the base case, the LSC of the IEEE 118-bus system is 5416 MW and the MIL is 1458 MW. The LSC of the RTS-96 system is 9825 MW and the MIL is 1228 MW. Table 13 presents the computing times for the calculation of the LSC-MIL bounds of the systems. In Figs. 11 and 12 the IEEE-118 LSC-MIL range in relation to summer and winter week-day load patterns is showed. In Figs. 13 and 14 the RTS-96 LSC-MIL range is illustrated. LLFR and Benders cuts upgrading results Table 14 presents computing times for the calculation of LLFR for the LSC demand levels for both IEEE-118 and RTS-96 systems. Benders cuts upgrading consists in the building of cuts by Ec. (58). This process do not posses a significant computing impact in relation to the other steps of BD initialization. Therefore, these computing times are not presented.
the number of iterations. Finally, the overall convergence patterns illustrated in Figs. 15–18 show the faster entrance into the pre-specified gap tolerance. Tables 19 and 20 illustrate how each initialization step impacts the global performance of BD based SCUC calculation, excluding the network pre-processing. The results presented here are consistent with previous work done in other study areas. [16,27] reported the improved general performance of BD with network problems. Additionally, [28] reported similar experience with facility location problems. Furthermore, [17] reported the reduction in the total computation time with aircraft networks. Finally, [18] reported the decrease in the number of iterations with fixed charge network problems. To summarize, it is important to remark that all the simulation instances did improve the convergence of BD algorithm. Although the results are preliminary, they seem to be very promissory and may support the development of more sophisticated initialization strategies applied to SCUC based BD.
Table 19 Initialization steps impact on global performance for IEEE-118. Step calculation
Load pattern
LSC [%]
Time [s]
Relative to initialized BD [%]
IEEE-118 LSC
Winter
60 80 60 80
0.56
1.43 1.10 1.48 1.16
60 80 60 80
3.79
9.25 7.12 9.73 7.55
60 80 60 80
1.60 1.87 1.72 1.75
4.07 3.65 4.54 3.62
60 80 60 80
0.49 0.54 0.51 0.50
1.25 1.05 1.35 1.03
Summer MIL
Winter Summer
Global results RHUC
Tables 15 and 16 summarize the results for summer and winter load patterns at 80% and 60% of LSC for the IEEE-118. Figs. 15 and 16 illustrate the convergence of Benders algorithm on the IEEE-118 for both summer and winter load patterns at 80% and 60% of LSC respectively. Tables 17 and 18 summarize the results for summer and winter load patterns at 80% and 60% of LSC for the RTS-96. Figs. 17 and 18 illustrate the convergence of Benders algorithm on the RTS-96 for both summer and winter load patterns at 80% and 60% of LSC. Computing times labeled as Initialized BD reported in Tables 15–18 include the time spent by the initialization procedure. Computing times, both Classic and Initialized BD, do not include the time spent by the pre-processing of redundant network constraints step, due to this is an external step. In addition, the network preprocessing and LSC-MIL bounds are applied to both, classic and initialized BD. Improvements in Tables 15–18 are calculated as the relation between Classic and Initialized BD, for both parameters, number of iterations and total time. Results of Benders based SCUC were validated against results calculated using PSSÒE. The comparisons are shown in Tables B.21 and B.22 in Appendix B. Also, Tables C.24 and C.25 of Appendix C correspond to iterative results of SCUC patterns. The results suggest that the initialization of Benders decomposition applied to SCUC problems has the possibility to improve the overall convergence of the algorithm. The percentage improvements reported in Tables 15–18 reveal the impact of the initialization upon the performance of BD algorithm. Additionally, the computing times reported in Tables 15–18 show the reduction in the total CPU solution times. Furthermore, the iterations reported in Tables 15–18 illustrate the decrease in
Winter Summer
LLFR
Winter Summer
Table 20 Initialization steps impact on global performance for RTS-96. Step calculation
Load pattern
LSC [%]
Time [s]
Relative to Initialized BD [%]
RTS-96 LSC
Winter
60 80 60 80
0.44
0.26 0.19 0.25 0.18
60 80 60 80
1.88
1.09 0.83 1.09 0.79
60 80 60 80
29 14 56 34
16.9 6.19 32.4 14.2
60 80 60 80
0.43 0.44 0.42 0.44
0.25 0.19 0.24 0.18
Summer MIL
Winter Summer
RHUC
Winter Summer
LLFR
Winter Summer
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J. Alemany, F. Magnago / Electrical Power and Energy Systems 66 (2015) 53–66 Table B.21 Branch loading checking: initial master.
Conclusion In this work, a new BD initialization methodology applied to SCUC problems has been proposed. The initialization was based on the addition of inexpensive cuts to the initial UC master problem. The initial cuts were obtained after the application of the following steps: calculation of LSC, calculation of MIL, calculation of relaxed horizon UC, elimination of redundant network constraints, calculation of LLF with rescheduling, and upgrading of Benders cuts. The results obtained by the application of the initialization methodology to the IEEE-118 and the RTS-96 systems have been presented. Based on these results, it is confirmed the possibility of improving the overall convergence of Benders algorithm. From a computational standpoint, it can be concluded that the procedure offers the potential for reducing the number of iterations, as well as the computing time required to reach a SCUC solution.
Load pattern
LSC [%]
Bus from
To
Loading [MW]
Rating [MW]
Perc. [%]
Classic BD Winter
60
5 17 26 5 17 23 26 37
8 30 30 8 30 25 30 38
232.0 197.9 220.1 271.2 247.2 160.4 252.4 230.4
175 175 175 175 175 140 175 175
132.5 113.1 125.8 154.9 141.2 114.6 144.2 131.7
5 17 26 5 30 23 26 37
8 30 30 8 247.2 25 30 38
231.9 198.7 212.6 271.2 175 160.4 252.4 230.4
175 175 175 175 141.2 140 175 175
132.5 113.5 121.5 154.9
5 17 26 5 17 23 26 37
8 30 30 8 30 25 30 38
221.3 196.4 210.6 268.4 235.7 158.5 239.0 214.2
175 175 175 175 175 140 175 175
126.4 112.2 120.3 153.4 134.7 113.2 136.6 122.4
5 17 26 5 17 23 26 37
8 30 30 8 30 25 30 38
221.3 196.4 210.6 268.4 235.7 158.3 238.7 214.2
175 175 175 175 175 140 175 175
126.4 112.2 120.3 153.4 134.7 113.1 136.4 122.4
80
Summer
60
80 17
Acknowledgements This work was funded by CONICET and SECyT-UNRC, to whom grateful acknowledgment is due.
Initialized BD Winter 60
Benders decomposition
80
In this appendix BD methodology is reviewed. Additional information about BD can be found in [3]. A general two-level optimization problem can be formulated mathematically as follows: Summer
min cT1 x1 þ cT2 x2
60
x1 ;x2
A1 x1 ¼ b1 B1 x1 þ A2 x2 ¼ b2
ðA:1Þ
where x1 is the vector of integer variables considered at the first level, and x2 is the vector of continuous variables considered at the second level. A1 ; A2 , and B1 are real matrices, c1 ; c2 ; b1 , and b2 are real vectors. An example of this kind of problem formulation is the SCUC, which is a large-scale optimization problem that must provide optimal decisions subject to unit, system and network constraints. Decomposition methods were proposed to solve this kind of optimization problems in an efficient way. These methods take advantage of the special structure of the problem solving iteratively smaller problems. Although they are general methods, the application of them depends mainly on the type of problem; if the problem is the variables or the restrictions. BD is normally applied when the problem is the restrictions. Hence, for SCUC problems BD method is the most used one. The BD method decomposes the two-level SCUC problem into a master problem and a sub-problem. The unit commitment master problem is represented in the first optimization level, and the network signals, known as cuts, derive from the sub-problems in the second optimization level. There are two main applications of Benders: when variables x1 make the problem too complex, and/or when the master and the sub-problems are different in nature. Mathematically, Eq. (A.1) can be reformulated as follows:
min x1 ;h2
cT1 x1
80
114.6 144.2 131.7
Table B.22 Branch loading checking: final master. Load pattern Classic BD Winter
Summer
LSC [%]
Bus from
To
Loading [MW]
Rating [MW]
Perc. [%]
60 80
17 8 17 30 37
30 30 30 38 38
175.1 175.5 179.9 187.0 180.2
175 175 175 175 175
100.1 100.3 102.8 106.9 103.0
60
17 26 5
30 30 8
175.4 176.7 175.2
175 175 175
100.2 101.0 100.1
60 80
17 17 26 37
30 30 30 38
175.1 175.6 177.0 178.9
175 175 175 175
100.1 100.3 101.2 102.3
60
5 17 5 37
8 30 8 38
175.2 175.3 176.4 175.8
175 175 175 175
100.1 100.2 100.8 100.4
80 Initialized BD Winter
Summer
80
þ h2 ðx1 Þ ðA:2Þ
A1 x1 ¼ b1 The function h2 ðx1 Þ represents the objective function of the following sub-problem:
h2 ðx1 Þ ¼ mincT2 x2 x2
A 2 x2 ¼ b 2 B 1 x1 : p 2
ðA:3Þ
Table C.23 Mapping between units and buses. Bus Unit
10 4
25 10
27 12
59 24
61 25
80 36
92 43
99 44
100 45
113 53
65
J. Alemany, F. Magnago / Electrical Power and Energy Systems 66 (2015) 53–66 Table C.24 Initialized Benders status. Gen
Hr 10
it1 4 10 12 24 25 36 43 44 45 53 it2 4 10 12 24 25 36 43 44 45 53 it3 4 10 12 24 25 36 43 44 45 53 it4 4 10 12 24 25 36 43 44 45 53 it5 4 10 12 24 25 36 43 44 45 53
Table C.25 Classic Benders status.
1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
Gen 11 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
12 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
13 1 0 0 1 1 1 1 1 1 0
14 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
15 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
16 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
17 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
18 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 1
1 0 0 1 1 1 1 1 1 1
19 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
20 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
21 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
22 1 0 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
1 0 0 1 1 1 1 1 1 0
where p2 are the constraints dual variables (known as shadow prices). Then, the original two-level problem represented by Eq. (A.1) can be reformulated as follows:
min cT1 x1 þ h2 x1 ;h2
A1 x1 ¼ b1 T
h2 P ðb2 B1 x1 Þ : p12 ... T
h2 P ðb2 B1 x1 Þ : pm2
ðA:4Þ
Hr 10
11
12
13
14
15
16
17
18
19
20
21
22
it1 4 5 10 24 25 36 43 44 45 53
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
it2 4 5 10 24 25 36 43 44 45 53
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 0
it3 4 5 10 24 25 36 43 44 45 53
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 0
it4 4 5 10 24 25 36 43 44 45 53
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
0 1 0 1 1 1 1 1 1 0
it5 4 5 10 24 25 36 43 44 45 53
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
it6 4 5 10 24 25 36 43 44 45 53
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
1 1 0 1 1 1 1 1 1 0
where m is the maximum number of cuts. Eqs. (A.4) represent the complete master problem since contains all possible cuts. Instead of including all cuts at the same time in the master problem, the BD algorithm permits including one cut per iteration.
66
J. Alemany, F. Magnago / Electrical Power and Energy Systems 66 (2015) 53–66
Results validation Tables B.21 and B.22 correspond to results of Benders based SCUC calculated using PSSÒE. SCUC patterns Table C.23 shows the mapping between buses and units. Tables C.24 and C.25 correspond to results of Benders based SCUC patterns for the summer 60% LSC. Patterns fully 0 or 1 are not presented. Hour 10–22, where the load peak is, are presented. References [1] Pinto H et al. Security constrained unit commitment: network modeling and solution issues. In: Power systems conference and exposition. IEEE PES; 2006. p. 1759–66. [2] Samiee M et al. Security constrained unit commitment of power systems by a new combinatorial solution strategy composed of enhanced harmony search algorithm and numerical optimization. Int J Electr Power Energy Syst 2013;44(1):471–81. [3] Conejo A et al. Decomposition techniques in mathematical programming: engineering and science applications. Berlin: Springer-Verlag; 2006. [4] Granville S. Mathematical decomposition techniques for power system expansion planning: decomposition methods and uses. In: Electric Power Research Institute and Stanford University, Systems Optimization Laboratory. The Institute Publisher; 1988. [5] Parastegari M et al. AC constrained hydro-thermal generation scheduling problem: application of Benders decomposition method improved by BFPSO. Int J Electr Power Energy Syst 2013;49:199–212. [6] Pereira M et al. A decomposition approach to automated generation/ transmission expansion planning. IEEE Trans Power Ap Syst 1985;PAS104(11):3074–83. [7] Asadamongkol S. Transmission expansion planning with AC model based on generalized Benders decomposition. Int J Electr Power Energy Syst 2013;47:402–7. [8] Sifuentes W et al. Hydrothermal scheduling using Benders decomposition: accelerating techniques. IEEE Trans Power Syst 2007;22(3):1351–9. [9] Sifuentes W et al. Short-term hydrothermal coordination considering an AC network modeling. Int J Electr Power Energy Syst 2007;29(6):488–96. [10] Alguacil N et al. Multiperiod optimal power flow using Benders decomposition. IEEE Trans Power Syst 2000;15(1):196–201.
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