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Efficient period elimination Benders decomposition for network‐constrained AC unit commitment

Efficient period elimination Benders decomposition for network‐constrained AC unit commitment NomenclatureIndicesg, NGindex and number of generation unitst, NTindex and number of time periodss, NSindex and number of generator bid stepsi, j, NIindices and number of busesk, m, NKindices and number of regionsnindex of Benders cuts iterationsParametersSUCgstart‐up cost of unit g ($)SDCgshutdown cost of unit g ($)SBg,t,sbid of step s of unit g at period t ($/MWh)LSg,slength of bid step s of unit g (MW)Pg,min, Pg,maxoperational limits of unit g active power (MW)Qg,min, Qg,maxoperational limits of unit g reactive power (MVar)URg, DRgup and down ramp rate limits of unit g (MW/h)Tgon, Tgoffminimum up and down times of unit g (h)Mgonnumber of initial periods, in which unit g must be on due to its minimum on timeMgoffnumber of initial periods, in which unit g must be off due to its minimum off timeVimin, Vimaxvoltage magnitude limits of bus i (pu)Sbsystem base MVAYi,t∠θi,tentry i–j of admittance matrix (pu)Pi,jmaxactive power limit of branch i–j (MW)ℏi,gbinary parameter equal to 1 if generator g is at bus iPDi,t, QDi,tactive and reactive loads of bus i at period t (MW, MVar)PRk,mmaxinter‐regional power limit of region k–m (MW)χi,j,k,mbinary parameter equal to 1 if branch i–j connects region k–mε1, ε2convergence tolerances of SP1 and SP2ρweighting factor of power mismatch in SP2VariablesFg,tcost of unit g at period t ($)Ig,tbinary on/off status of unit g at period tSUg,tbinary start‐up status of unit g at period tSDg,tbinary shutdown status of unit g at period tPg,t, Qg,ttotal active and reactive powers of unit g at period t (MW, MVar)CSg,t,scommitted step s of unit g at period t (MW)αg,t,sbinary status of step s of unit g at period tβg,t,sbinary variable equal to 1 if step s of unit g at period t is fully committedVi,t∠δi,tvoltage phasor of bus i at period t (pu)Pi,j,t, Qi,j,tactive and reactive powers of branch i–j at period t (MW, MVar)PRk,m,tpower flowing in region k–m at period t (MW)Ai,t+, Ai,t−positive slack variables to model active power mismatch of buses (MW)Bi,t+, Bi,t−positive slack variables to model reactive power mismatch of buses (MVar)γg,t, πg,tmarginal value of unit active powers in SP1 and SP2φ_g,t, φ¯g,tmarginal values of unit reactive limits in SP1Ci,j,t+, Ci,j,t−positive slack variables to model branch flow violations (MW)Dk,m,t+, Dk,m,t−positive slack variables to model inter‐regional transaction violations (MW)μ_g,t, μ¯g,tmarginal values of unit reactive limits in SP2cos¯i,j,tlinear approximation of the cosine function bounded to [0,1]IntroductionMotivation and aimNetwork‐constrained unit commitment (NCUC) is one of the computationally challenging algorithms used by the independent system operator in electricity markets to determine an economical operating point with power system constraints. The prevailing constraints of the NCUC optimisation problem include bus power balances, units’ minimum on/off times and ramping limits, and system spinning reserve requirements. The NCUC problem is generally solved using Lagrangian relaxation or mixed‐integer programming techniques, where each technique has its advantages and disadvantages [1]. Since NCUC is a large‐scale operational problem, in practical applications it needs computationally efficient methods with enough accuracy. Benders decomposition (BD) is an efficient and popular mathematical algorithm that has been applied to solve NCUC problems [2]. Consequently, a modified BD with enhanced speed will be valuable in practical applications of AC NCUC problems.To increase the solution speed of NCUC, different models can also be employed. One of them that is widely used in the literature is to employ the DC model of the network, which is simple and easy to solve. Nevertheless, the accuracy may be sacrificed in the DC model due to the elimination of voltage and reactive powers [3, 4]. In addition, the DC model cannot evaluate power losses due to using a lossless model. On the other hand, the network full AC model can be used for a more precise solution at higher computational cost because the AC formulation presents a non‐convex mixed‐integer non‐linear programming (MINLP) problem [4], which is time‐consuming and hard‐to‐solve. To cope with this, the AC model is linearised in some works such as [5–7] to preserve accuracy and speed.Literature reviewNCUC/security‐CUC (SCUC) is researched from different aspects in literature employing BD as the solution algorithm. For instance, in [8], transient stability constraints are added to the standard SCUC without differential‐algebraic equations. Stabilisation Benders cuts are introduced to model transient stability in subproblems (SPs). In [9], utility‐scale energy storage (such as batteries) is included in the corrective SCUC as almost instant injections following a contingency to give time to slow‐ramped units to adjust their outputs. In [10], SCUC with robust optimisation is proposed to model wind and load uncertainties using an uncertain interval concept. It eliminates less probable scenarios to further limit the conservatism level of the robust solution. In [11], the NCUC is solved using a tri‐level decomposition algorithm with primal and dual cutting planes to schedule uncertain wind power. In [12], a two‐stage approach is proposed to solve the NCUC for day‐ahead energy scheduling with demand response.The speed enhancement of BD is also addressed in the literature. In [13], a two‐stage robust SCUC is proposed to model uncertainties of wind power and contingencies using a modified BD. The modified BD uses a relaxed linear programming (LP) SP instead of the original mixed‐integer LP (MILP) SP to generate valid Benders cuts. Dai et al. in [2] accelerated BD by generating multiple Benders cuts to further narrow the feasible region of the master problem (MP) by identifying worst‐case realisations. The method in [14] accelerated the BD by proposing higher‐density multiple strong Benders cuts, which are Pareto optimal, among candidates from multiple dual optimal solutions. In [15], local branching is used to accelerate the classical BD algorithm by improving both lower and upper bounds. Alemany and Magnago in [16] proposed a methodology to initialise BD by including inexpensive network signals that can be added during the initial MP. In [17], a BD decomposition is proposed to solve the NCUC problem by embedding network evaluation SPs into the branch‐and‐bound/branch‐and‐cut procedure of the unit commitment (UC) master SP. However, the literature works, some of which are reviewed above, did not focus on the BD procedure to enhance its computational efficiency by eliminating unnecessary calculations.Although the AC model is used in some works such as [18–20], it offers a non‐convex problem with a challenging computational burden. To this end, some other works linearised the AC model to derive a convex model for higher speed; for instance, in [4], a stochastic SCUC is proposed to increase the penetration level of wind energy by using a linearised AC optimal power flow (OPF) model for calculation of power losses and bus voltages. Or, Nick et al. in [7] presented an SCUC integrated with a dynamic line rating. The AC OPF is linearised and heat‐balance equations of overhead lines are taken into account. In [6], a piecewise linear model of AC power flow is applied to model the voltage and reactive power in power system islanding.Contributions and paper organisationIn this paper, we propose a modified version of BD to reduce the computational burden of the AC NCUC problem. The main non‐convex problem is convexified using Taylor series and piecewise linear approximations, and then the main problem is decomposed into an MP – formulated as MILP problem for the UC – and two SPs – formulated as LP problems for branch rating checks. Benders cuts are generated from violations of SPs to be supplied back into the MP in next iterations.It is noted that in the original BD, all time periods are solved together during iterations until the algorithm has converged. However, most time periods become without violations after early iterations up to the convergence. In fact, the procedure of solving all time periods together not only wastes CPU time, but also makes it harder to find the solution by lengthening its procedure especially when the AC formulation is considered for large‐scale power systems [4].On the other hand, in our proposed method called here period‐elimination BD (PEBD), we have one MP for UC, subproblem 1 (SP1) to ensure power balance at buses, and subproblem 2 (SP2) to ensure branch security limits. In the classical BD, SP1 and SP2 are solved for all time periods at each iteration. However, in our PEBD algorithm, a period that has converged in an iteration of SP1 is eliminated from next iterations of SP1, and therefore only remaining time periods are solved until all time periods become without violations. When a time period satisfies SP1, it is solved for SP2. To keep the SP2 solution satisfy SP1, the objective function of SP1 is added to SP2. Therefore, the satisfaction of SP2 implies that all time periods are satisfied in both SP1 and SP2. Each time period that is satisfied in SP2 is eliminated from the SP2 solution procedure. Finally, since time periods are interrelated by generator up/down ramp rates, minimum up/down times, as well as start‐up/shutdown costs, it is necessary to check all of them with full periods accounted after satisfying SP1 and SP2 because some of the time periods that were removed from the solution procedure may not be satisfied due to interrelation constraints. Then, the obtained solution may need a few more iterations to fully converge. As it will be shown by numerical results in Section 4, the proposed PEBD algorithm is more efficient than the classical BD and saves CPU time. This time saving is critical for operational problems, where the operator should get an answer in a given time. We here test the PEBD in the NCUC with the stepwise model for generator bidding.It is noted that though we do not consider contingencies, the proposed PEBD algorithm can be easily extended to SCUC applications. Also, in the literature, the SCUC/NCUC problem is studied both in deterministic frameworks (such as [7–9]) or in stochastic frameworks to consider uncertainty in renewable energies or power system components (such as [4, 13, 21]). Or, the SCUC/NCUC can be formulated to consider power system stability [8]. However, we here formulate the proposed PEBD in a basic NCUC framework to better focus on its features. Nevertheless, the PEBD approach can be applied to more complex applications, where there is a higher need for faster methods due to the larger dimensionality of optimisation problems.It should be noted that there are a variety of Benders methods in the literature [22]. Stabilisation [23] and/or inexact cuts [24] speed up the convergence especially in large‐scale SPs [25]. However, we here apply the period‐elimination procedure to the classical BD in order to focus on PEBD features. The proposed PEBD algorithm can also be implemented with stabilisation and/or inexact cuts to further speed up the NCUC problem.The rest of this paper is organised as follows. In Section 2, the non‐convex model of the original AC NCUC formulation is expressed. In Section 3, the proposed PEBD method is delineated with the convexified model of NCUC. In Section 4, the proposed method is tested on two test systems and its efficiency is discussed. Finally, Section 5 concludes this paper.Non‐convex AC NCUC problemThe non‐convex AC NCUC is modelled using (1)–(30). We convexify this model in the next section to implement the proposed PEBD algorithm1Min∑g=1NG∑t=1NTFg,t+SUCg⋅SUg,t+SDCg⋅SDg,t2Fg,t=∑s=1NSSBg,t,s⋅CSg,t,s,∀g,∀t3Pg,t=∑s=1NSCSg,t,s,∀g,∀t40≤CSg,t,s≤LSg,s⋅αg,t,s,∀g,∀t,∀s5αg,t,s+1≤αg,t,s,∀g,∀t,∀s6βg,t,s≤αg,t,s,∀g,∀t,∀s7CSg,t,s≥LSg,s⋅βg,t,s,∀g,∀t,∀s8βg,t,s≤12αg,t,s+αg,t,s+1,∀g,∀t,∀s9βg,t,s≥αg,t,s+αg,t,s+1−1,∀g,∀t,∀s10Ig,t=αg,t,s1,∀g,∀t11SUg,t−SDg,t=Ig,t−Ig,t−1,∀g,∀t12SUg,t+SDg,t≤1,∀g,∀t13Pg,minIg,t≤Pg,t≤Pg,maxIg,t,∀g,∀t14Qg,minIg,t≤Qg,t≤Qg,maxIg,t,∀g,∀t15Pg,t−Pg,t−1≤URg,∀g,∀t16Pg,t−1−Pg,t≤DRg,∀g,∀t17∑t=1Mgon1−Ig,t=0,∀g18∑τ=tt+Tgon−1Ig,τ≥TgonIg,t−Ig,t−1,∀g,∀t∈Mgon+1,NT−Tgon+119∑τ=tNTIg,τ−Ig,t−Ig,t−1≥0,∀g,∀t∈NT−Tgon+2,NT20∑t=1MgoffIg,t=0,∀g21∑τ=tt+Tgoff−11−Ig,τ≥TgoffIg,t−1−Ig,t∀g,∀t∈Mgoff+1,NT−Tgoff+122∑τ=tNT1−Ig,τ−Ig,t−1−Ig,t≥0,∀g,∀t∈NT−Tgoff+2,NT23Vimin≤Vi,t≤Vimax,∀i,∀t24Pi,j,t=SbGi,jVi,t2−Gi,jVi,tVj,tcosδi,t−δj,t−Bi,jVi,tVj,tsinδi,t−δj,t,∀i,∀j,∀t25Qi,j,t=Sb−Bi,jVi,t2+Bi,jVi,tVj,tcosδi,t−δj,t−Gi,jVi,tVj,tsinδi,t−δj,t,∀i,∀j,∀t26−Pi,jmax≤Pi,j,t≤Pi,jmax,∀i,∀j,∀t27∑g=1NGPg,t⋅ℏi,g=PDi,t+∑j=1NIPi,j,t,∀i,∀t28∑g=1NGQg,t⋅ℏi,g=QDi,t+∑j=1NIQi,j,t,∀i,∀t29PRk,m,t=∑i=1NI∑j=1NIPi,j,t⋅χi,j,k,m,∀k,∀m,∀t30−PRk,mmax≤PRk,m,t≤PRk.mmax∀k,∀m,∀t.The cost‐objective function is minimised in (1) including the bid of units as well as start‐up and shutdown costs. The total cost of each unit is given by (2) as the sum of committed bid steps. Active power of units is obtained in (3) as the sum of their committed bid steps using stepwise bids. Constraint (4) ensures that the committed bid steps of units be less than their step length. Equation (5) states that the next bid step can be committed if its preceding bid step is committed. In a unit, only the last bid step can be partially committed; this is constrained by (6) and (7). Logical constraints between α and β are modelled by (8) and (9). Commitment state of a unit at each period is obtained from its first bid step commitment in (10). Logical constraints between start‐up and shutdown status of units are modelled by (11) and (12). Equations (13) and (14) constrain active and reactive powers of units to their operational limits. Constraints (15) and (16) consider ramp up/down limits of units, respectively. Minimum up time of units is modelled by (17)–(19) [26]. The initial periods during which unit g must be online is constrained by (17). Also, (18) and (19) observe up time of units for middle and last time periods, respectively. Similarly, (20)–(22) impose minimum down time of units. Up/down times of units can also be modelled using alternative approaches [2]. Voltage limits of buses are dictated by (23). Active and reactive powers flowing through branches are calculated by (24) and (25); (26) constraint branch flows to its operational limits. Active and reactive power balances at every bus is constrained by (27) and (28), respectively. Power exchange between regions is calculated by (29) and is limited to their permitted values by (30).Convexified AC NCUC and the proposed PEBDWe here introduce the PEBD algorithm as an enhanced version of BD to more efficiently solve NCUC problems. The PEBD consists of an MP and two SPs, where the SP1 imposes AC power flow balance and the SP2 imposes rating of inter‐regional power transactions. Both SPs are convexified in the proposed approach since a linear/convex model offers diverse advantages over non‐linear models such as: (i) the elapsed solution time by a linear model is less than non‐linear ones and this is vital in operational problems such as NCUC; (ii) non‐linear models may be intractable in large‐scale power systems; and (iii) BD generally needs a convex model.Master problemThe MP is solved for optimal generation and commitment of units in the planning horizon. At each iteration, MP solves UC subject to unit constraints and Benders cuts produced from SPs. Network constraints and unit reactive powers are not modelled in the MP. The output solution of MP is then transferred to SPs to check power balance at buses and branch and inter‐area ratings. The MP is formulated as an MILP optimisation problem with the objective function of (1) subject to (2)–(13) and (15)–(22). Also, the generation adequacy is added to the MP as a whole regardless of network configuration31∑g=1NGPg,t≥∑i=1NIPDi,t,∀t.Subproblem 1This SP ensures power balance at every bus for all time periods. Inasmuch as (24) and (25) have non‐linear terms, we here convexify them [6] to make their use possible in BD SPs32Vi,t2≃2Vi,t−1,∀i,∀t33Vi,tVj,tsinδi,t−δj,t≃δi,t−δj,t,∀i,∀j,∀t34Vi,tVj,tcosδi,t−δj,t≃Vi,t+Vj,t+cos¯i,j,t−2,∀i,∀j,∀tIn (32), quadratic voltage is linearised using Taylor series expansion around the voltage of 1 pu [6]. The sinus term in (33) is expanded around the angle of zero [6] since the phase angle difference of branches is typically very small. In (34), the non‐linear cosδi,t−δj,t function is approximated by a new decision variable cos¯i,j,t through relaxing the cosine function from the equality constraint to a set of inequality constraints. Under this relaxation, binary variables are avoided. As shown in Fig. 1, the relaxation is done by a number of hyperplanes (eight lines in this figure) that are tangent to the cosine function: Lh (·) represents h th hyperplane tangent line [27]. Since phase angle differences are small, it is reasonable to restrict their range to (−π/2, π/2) as shown in Fig. 1. The relaxed feasible region is constrained by inequality constraints of the decision variable cos¯i,j,t [7]35cos¯i,j,t≤Lhδi,t−δj,t,∀i,∀j,∀t,∀h1Fig.Linearisation of the non‐linear cosine functionIn Fig. 1, the solid black curve shows the cos(·) function, the thinner lines are tangents, and the shaded area is the feasible region that is constrained by polyhedral relaxations. To increase the accuracy of relaxation, the relaxed cos¯i,j,t value should be maximised to be close to the non‐linear cosine function. To this end, the maximisation ∑∀i,∀j,∀tcos¯i,j,t can be added to the objective function to select the ideal value in the convex hull of the relaxed cosine function [27]. However, some classic objective functions such as minimising generation costs or branch losses have a side effect on minimising the cosine variable [6, 27]. In other words, such classic objective functions intend to choose the ideal value in the relaxed area of Fig. 1. By this assumption, we can employ the relaxation of cosine function in our model since our objective function is the generation cost.As a result of substituting the linearised terms, non‐convex equations of (24) and (25) can be linearised as36Pi,j,t≃SbGi,j2Vi,t−1−Bi,jδi,t−δj,t−Gi,jVi,t+Vj,t+cos¯i,j,t−2∀i,∀j,∀t37Qi,j,t≃Sb−Bi,j2Vi,t−1−Gi,jδi,t−δj,t+Bi,jVi,t+Vj,t+cos¯i,j,t−2∀i,∀j,∀tTherefore, the non‐linear terms are linearised and the convex form of the SP1 optimisation problem is formulated as38MinωtI^g,t,P^g,t=∑i=1NIAi,t++Ai,t−+Bi,t++Bi,t−,∀t39Pg,t=P^g,t→γg,t,∀g,∀t40∑g=1NGPg,t⋅ℏi,g+Ai,t−=Ai,t++PDi,t+∑j=1NIPi,j,t,∀i,∀t41∑g=1NGQg,t⋅ℏi,g+Bi,t−=Bi,t++QDi,t+∑j=1NIQi,j,t,∀i,∀t42Qg,minI^g,t≤Qg,t≤Qg,maxI^g,t→φ_g,t,φ¯g,t,∀g,∀tand (23), (35)–(37).In (38), the total mismatch of active and reactive powers at buses are minimised as the sum of positive slack variables. Equation (39) sets generation of units as obtained from the MP; the ^ symbol indicates the values obtained from the solution of MP and the → symbol represents the marginal value of constraints that will be used later in constructing Benders cuts. In (40), the left‐hand side summation represents the total generation at bus i, whereas the right‐hand side gives the active power being supplied to demand and leaving the bus. Positive slack variables of A− and A+ specify active generation deficiency and surplus, respectively. For instance, if the generation is deficient in (40), a positive non‐zero Ai,t− adjusts equality. Equation (41) imposes similar constraints for reactive powers of buses with slack variables of B+ and B−. Equation (42) imposes reactive power limits of generators.After solving SP1, if its objective function for an individual time period exceeds the tolerance of ɛ1, it means that SP1 is not yet satisfied for that period. Therefore, its corresponding Benders cuts are generated as follows to be added to the MP in the next iterations to reschedule the commitment and generation of units to satisfy SP1: (see (43)).43SP1Cutt,n:ω^t,n+∑g=1NGγg,tnPg,t−P^g,t+∑g=1NGφ¯g,tnQg,maxIg,t−I^g,t−∑g=1NGφ_g,tnQg,minIg,t−I^g,t≤0,∀t,∀nOnce SP1 is satisfied for a period, flow constraints are checked by SP2. All Benders cuts are saved to be added to the next iteration of MP. Bus voltages obtained from SP1 are also saved to be used as the initial point for SP2 to accelerate its procedure.Subproblem 2SP2, which ensures branch flow and inter‐regional power transaction limits, is formulated as44MinξtI^g,t,P^g,t=∑i=1NI∑j=1NICi,j,t++Ci,j,t−+∑k=1NK∑m=1NKDk,m,t++Dk,m,t−+ρ∑i=1NIAi,t++Ai,t−+Bi,t++Bi,t−,∀t45Pg,t=P^g,t→πg,t,∀g,∀t46Pi,j,t−Ci,j,t+≤Pi,jmax,∀i,∀j,∀t47Pi,j,t+Ci,j,t−≥−Pi,jmax,∀i,∀j,∀t48PRk,m,t−Dk,m,t+≤PRk,mmax,∀k,∀m,∀t49PRk,m,t+Dk,m,t−≥−PRk,mmax,∀k,∀m,∀t50Qg,minI^g,t≤Qg,t≤Qg,maxI^g,t→μ_g,t,μ¯g,t,∀g,∀tand (23), (29), (35)–(37), (40), and (41).The objective function in (44) minimises violations of branch flows and inter‐regional powers by modelling violations as positive slack variables. Power mismatch of SP1 are also added to (44) in the form of a penalty term (the last summation) in order to remove the need to solve SP1 when SP2 is solved. In other words, SP2 is impacted by SP1 due to the penalty term. Equation (45) uses generation of units as obtained from MP. Violations of branch flow limits are modelled by positive slack variables by (46) and (47) when line flow is positive and negative, respectively. Similarly, constraints (48) and (49) model inter‐regional power flows. Constraint (50) sets reactive limits of units.For an individual time period, if the objective function SP2 is obtained less than its tolerance (ε2), SP2 is satisfied implying that all violations of the branch and inter‐regional flows are mitigated. Otherwise, Benders cuts are generated for the individual time period to be added to the MP in next iterations (see (51)).51SP2Cutt,n:ξ^t,n+∑g=1NGπg,tnPg,t−P^g,t+∑g=1NGμ¯g,tnQg,maxIg,t−I^g,t−∑g=1NGμ_g,tnQg,minIg,t−I^g,t≤0∀t,∀n.Bus voltages at each iteration are saved to be used in the next iteration as a starting point to accelerate the solution procedure.Algorithm of the proposed PEBDThe whole algorithm of the proposed PEBD is depicted in Fig. 2 with its main blocks of MP, SP1, SP2, and interrelation check. At the first stage, the MP block is solved. Then, the power balance of buses is checked by SP1 for each time period. If SP1 is not satisfied (i.e. there are power mismatch at buses), its Benders cut is generated and this process continues for all time periods (the number of time periods is considered as 24 h in the algorithm). Each time period that satisfies SP1 is eliminated from the SP1 solution procedure to enhance computational efficiency. Probable power mismatch of next iterations are mitigated along with branch violations using the penalty term in (44). For an individual period that is eliminated from SP1, SP2 is checked. This means that SP2 is not solved for a period before satisfying SP1. After solving SP2, if it is not satisfied, its Benders cut is generated to be added to MP at the next iteration. Otherwise, if SP2 is satisfied for a time period, it is eliminated from next iterations to speed up calculations. As a result of these eliminations, the computational efficiency of PEBD is enhanced. When SP2 is satisfied for all periods, there is no need to check SP1 because SP2 has a penalty term from SP1 and the satisfaction of SP2 implies that SP1 is also satisfied. After satisfying all time periods in SP2, the interrelation check block is executed. It is noted that Benders cuts of remaining periods may alter the schedule of previously eliminated periods due to coupling constraints between time periods (ramp up/down, minimum up/down times, and start‐up/shutdown cost of units) in the MP. Therefore, the last block of the flowchart is run to mitigate probable violations considering ALL time periods together by solving SP2. A few more iterations may be required to finalise the solution to satisfy interrelation constraints of time periods. About the convergence of PEBD, it is noted that in the last stage of the flowchart of Fig. 2 (interrelation check), no period elimination is done and the problem is run such as a classical BD; however, the problem has a good initial condition from the BD cuts obtained in preceding iterations. As a result, the convergence of the PEBD at last iterations acts as the classical BD.2Fig.Proposed PEBD algorithmIn another simpler version of PEBD, which we call it here simple PEBD (SPEBD), SP1 is eliminated from a period only if SP2 is eliminated. In a case where SP2 is not satisfied for a time period, SP1 has to be solved even if SP1 was satisfied in previous iterations. In addition, the SP2 objective function of SPEBD does not have the penalty term as shown in (44). Finally, the interrelation check in the SPEBD is done using classical BD. In the next section, we compare the performance of PEBD with SPEBD and classical BD.The PEBD could also be designed alternatively to have one joint SP as we call it joint PEBD (JPEBD). In JPEBD, the objective function of the joint SP includes the summation of all slack variables from SP1 and SP2. The joint SP is optimised subject to (23), (29), (35)–(37), (40), (41), and (45)–(50). We later compare the performance of JPEBD with PEDB.Application of PEBD to more complex problemsAs mentioned earlier, the PEBD approach can be applied to more complex BD‐based applications than the basic NCUC problem. For instance, Nasri et al. [21] propose a scenario‐based stochastic NCUC based on BD considering wind generation uncertainty. The original problem is decomposed into an MP and a few SPs, which are solved for each scenario and time period. If all time periods in all scenarios are satisfied in BD iterations, the problem is solved. Otherwise, corresponding Benders cuts are constructed for unsatisfied periods/scenarios to be applied to the MP in next BD iterations. By applying the PEBD to the method of [21], satisfied periods/scenarios are eliminated from BD iterations. The elimination procedure continues when all periods/scenarios are removed. Afterwards, the interrelation is checked for periods/scenarios according to Fig. 2 to consider all periods/scenarios simultaneously for inter‐period constraints. As seen, it is possible to apply the PEBD framework to stochastic NCUC problems.Or as another example, Xu et al. [8] presents a transient stability‐constrained UC based on BD using an MP for UC and a few SPs for network steady‐state security evaluation (NSE) and transient stability assessment (TSA). SPs are solved for each time period and contingency. If all contingencies are both NSE and TSA stable, the problem is solved. Otherwise, BD cuts for NSE and/or TSA are constructed to be added to the MP at next iterations. This procedure continues so that solutions of NSE and TSA become stable for all time periods and contingencies. By applying the PEBD, time periods with stable NSE and TSA are eliminated from BD iterations to accelerate the procedure. Subsequently, NSE and TSA are solved considering all time periods to check interrelation constraints. If NSE and/or TSA are/is not stable for some time periods, Benders cuts and/or stability cuts are/is constructed. After all, contingencies become stable for all periods, the procedure is terminated.As seen from the above more complex methods [8, 21], the proposed PEBD approach can be applied to all problems that are based on BD in order to accelerate their calculations.Case studies and numerical resultsThe proposed approach is tested on the 9‐bus and three‐region IEEE 118‐bus test systems. The 9‐bus test system is selected because of its small size as an illustration just to better explain the PEBD algorithm in detail. Optimisations are implemented using the CPLEX 12.8 solver of GAMS 24.1.2 [28]. It is noted that since the running time of methods may differ at each run, they are executed for 30 times and the average elapsed time is reported in next results. The technical specifications of the computer used for simulations were 2.6 GHz Intel(R) Core(TM) i5‐3230 CPU with 4 GB of RAM.9‐Bus test systemThis test system consists of three generation units, nine branches, and three loads [29]. Stepwise bidding is considered in ten steps for generation units as depicted in Fig. 3. The convergence thresholds of SP1 and SP2 in simulations are considered as ε1 = 0.1 MW and ε2 = 1 MW, respectively.3Fig.Stepwise bidding curves of the 9‐bus test system unitsThe proposed PEBD method converges elapsing an average execution time of 25.13 s with the standard deviation of 0.2664 s. As shown in Fig. 4, the PEBD is faster than SPEBD and classical BD. Although PEBD needs more iterations to converge (nine iterations against eight), it needs less elapsed time. This is due to the fact that iterations of PEBD do not involve all time periods of NCUC and the satisfied periods are eliminated from its solution procedure. As seen from Fig. 4, CPU time in the BD curve increases almost linearly with iterations implying that its iterations need almost equal CPU time; this is expected since the same problem with a fixed problem size is solved over iterations for BD. However, in period‐elimination‐based BD methods especially PEBD, after the early iterations, each iteration needs lower CPU time (the slope of curves gets flattered) because of eliminating satisfied time periods from the solution process. In fact, all time periods in SPEBD and PEBD are satisfied at iterations eight and seven, respectively. In the last two iterations, all time periods are checked due to interrelation among periods, and then these iterations elapse more CPU resource (the slope gets steeper in the figure in the last two iterations). However, the ultimate outcomes of PEBD and SPEBD are that they lead to fewer execution times (25.13 and 38.23 s, respectively) compared with classical BD (41.18 s). Assuming the BD as the benchmark approach, SPEBD and PEBD are 7.1 and 38.9%, respectively, faster than the BD method. The final operation cost of NCUC by BD, SPEBD, and PEBD algorithms is obtained as $202,805.38, $202,805.93, and $202,833, respectively, which are similar. The slight difference happens due to differently achieving mismatch under the convergence thresholds.4Fig.Execution times of PEBD, SPEBD, and classical BD methods in the 9‐bus test systemFig. 5a demonstrates the period‐elimination procedure and how the number of satisfied and checked time periods change with iterations. Numerical values of bars are indicated on top of them for convenience. At the first iteration, all 24 time periods are solved in SP1 to calculate mismatch. Since total mismatch exceeds the threshold of ε1, Benders cuts are calculated to be added to the MP at the next iteration. At iteration 2, total mismatch becomes less than ε1 for 19 periods out of 24 periods. Then, SP2 is solved for these periods and they are eliminated from SP1; mismatch and flow violations will be simultaneously minimised by SP2 in the next iterations for these satisfied periods. Since the objective function of SP2 is obtained larger than ɛ2 for all checked periods, all of these periods are considered at the next iteration of SP2, and then no period elimination happens. Consequently, corresponding mismatch and flow violation cuts are generated and added to iteration 3 of MP. At iteration 3, generation schedule is improved and total mismatch becomes less than ɛ1 for remaining periods. Therefore, at the next iterations of the algorithm, SP1 is neglected for all periods. Thus, SP2 is solved for 24 satisfied periods (19 periods of iteration 2 plus 5 periods of iteration 3). None of 24 periods satisfies SP2, and then no period elimination occurs. Corresponding Benders cuts are constructed to be added to the next iteration of MP. At iteration 4, generation schedule is improved and total mismatch becomes less than ɛ1 for remaining periods. Therefore, at the next iterations of the algorithm, SP1 is neglected for all periods. Since the objective function of SP2 becomes less than ɛ2 for six periods, these periods are eliminated from SP2 at next iterations. Moreover, Benders cuts are built and added to iteration 5 of MP. In iterations 5 and 6, transmission flow violation becomes less than ɛ2 for 14 and 1 periods, respectively. Therefore, these periods are ignored at the next iterations of SP2 and Benders cuts are formed and added to the next iteration of MP. At iteration 7, the objective function of SP2 reaches less than ɛ2 for three remaining periods. However, concerning the effect of coupling constraints on the generation schedule of previously removed periods, all periods are considered at iteration 8. In this iteration, two periods are not yet satisfied, and then their Benders cuts are calculated and added to the MP at the next iteration. Finally, at iteration 9, the optimal generation schedule is obtained and the objective function of SP2 is satisfied for all periods.5Fig.Convergence of the PEBD algorithm in the 9‐bus test system(a) Number of satisfied and checked time periods, (b) Convergence of SP1 and SP2Fig. 5b shows the variations of SP1 and SP2 objective functions over iterations for the worst time period. As seen, SP1 is satisfied at iteration 3 for the worst time period with the mismatch of 0.038, which is smaller than ε1 = 0.1 MW. Therefore, from iteration 3 going on, SP1 is not solved for time periods. Also, SP2 is satisfied at iteration 9 with the objective function of 0.39, which is smaller than ε2 = 1 MW.The final dispatched power of three generation units obtained by the BD and PEBD algorithms is shown in Fig. 6. As seen, the two algorithms result in similar hourly generation dispatches. This is because the PEBD does not alter the solution trend of BD, but it just accelerates the solution process. Total energy supplied by units 1–3 over 24 h is equal to 4516.36, 3180.36, and 2819.50 MWh for BD and 4532.83, 3164.69, and 2819.18 MWh for PEBD, respectively. Unit 1 supplies more energy because of its lower bids (as seen in Fig. 3). Total energy supplied by generators is 10,516.2 MWh for BD and 10,516.7 MWh for PEBD, while the demand required energy is 10,275.3 MWh; the difference of 240.9 MWh (2.34%) and 241.4 MWh (2.35%) is the network energy losses for BD and PEBD algorithms, respectively.6Fig.Optimal generation of units obtained by the BD and PEBD methods in the 9‐bus test systemIEEE 118‐bus test systemIt is expected that the computational advantage of the proposed PEBD algorithm is higher in the larger test system of 118 bus. This test system [29] has three regions (zones) as shown in Fig. 7 with 54 units, 186 lines, 99 loads, and peak load of 4242 MW at hour 20. We here investigate the performance of SPEBD and PEBD algorithms in two cases of NCUC without inter‐regional power limits (case 1) and with inter‐regional power limits (case 2). The convergence thresholds of SP1 and SP2 in simulations are considered as ε1 = 1 MW and ε2 = 3 MW, respectively.7Fig.One‐line diagram of the IEEE 118‐bus test system with three regions8Fig.Execution times of PEBD, SPEBD, and classical BD methods in the 118‐bus test system(a) Case 1, (b) Case 2Fig. 8a compares the execution times of classical BD, SPEBD, and PEBD algorithms in the IEEE 118 bus (case 1). As seen, the BD approaches its solution in 18 iterations elapsing 1091.0 s. However, the SPEBD and PEBD algorithms approach to their solution with 15 iterations (430.5 s) and 13 iterations (277.6 s), respectively. The PEBD and SPEBD algorithms present CPU time saving as much as 74.6 and 60.5% compared with the classical BD algorithm. That is, the computational efficiency of the proposed PEBD method becomes higher in larger test systems (74.6% in the IEEE 118 bus against 38.9% in the 9‐bus). The NCUC operation cost is obtained by BD, SPEBD, and PEBD algorithms as $972,273.62, $972,145.87, and $972,119.29, respectively. All of these solutions are acceptable and their differences are due to convergence tolerance. In a comparison of the JPEBD and PEBD algorithms in case 1, the JPEBD needs 344.0 s to converge, whereas the PEBD converges in 277.6 s. This implies that having two SPs enhances computational efficiency more than one joint SP. Fig. 8b represents the execution times of the three algorithms in case 2, where inter‐regional powers are constrained as 80 MW for regions 1 and 2 and 40 MW for regions 2 and 3. As seen from this figure, the PEBD algorithms converge in 15 iterations which is faster than others. The PEBD and SPEBD algorithms are faster than the classical BD as much as 70.8 and 61.0%, respectively. The operating cost of BD, SPEBD, and PEBD algorithms are $973,127.36, $973,031.50, and $972,949.23, respectively. The costs of case 2 are higher than those of case 1 due to additional inter‐regional constraints. Similar to results of the 9‐bus test system in Section 4.1, the slope of PEBD in Fig. 8 decreases by iterations implying less computational time for last iterations due to the elimination of time periods. The slope increases in one last iteration of SPEBD and in two last iterations of PEBD due to considering all time periods for correlation of interrelation constraints. In case 2, PEBD offers also a faster convergence than JPEBD: JPEBD elapses 474.3 s against 397.7 s of PEBD.The period‐elimination process is plotted in Fig. 9a for the PEBD algorithm in case 2. As seen, the SP1 objective function has reached less than ɛ1 for all periods at iteration 7, and then from iteration 8 up to the end, SP1 is neglected (corresponding bars have zero height in the figure). At iteration 13, the SP2 objective function becomes less than ɛ2 for all time periods. Therefore, all periods are checked at iteration 14 and due to interrelations constraints, the objective function of SP2 exceeds ɛ2 for two periods. Consequently, corresponding Benders cuts are generated and the process continues to iteration 15 to mitigate violations. At this iteration, the objective function of SP2 becomes less than ɛ2 for all periods, and finally, the optimal solution is found.9Fig.Convergence of the PEBD algorithm in the 118‐bus test system in case 2(a) Period‐elimination process, (b) Convergences of SP1 and SP2Fig. 9b demonstrates the convergences of SP1 and SP2 for the PEBD algorithm for the worst period in case 2. As shown, SP1 is satisfied at iteration 7 for all time periods, and then it is ignored in next iterations. The SP2 is satisfied at iteration 13 for all periods, and then all periods are considered at iteration 14 to check the interrelation of periods. Finally, the algorithm converges at iteration 15 with the violation of 2.97 for the worst time period.To observe the sensitivity of execution time with respect to convergence tolerances of SP1 and SP2, values of ɛ1 = 1 MW and ɛ2 = 3 MW are altered by a coefficient and results are depicted in Fig. 10 for case 1 of the IEEE 118‐bus test system. As seen from this figure, the PEBD algorithm outperforms the other algorithms for all tolerance values. By increasing tolerances, the algorithms are more relaxed and execution times decrease. Also, the effect of period elimination becomes less effective in larger tolerances as the difference of BD with the three period‐elimination‐based algorithms (PEBD, JPEBD, and SPEBD) decrease. However, in lower tolerances (where the accuracy of solutions is higher), period‐elimination‐based algorithms are more advantageous. In other words, the efficiency of period elimination becomes higher in tighter tolerances due to the larger number of iterations, in which the elimination of satisfied time periods can save the CPU time more. In all values of convergence tolerance, the JPEBD algorithms performance is close to the PEBD.10Fig.Sensitivity of execution time with respect to SP1 and SP2 tolerancesConclusionsIn this paper, a modified version of BD called PEBD is introduced with enhanced computational efficiency to solve NCUC problems. To speed up the process, the time periods that are satisfied are eliminated from next iterations of the PEBD. A convex model of NCUC with inter‐regional power constraints and generator stepwise bidding is used to evaluate the proposed PEBD algorithm. It is found that the PEBD algorithm saves the execution time of NCUC with respect to the classical BD as much as 38.9 and 74.6% in the two test systems used in simulations, while it approaches the same solution of the classical BD. It is expected that the proposed method has even higher efficiency in larger test systems. This CPU time saving is of vital importance for practical applications of NCUC in electricity markets. The current research is going on to use more efficient BD algorithms such as stability and inexact cuts along with the PEBD to enhance further computational efficiency especially in more sophisticated applications such as renewable energies with uncertainties and stability‐constrained problems.6 References1Fu, Y., Li, Z., Wu, L.: ‘Modeling and solution of the large‐scale security‐constrained unit commitment’, IEEE Trans. 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Efficient period elimination Benders decomposition for network‐constrained AC unit commitment

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© The Authors. IET Generation, Transmission & Distribution published by John Wiley & Sons, Ltd. on behalf of The Institution of Engineering and Technology
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10.1049/iet-gtd.2018.5409
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NomenclatureIndicesg, NGindex and number of generation unitst, NTindex and number of time periodss, NSindex and number of generator bid stepsi, j, NIindices and number of busesk, m, NKindices and number of regionsnindex of Benders cuts iterationsParametersSUCgstart‐up cost of unit g ($)SDCgshutdown cost of unit g ($)SBg,t,sbid of step s of unit g at period t ($/MWh)LSg,slength of bid step s of unit g (MW)Pg,min, Pg,maxoperational limits of unit g active power (MW)Qg,min, Qg,maxoperational limits of unit g reactive power (MVar)URg, DRgup and down ramp rate limits of unit g (MW/h)Tgon, Tgoffminimum up and down times of unit g (h)Mgonnumber of initial periods, in which unit g must be on due to its minimum on timeMgoffnumber of initial periods, in which unit g must be off due to its minimum off timeVimin, Vimaxvoltage magnitude limits of bus i (pu)Sbsystem base MVAYi,t∠θi,tentry i–j of admittance matrix (pu)Pi,jmaxactive power limit of branch i–j (MW)ℏi,gbinary parameter equal to 1 if generator g is at bus iPDi,t, QDi,tactive and reactive loads of bus i at period t (MW, MVar)PRk,mmaxinter‐regional power limit of region k–m (MW)χi,j,k,mbinary parameter equal to 1 if branch i–j connects region k–mε1, ε2convergence tolerances of SP1 and SP2ρweighting factor of power mismatch in SP2VariablesFg,tcost of unit g at period t ($)Ig,tbinary on/off status of unit g at period tSUg,tbinary start‐up status of unit g at period tSDg,tbinary shutdown status of unit g at period tPg,t, Qg,ttotal active and reactive powers of unit g at period t (MW, MVar)CSg,t,scommitted step s of unit g at period t (MW)αg,t,sbinary status of step s of unit g at period tβg,t,sbinary variable equal to 1 if step s of unit g at period t is fully committedVi,t∠δi,tvoltage phasor of bus i at period t (pu)Pi,j,t, Qi,j,tactive and reactive powers of branch i–j at period t (MW, MVar)PRk,m,tpower flowing in region k–m at period t (MW)Ai,t+, Ai,t−positive slack variables to model active power mismatch of buses (MW)Bi,t+, Bi,t−positive slack variables to model reactive power mismatch of buses (MVar)γg,t, πg,tmarginal value of unit active powers in SP1 and SP2φ_g,t, φ¯g,tmarginal values of unit reactive limits in SP1Ci,j,t+, Ci,j,t−positive slack variables to model branch flow violations (MW)Dk,m,t+, Dk,m,t−positive slack variables to model inter‐regional transaction violations (MW)μ_g,t, μ¯g,tmarginal values of unit reactive limits in SP2cos¯i,j,tlinear approximation of the cosine function bounded to [0,1]IntroductionMotivation and aimNetwork‐constrained unit commitment (NCUC) is one of the computationally challenging algorithms used by the independent system operator in electricity markets to determine an economical operating point with power system constraints. The prevailing constraints of the NCUC optimisation problem include bus power balances, units’ minimum on/off times and ramping limits, and system spinning reserve requirements. The NCUC problem is generally solved using Lagrangian relaxation or mixed‐integer programming techniques, where each technique has its advantages and disadvantages [1]. Since NCUC is a large‐scale operational problem, in practical applications it needs computationally efficient methods with enough accuracy. Benders decomposition (BD) is an efficient and popular mathematical algorithm that has been applied to solve NCUC problems [2]. Consequently, a modified BD with enhanced speed will be valuable in practical applications of AC NCUC problems.To increase the solution speed of NCUC, different models can also be employed. One of them that is widely used in the literature is to employ the DC model of the network, which is simple and easy to solve. Nevertheless, the accuracy may be sacrificed in the DC model due to the elimination of voltage and reactive powers [3, 4]. In addition, the DC model cannot evaluate power losses due to using a lossless model. On the other hand, the network full AC model can be used for a more precise solution at higher computational cost because the AC formulation presents a non‐convex mixed‐integer non‐linear programming (MINLP) problem [4], which is time‐consuming and hard‐to‐solve. To cope with this, the AC model is linearised in some works such as [5–7] to preserve accuracy and speed.Literature reviewNCUC/security‐CUC (SCUC) is researched from different aspects in literature employing BD as the solution algorithm. For instance, in [8], transient stability constraints are added to the standard SCUC without differential‐algebraic equations. Stabilisation Benders cuts are introduced to model transient stability in subproblems (SPs). In [9], utility‐scale energy storage (such as batteries) is included in the corrective SCUC as almost instant injections following a contingency to give time to slow‐ramped units to adjust their outputs. In [10], SCUC with robust optimisation is proposed to model wind and load uncertainties using an uncertain interval concept. It eliminates less probable scenarios to further limit the conservatism level of the robust solution. In [11], the NCUC is solved using a tri‐level decomposition algorithm with primal and dual cutting planes to schedule uncertain wind power. In [12], a two‐stage approach is proposed to solve the NCUC for day‐ahead energy scheduling with demand response.The speed enhancement of BD is also addressed in the literature. In [13], a two‐stage robust SCUC is proposed to model uncertainties of wind power and contingencies using a modified BD. The modified BD uses a relaxed linear programming (LP) SP instead of the original mixed‐integer LP (MILP) SP to generate valid Benders cuts. Dai et al. in [2] accelerated BD by generating multiple Benders cuts to further narrow the feasible region of the master problem (MP) by identifying worst‐case realisations. The method in [14] accelerated the BD by proposing higher‐density multiple strong Benders cuts, which are Pareto optimal, among candidates from multiple dual optimal solutions. In [15], local branching is used to accelerate the classical BD algorithm by improving both lower and upper bounds. Alemany and Magnago in [16] proposed a methodology to initialise BD by including inexpensive network signals that can be added during the initial MP. In [17], a BD decomposition is proposed to solve the NCUC problem by embedding network evaluation SPs into the branch‐and‐bound/branch‐and‐cut procedure of the unit commitment (UC) master SP. However, the literature works, some of which are reviewed above, did not focus on the BD procedure to enhance its computational efficiency by eliminating unnecessary calculations.Although the AC model is used in some works such as [18–20], it offers a non‐convex problem with a challenging computational burden. To this end, some other works linearised the AC model to derive a convex model for higher speed; for instance, in [4], a stochastic SCUC is proposed to increase the penetration level of wind energy by using a linearised AC optimal power flow (OPF) model for calculation of power losses and bus voltages. Or, Nick et al. in [7] presented an SCUC integrated with a dynamic line rating. The AC OPF is linearised and heat‐balance equations of overhead lines are taken into account. In [6], a piecewise linear model of AC power flow is applied to model the voltage and reactive power in power system islanding.Contributions and paper organisationIn this paper, we propose a modified version of BD to reduce the computational burden of the AC NCUC problem. The main non‐convex problem is convexified using Taylor series and piecewise linear approximations, and then the main problem is decomposed into an MP – formulated as MILP problem for the UC – and two SPs – formulated as LP problems for branch rating checks. Benders cuts are generated from violations of SPs to be supplied back into the MP in next iterations.It is noted that in the original BD, all time periods are solved together during iterations until the algorithm has converged. However, most time periods become without violations after early iterations up to the convergence. In fact, the procedure of solving all time periods together not only wastes CPU time, but also makes it harder to find the solution by lengthening its procedure especially when the AC formulation is considered for large‐scale power systems [4].On the other hand, in our proposed method called here period‐elimination BD (PEBD), we have one MP for UC, subproblem 1 (SP1) to ensure power balance at buses, and subproblem 2 (SP2) to ensure branch security limits. In the classical BD, SP1 and SP2 are solved for all time periods at each iteration. However, in our PEBD algorithm, a period that has converged in an iteration of SP1 is eliminated from next iterations of SP1, and therefore only remaining time periods are solved until all time periods become without violations. When a time period satisfies SP1, it is solved for SP2. To keep the SP2 solution satisfy SP1, the objective function of SP1 is added to SP2. Therefore, the satisfaction of SP2 implies that all time periods are satisfied in both SP1 and SP2. Each time period that is satisfied in SP2 is eliminated from the SP2 solution procedure. Finally, since time periods are interrelated by generator up/down ramp rates, minimum up/down times, as well as start‐up/shutdown costs, it is necessary to check all of them with full periods accounted after satisfying SP1 and SP2 because some of the time periods that were removed from the solution procedure may not be satisfied due to interrelation constraints. Then, the obtained solution may need a few more iterations to fully converge. As it will be shown by numerical results in Section 4, the proposed PEBD algorithm is more efficient than the classical BD and saves CPU time. This time saving is critical for operational problems, where the operator should get an answer in a given time. We here test the PEBD in the NCUC with the stepwise model for generator bidding.It is noted that though we do not consider contingencies, the proposed PEBD algorithm can be easily extended to SCUC applications. Also, in the literature, the SCUC/NCUC problem is studied both in deterministic frameworks (such as [7–9]) or in stochastic frameworks to consider uncertainty in renewable energies or power system components (such as [4, 13, 21]). Or, the SCUC/NCUC can be formulated to consider power system stability [8]. However, we here formulate the proposed PEBD in a basic NCUC framework to better focus on its features. Nevertheless, the PEBD approach can be applied to more complex applications, where there is a higher need for faster methods due to the larger dimensionality of optimisation problems.It should be noted that there are a variety of Benders methods in the literature [22]. Stabilisation [23] and/or inexact cuts [24] speed up the convergence especially in large‐scale SPs [25]. However, we here apply the period‐elimination procedure to the classical BD in order to focus on PEBD features. The proposed PEBD algorithm can also be implemented with stabilisation and/or inexact cuts to further speed up the NCUC problem.The rest of this paper is organised as follows. In Section 2, the non‐convex model of the original AC NCUC formulation is expressed. In Section 3, the proposed PEBD method is delineated with the convexified model of NCUC. In Section 4, the proposed method is tested on two test systems and its efficiency is discussed. Finally, Section 5 concludes this paper.Non‐convex AC NCUC problemThe non‐convex AC NCUC is modelled using (1)–(30). We convexify this model in the next section to implement the proposed PEBD algorithm1Min∑g=1NG∑t=1NTFg,t+SUCg⋅SUg,t+SDCg⋅SDg,t2Fg,t=∑s=1NSSBg,t,s⋅CSg,t,s,∀g,∀t3Pg,t=∑s=1NSCSg,t,s,∀g,∀t40≤CSg,t,s≤LSg,s⋅αg,t,s,∀g,∀t,∀s5αg,t,s+1≤αg,t,s,∀g,∀t,∀s6βg,t,s≤αg,t,s,∀g,∀t,∀s7CSg,t,s≥LSg,s⋅βg,t,s,∀g,∀t,∀s8βg,t,s≤12αg,t,s+αg,t,s+1,∀g,∀t,∀s9βg,t,s≥αg,t,s+αg,t,s+1−1,∀g,∀t,∀s10Ig,t=αg,t,s1,∀g,∀t11SUg,t−SDg,t=Ig,t−Ig,t−1,∀g,∀t12SUg,t+SDg,t≤1,∀g,∀t13Pg,minIg,t≤Pg,t≤Pg,maxIg,t,∀g,∀t14Qg,minIg,t≤Qg,t≤Qg,maxIg,t,∀g,∀t15Pg,t−Pg,t−1≤URg,∀g,∀t16Pg,t−1−Pg,t≤DRg,∀g,∀t17∑t=1Mgon1−Ig,t=0,∀g18∑τ=tt+Tgon−1Ig,τ≥TgonIg,t−Ig,t−1,∀g,∀t∈Mgon+1,NT−Tgon+119∑τ=tNTIg,τ−Ig,t−Ig,t−1≥0,∀g,∀t∈NT−Tgon+2,NT20∑t=1MgoffIg,t=0,∀g21∑τ=tt+Tgoff−11−Ig,τ≥TgoffIg,t−1−Ig,t∀g,∀t∈Mgoff+1,NT−Tgoff+122∑τ=tNT1−Ig,τ−Ig,t−1−Ig,t≥0,∀g,∀t∈NT−Tgoff+2,NT23Vimin≤Vi,t≤Vimax,∀i,∀t24Pi,j,t=SbGi,jVi,t2−Gi,jVi,tVj,tcosδi,t−δj,t−Bi,jVi,tVj,tsinδi,t−δj,t,∀i,∀j,∀t25Qi,j,t=Sb−Bi,jVi,t2+Bi,jVi,tVj,tcosδi,t−δj,t−Gi,jVi,tVj,tsinδi,t−δj,t,∀i,∀j,∀t26−Pi,jmax≤Pi,j,t≤Pi,jmax,∀i,∀j,∀t27∑g=1NGPg,t⋅ℏi,g=PDi,t+∑j=1NIPi,j,t,∀i,∀t28∑g=1NGQg,t⋅ℏi,g=QDi,t+∑j=1NIQi,j,t,∀i,∀t29PRk,m,t=∑i=1NI∑j=1NIPi,j,t⋅χi,j,k,m,∀k,∀m,∀t30−PRk,mmax≤PRk,m,t≤PRk.mmax∀k,∀m,∀t.The cost‐objective function is minimised in (1) including the bid of units as well as start‐up and shutdown costs. The total cost of each unit is given by (2) as the sum of committed bid steps. Active power of units is obtained in (3) as the sum of their committed bid steps using stepwise bids. Constraint (4) ensures that the committed bid steps of units be less than their step length. Equation (5) states that the next bid step can be committed if its preceding bid step is committed. In a unit, only the last bid step can be partially committed; this is constrained by (6) and (7). Logical constraints between α and β are modelled by (8) and (9). Commitment state of a unit at each period is obtained from its first bid step commitment in (10). Logical constraints between start‐up and shutdown status of units are modelled by (11) and (12). Equations (13) and (14) constrain active and reactive powers of units to their operational limits. Constraints (15) and (16) consider ramp up/down limits of units, respectively. Minimum up time of units is modelled by (17)–(19) [26]. The initial periods during which unit g must be online is constrained by (17). Also, (18) and (19) observe up time of units for middle and last time periods, respectively. Similarly, (20)–(22) impose minimum down time of units. Up/down times of units can also be modelled using alternative approaches [2]. Voltage limits of buses are dictated by (23). Active and reactive powers flowing through branches are calculated by (24) and (25); (26) constraint branch flows to its operational limits. Active and reactive power balances at every bus is constrained by (27) and (28), respectively. Power exchange between regions is calculated by (29) and is limited to their permitted values by (30).Convexified AC NCUC and the proposed PEBDWe here introduce the PEBD algorithm as an enhanced version of BD to more efficiently solve NCUC problems. The PEBD consists of an MP and two SPs, where the SP1 imposes AC power flow balance and the SP2 imposes rating of inter‐regional power transactions. Both SPs are convexified in the proposed approach since a linear/convex model offers diverse advantages over non‐linear models such as: (i) the elapsed solution time by a linear model is less than non‐linear ones and this is vital in operational problems such as NCUC; (ii) non‐linear models may be intractable in large‐scale power systems; and (iii) BD generally needs a convex model.Master problemThe MP is solved for optimal generation and commitment of units in the planning horizon. At each iteration, MP solves UC subject to unit constraints and Benders cuts produced from SPs. Network constraints and unit reactive powers are not modelled in the MP. The output solution of MP is then transferred to SPs to check power balance at buses and branch and inter‐area ratings. The MP is formulated as an MILP optimisation problem with the objective function of (1) subject to (2)–(13) and (15)–(22). Also, the generation adequacy is added to the MP as a whole regardless of network configuration31∑g=1NGPg,t≥∑i=1NIPDi,t,∀t.Subproblem 1This SP ensures power balance at every bus for all time periods. Inasmuch as (24) and (25) have non‐linear terms, we here convexify them [6] to make their use possible in BD SPs32Vi,t2≃2Vi,t−1,∀i,∀t33Vi,tVj,tsinδi,t−δj,t≃δi,t−δj,t,∀i,∀j,∀t34Vi,tVj,tcosδi,t−δj,t≃Vi,t+Vj,t+cos¯i,j,t−2,∀i,∀j,∀tIn (32), quadratic voltage is linearised using Taylor series expansion around the voltage of 1 pu [6]. The sinus term in (33) is expanded around the angle of zero [6] since the phase angle difference of branches is typically very small. In (34), the non‐linear cosδi,t−δj,t function is approximated by a new decision variable cos¯i,j,t through relaxing the cosine function from the equality constraint to a set of inequality constraints. Under this relaxation, binary variables are avoided. As shown in Fig. 1, the relaxation is done by a number of hyperplanes (eight lines in this figure) that are tangent to the cosine function: Lh (·) represents h th hyperplane tangent line [27]. Since phase angle differences are small, it is reasonable to restrict their range to (−π/2, π/2) as shown in Fig. 1. The relaxed feasible region is constrained by inequality constraints of the decision variable cos¯i,j,t [7]35cos¯i,j,t≤Lhδi,t−δj,t,∀i,∀j,∀t,∀h1Fig.Linearisation of the non‐linear cosine functionIn Fig. 1, the solid black curve shows the cos(·) function, the thinner lines are tangents, and the shaded area is the feasible region that is constrained by polyhedral relaxations. To increase the accuracy of relaxation, the relaxed cos¯i,j,t value should be maximised to be close to the non‐linear cosine function. To this end, the maximisation ∑∀i,∀j,∀tcos¯i,j,t can be added to the objective function to select the ideal value in the convex hull of the relaxed cosine function [27]. However, some classic objective functions such as minimising generation costs or branch losses have a side effect on minimising the cosine variable [6, 27]. In other words, such classic objective functions intend to choose the ideal value in the relaxed area of Fig. 1. By this assumption, we can employ the relaxation of cosine function in our model since our objective function is the generation cost.As a result of substituting the linearised terms, non‐convex equations of (24) and (25) can be linearised as36Pi,j,t≃SbGi,j2Vi,t−1−Bi,jδi,t−δj,t−Gi,jVi,t+Vj,t+cos¯i,j,t−2∀i,∀j,∀t37Qi,j,t≃Sb−Bi,j2Vi,t−1−Gi,jδi,t−δj,t+Bi,jVi,t+Vj,t+cos¯i,j,t−2∀i,∀j,∀tTherefore, the non‐linear terms are linearised and the convex form of the SP1 optimisation problem is formulated as38MinωtI^g,t,P^g,t=∑i=1NIAi,t++Ai,t−+Bi,t++Bi,t−,∀t39Pg,t=P^g,t→γg,t,∀g,∀t40∑g=1NGPg,t⋅ℏi,g+Ai,t−=Ai,t++PDi,t+∑j=1NIPi,j,t,∀i,∀t41∑g=1NGQg,t⋅ℏi,g+Bi,t−=Bi,t++QDi,t+∑j=1NIQi,j,t,∀i,∀t42Qg,minI^g,t≤Qg,t≤Qg,maxI^g,t→φ_g,t,φ¯g,t,∀g,∀tand (23), (35)–(37).In (38), the total mismatch of active and reactive powers at buses are minimised as the sum of positive slack variables. Equation (39) sets generation of units as obtained from the MP; the ^ symbol indicates the values obtained from the solution of MP and the → symbol represents the marginal value of constraints that will be used later in constructing Benders cuts. In (40), the left‐hand side summation represents the total generation at bus i, whereas the right‐hand side gives the active power being supplied to demand and leaving the bus. Positive slack variables of A− and A+ specify active generation deficiency and surplus, respectively. For instance, if the generation is deficient in (40), a positive non‐zero Ai,t− adjusts equality. Equation (41) imposes similar constraints for reactive powers of buses with slack variables of B+ and B−. Equation (42) imposes reactive power limits of generators.After solving SP1, if its objective function for an individual time period exceeds the tolerance of ɛ1, it means that SP1 is not yet satisfied for that period. Therefore, its corresponding Benders cuts are generated as follows to be added to the MP in the next iterations to reschedule the commitment and generation of units to satisfy SP1: (see (43)).43SP1Cutt,n:ω^t,n+∑g=1NGγg,tnPg,t−P^g,t+∑g=1NGφ¯g,tnQg,maxIg,t−I^g,t−∑g=1NGφ_g,tnQg,minIg,t−I^g,t≤0,∀t,∀nOnce SP1 is satisfied for a period, flow constraints are checked by SP2. All Benders cuts are saved to be added to the next iteration of MP. Bus voltages obtained from SP1 are also saved to be used as the initial point for SP2 to accelerate its procedure.Subproblem 2SP2, which ensures branch flow and inter‐regional power transaction limits, is formulated as44MinξtI^g,t,P^g,t=∑i=1NI∑j=1NICi,j,t++Ci,j,t−+∑k=1NK∑m=1NKDk,m,t++Dk,m,t−+ρ∑i=1NIAi,t++Ai,t−+Bi,t++Bi,t−,∀t45Pg,t=P^g,t→πg,t,∀g,∀t46Pi,j,t−Ci,j,t+≤Pi,jmax,∀i,∀j,∀t47Pi,j,t+Ci,j,t−≥−Pi,jmax,∀i,∀j,∀t48PRk,m,t−Dk,m,t+≤PRk,mmax,∀k,∀m,∀t49PRk,m,t+Dk,m,t−≥−PRk,mmax,∀k,∀m,∀t50Qg,minI^g,t≤Qg,t≤Qg,maxI^g,t→μ_g,t,μ¯g,t,∀g,∀tand (23), (29), (35)–(37), (40), and (41).The objective function in (44) minimises violations of branch flows and inter‐regional powers by modelling violations as positive slack variables. Power mismatch of SP1 are also added to (44) in the form of a penalty term (the last summation) in order to remove the need to solve SP1 when SP2 is solved. In other words, SP2 is impacted by SP1 due to the penalty term. Equation (45) uses generation of units as obtained from MP. Violations of branch flow limits are modelled by positive slack variables by (46) and (47) when line flow is positive and negative, respectively. Similarly, constraints (48) and (49) model inter‐regional power flows. Constraint (50) sets reactive limits of units.For an individual time period, if the objective function SP2 is obtained less than its tolerance (ε2), SP2 is satisfied implying that all violations of the branch and inter‐regional flows are mitigated. Otherwise, Benders cuts are generated for the individual time period to be added to the MP in next iterations (see (51)).51SP2Cutt,n:ξ^t,n+∑g=1NGπg,tnPg,t−P^g,t+∑g=1NGμ¯g,tnQg,maxIg,t−I^g,t−∑g=1NGμ_g,tnQg,minIg,t−I^g,t≤0∀t,∀n.Bus voltages at each iteration are saved to be used in the next iteration as a starting point to accelerate the solution procedure.Algorithm of the proposed PEBDThe whole algorithm of the proposed PEBD is depicted in Fig. 2 with its main blocks of MP, SP1, SP2, and interrelation check. At the first stage, the MP block is solved. Then, the power balance of buses is checked by SP1 for each time period. If SP1 is not satisfied (i.e. there are power mismatch at buses), its Benders cut is generated and this process continues for all time periods (the number of time periods is considered as 24 h in the algorithm). Each time period that satisfies SP1 is eliminated from the SP1 solution procedure to enhance computational efficiency. Probable power mismatch of next iterations are mitigated along with branch violations using the penalty term in (44). For an individual period that is eliminated from SP1, SP2 is checked. This means that SP2 is not solved for a period before satisfying SP1. After solving SP2, if it is not satisfied, its Benders cut is generated to be added to MP at the next iteration. Otherwise, if SP2 is satisfied for a time period, it is eliminated from next iterations to speed up calculations. As a result of these eliminations, the computational efficiency of PEBD is enhanced. When SP2 is satisfied for all periods, there is no need to check SP1 because SP2 has a penalty term from SP1 and the satisfaction of SP2 implies that SP1 is also satisfied. After satisfying all time periods in SP2, the interrelation check block is executed. It is noted that Benders cuts of remaining periods may alter the schedule of previously eliminated periods due to coupling constraints between time periods (ramp up/down, minimum up/down times, and start‐up/shutdown cost of units) in the MP. Therefore, the last block of the flowchart is run to mitigate probable violations considering ALL time periods together by solving SP2. A few more iterations may be required to finalise the solution to satisfy interrelation constraints of time periods. About the convergence of PEBD, it is noted that in the last stage of the flowchart of Fig. 2 (interrelation check), no period elimination is done and the problem is run such as a classical BD; however, the problem has a good initial condition from the BD cuts obtained in preceding iterations. As a result, the convergence of the PEBD at last iterations acts as the classical BD.2Fig.Proposed PEBD algorithmIn another simpler version of PEBD, which we call it here simple PEBD (SPEBD), SP1 is eliminated from a period only if SP2 is eliminated. In a case where SP2 is not satisfied for a time period, SP1 has to be solved even if SP1 was satisfied in previous iterations. In addition, the SP2 objective function of SPEBD does not have the penalty term as shown in (44). Finally, the interrelation check in the SPEBD is done using classical BD. In the next section, we compare the performance of PEBD with SPEBD and classical BD.The PEBD could also be designed alternatively to have one joint SP as we call it joint PEBD (JPEBD). In JPEBD, the objective function of the joint SP includes the summation of all slack variables from SP1 and SP2. The joint SP is optimised subject to (23), (29), (35)–(37), (40), (41), and (45)–(50). We later compare the performance of JPEBD with PEDB.Application of PEBD to more complex problemsAs mentioned earlier, the PEBD approach can be applied to more complex BD‐based applications than the basic NCUC problem. For instance, Nasri et al. [21] propose a scenario‐based stochastic NCUC based on BD considering wind generation uncertainty. The original problem is decomposed into an MP and a few SPs, which are solved for each scenario and time period. If all time periods in all scenarios are satisfied in BD iterations, the problem is solved. Otherwise, corresponding Benders cuts are constructed for unsatisfied periods/scenarios to be applied to the MP in next BD iterations. By applying the PEBD to the method of [21], satisfied periods/scenarios are eliminated from BD iterations. The elimination procedure continues when all periods/scenarios are removed. Afterwards, the interrelation is checked for periods/scenarios according to Fig. 2 to consider all periods/scenarios simultaneously for inter‐period constraints. As seen, it is possible to apply the PEBD framework to stochastic NCUC problems.Or as another example, Xu et al. [8] presents a transient stability‐constrained UC based on BD using an MP for UC and a few SPs for network steady‐state security evaluation (NSE) and transient stability assessment (TSA). SPs are solved for each time period and contingency. If all contingencies are both NSE and TSA stable, the problem is solved. Otherwise, BD cuts for NSE and/or TSA are constructed to be added to the MP at next iterations. This procedure continues so that solutions of NSE and TSA become stable for all time periods and contingencies. By applying the PEBD, time periods with stable NSE and TSA are eliminated from BD iterations to accelerate the procedure. Subsequently, NSE and TSA are solved considering all time periods to check interrelation constraints. If NSE and/or TSA are/is not stable for some time periods, Benders cuts and/or stability cuts are/is constructed. After all, contingencies become stable for all periods, the procedure is terminated.As seen from the above more complex methods [8, 21], the proposed PEBD approach can be applied to all problems that are based on BD in order to accelerate their calculations.Case studies and numerical resultsThe proposed approach is tested on the 9‐bus and three‐region IEEE 118‐bus test systems. The 9‐bus test system is selected because of its small size as an illustration just to better explain the PEBD algorithm in detail. Optimisations are implemented using the CPLEX 12.8 solver of GAMS 24.1.2 [28]. It is noted that since the running time of methods may differ at each run, they are executed for 30 times and the average elapsed time is reported in next results. The technical specifications of the computer used for simulations were 2.6 GHz Intel(R) Core(TM) i5‐3230 CPU with 4 GB of RAM.9‐Bus test systemThis test system consists of three generation units, nine branches, and three loads [29]. Stepwise bidding is considered in ten steps for generation units as depicted in Fig. 3. The convergence thresholds of SP1 and SP2 in simulations are considered as ε1 = 0.1 MW and ε2 = 1 MW, respectively.3Fig.Stepwise bidding curves of the 9‐bus test system unitsThe proposed PEBD method converges elapsing an average execution time of 25.13 s with the standard deviation of 0.2664 s. As shown in Fig. 4, the PEBD is faster than SPEBD and classical BD. Although PEBD needs more iterations to converge (nine iterations against eight), it needs less elapsed time. This is due to the fact that iterations of PEBD do not involve all time periods of NCUC and the satisfied periods are eliminated from its solution procedure. As seen from Fig. 4, CPU time in the BD curve increases almost linearly with iterations implying that its iterations need almost equal CPU time; this is expected since the same problem with a fixed problem size is solved over iterations for BD. However, in period‐elimination‐based BD methods especially PEBD, after the early iterations, each iteration needs lower CPU time (the slope of curves gets flattered) because of eliminating satisfied time periods from the solution process. In fact, all time periods in SPEBD and PEBD are satisfied at iterations eight and seven, respectively. In the last two iterations, all time periods are checked due to interrelation among periods, and then these iterations elapse more CPU resource (the slope gets steeper in the figure in the last two iterations). However, the ultimate outcomes of PEBD and SPEBD are that they lead to fewer execution times (25.13 and 38.23 s, respectively) compared with classical BD (41.18 s). Assuming the BD as the benchmark approach, SPEBD and PEBD are 7.1 and 38.9%, respectively, faster than the BD method. The final operation cost of NCUC by BD, SPEBD, and PEBD algorithms is obtained as $202,805.38, $202,805.93, and $202,833, respectively, which are similar. The slight difference happens due to differently achieving mismatch under the convergence thresholds.4Fig.Execution times of PEBD, SPEBD, and classical BD methods in the 9‐bus test systemFig. 5a demonstrates the period‐elimination procedure and how the number of satisfied and checked time periods change with iterations. Numerical values of bars are indicated on top of them for convenience. At the first iteration, all 24 time periods are solved in SP1 to calculate mismatch. Since total mismatch exceeds the threshold of ε1, Benders cuts are calculated to be added to the MP at the next iteration. At iteration 2, total mismatch becomes less than ε1 for 19 periods out of 24 periods. Then, SP2 is solved for these periods and they are eliminated from SP1; mismatch and flow violations will be simultaneously minimised by SP2 in the next iterations for these satisfied periods. Since the objective function of SP2 is obtained larger than ɛ2 for all checked periods, all of these periods are considered at the next iteration of SP2, and then no period elimination happens. Consequently, corresponding mismatch and flow violation cuts are generated and added to iteration 3 of MP. At iteration 3, generation schedule is improved and total mismatch becomes less than ɛ1 for remaining periods. Therefore, at the next iterations of the algorithm, SP1 is neglected for all periods. Thus, SP2 is solved for 24 satisfied periods (19 periods of iteration 2 plus 5 periods of iteration 3). None of 24 periods satisfies SP2, and then no period elimination occurs. Corresponding Benders cuts are constructed to be added to the next iteration of MP. At iteration 4, generation schedule is improved and total mismatch becomes less than ɛ1 for remaining periods. Therefore, at the next iterations of the algorithm, SP1 is neglected for all periods. Since the objective function of SP2 becomes less than ɛ2 for six periods, these periods are eliminated from SP2 at next iterations. Moreover, Benders cuts are built and added to iteration 5 of MP. In iterations 5 and 6, transmission flow violation becomes less than ɛ2 for 14 and 1 periods, respectively. Therefore, these periods are ignored at the next iterations of SP2 and Benders cuts are formed and added to the next iteration of MP. At iteration 7, the objective function of SP2 reaches less than ɛ2 for three remaining periods. However, concerning the effect of coupling constraints on the generation schedule of previously removed periods, all periods are considered at iteration 8. In this iteration, two periods are not yet satisfied, and then their Benders cuts are calculated and added to the MP at the next iteration. Finally, at iteration 9, the optimal generation schedule is obtained and the objective function of SP2 is satisfied for all periods.5Fig.Convergence of the PEBD algorithm in the 9‐bus test system(a) Number of satisfied and checked time periods, (b) Convergence of SP1 and SP2Fig. 5b shows the variations of SP1 and SP2 objective functions over iterations for the worst time period. As seen, SP1 is satisfied at iteration 3 for the worst time period with the mismatch of 0.038, which is smaller than ε1 = 0.1 MW. Therefore, from iteration 3 going on, SP1 is not solved for time periods. Also, SP2 is satisfied at iteration 9 with the objective function of 0.39, which is smaller than ε2 = 1 MW.The final dispatched power of three generation units obtained by the BD and PEBD algorithms is shown in Fig. 6. As seen, the two algorithms result in similar hourly generation dispatches. This is because the PEBD does not alter the solution trend of BD, but it just accelerates the solution process. Total energy supplied by units 1–3 over 24 h is equal to 4516.36, 3180.36, and 2819.50 MWh for BD and 4532.83, 3164.69, and 2819.18 MWh for PEBD, respectively. Unit 1 supplies more energy because of its lower bids (as seen in Fig. 3). Total energy supplied by generators is 10,516.2 MWh for BD and 10,516.7 MWh for PEBD, while the demand required energy is 10,275.3 MWh; the difference of 240.9 MWh (2.34%) and 241.4 MWh (2.35%) is the network energy losses for BD and PEBD algorithms, respectively.6Fig.Optimal generation of units obtained by the BD and PEBD methods in the 9‐bus test systemIEEE 118‐bus test systemIt is expected that the computational advantage of the proposed PEBD algorithm is higher in the larger test system of 118 bus. This test system [29] has three regions (zones) as shown in Fig. 7 with 54 units, 186 lines, 99 loads, and peak load of 4242 MW at hour 20. We here investigate the performance of SPEBD and PEBD algorithms in two cases of NCUC without inter‐regional power limits (case 1) and with inter‐regional power limits (case 2). The convergence thresholds of SP1 and SP2 in simulations are considered as ε1 = 1 MW and ε2 = 3 MW, respectively.7Fig.One‐line diagram of the IEEE 118‐bus test system with three regions8Fig.Execution times of PEBD, SPEBD, and classical BD methods in the 118‐bus test system(a) Case 1, (b) Case 2Fig. 8a compares the execution times of classical BD, SPEBD, and PEBD algorithms in the IEEE 118 bus (case 1). As seen, the BD approaches its solution in 18 iterations elapsing 1091.0 s. However, the SPEBD and PEBD algorithms approach to their solution with 15 iterations (430.5 s) and 13 iterations (277.6 s), respectively. The PEBD and SPEBD algorithms present CPU time saving as much as 74.6 and 60.5% compared with the classical BD algorithm. That is, the computational efficiency of the proposed PEBD method becomes higher in larger test systems (74.6% in the IEEE 118 bus against 38.9% in the 9‐bus). The NCUC operation cost is obtained by BD, SPEBD, and PEBD algorithms as $972,273.62, $972,145.87, and $972,119.29, respectively. All of these solutions are acceptable and their differences are due to convergence tolerance. In a comparison of the JPEBD and PEBD algorithms in case 1, the JPEBD needs 344.0 s to converge, whereas the PEBD converges in 277.6 s. This implies that having two SPs enhances computational efficiency more than one joint SP. Fig. 8b represents the execution times of the three algorithms in case 2, where inter‐regional powers are constrained as 80 MW for regions 1 and 2 and 40 MW for regions 2 and 3. As seen from this figure, the PEBD algorithms converge in 15 iterations which is faster than others. The PEBD and SPEBD algorithms are faster than the classical BD as much as 70.8 and 61.0%, respectively. The operating cost of BD, SPEBD, and PEBD algorithms are $973,127.36, $973,031.50, and $972,949.23, respectively. The costs of case 2 are higher than those of case 1 due to additional inter‐regional constraints. Similar to results of the 9‐bus test system in Section 4.1, the slope of PEBD in Fig. 8 decreases by iterations implying less computational time for last iterations due to the elimination of time periods. The slope increases in one last iteration of SPEBD and in two last iterations of PEBD due to considering all time periods for correlation of interrelation constraints. In case 2, PEBD offers also a faster convergence than JPEBD: JPEBD elapses 474.3 s against 397.7 s of PEBD.The period‐elimination process is plotted in Fig. 9a for the PEBD algorithm in case 2. As seen, the SP1 objective function has reached less than ɛ1 for all periods at iteration 7, and then from iteration 8 up to the end, SP1 is neglected (corresponding bars have zero height in the figure). At iteration 13, the SP2 objective function becomes less than ɛ2 for all time periods. Therefore, all periods are checked at iteration 14 and due to interrelations constraints, the objective function of SP2 exceeds ɛ2 for two periods. Consequently, corresponding Benders cuts are generated and the process continues to iteration 15 to mitigate violations. At this iteration, the objective function of SP2 becomes less than ɛ2 for all periods, and finally, the optimal solution is found.9Fig.Convergence of the PEBD algorithm in the 118‐bus test system in case 2(a) Period‐elimination process, (b) Convergences of SP1 and SP2Fig. 9b demonstrates the convergences of SP1 and SP2 for the PEBD algorithm for the worst period in case 2. As shown, SP1 is satisfied at iteration 7 for all time periods, and then it is ignored in next iterations. The SP2 is satisfied at iteration 13 for all periods, and then all periods are considered at iteration 14 to check the interrelation of periods. Finally, the algorithm converges at iteration 15 with the violation of 2.97 for the worst time period.To observe the sensitivity of execution time with respect to convergence tolerances of SP1 and SP2, values of ɛ1 = 1 MW and ɛ2 = 3 MW are altered by a coefficient and results are depicted in Fig. 10 for case 1 of the IEEE 118‐bus test system. As seen from this figure, the PEBD algorithm outperforms the other algorithms for all tolerance values. By increasing tolerances, the algorithms are more relaxed and execution times decrease. Also, the effect of period elimination becomes less effective in larger tolerances as the difference of BD with the three period‐elimination‐based algorithms (PEBD, JPEBD, and SPEBD) decrease. However, in lower tolerances (where the accuracy of solutions is higher), period‐elimination‐based algorithms are more advantageous. In other words, the efficiency of period elimination becomes higher in tighter tolerances due to the larger number of iterations, in which the elimination of satisfied time periods can save the CPU time more. In all values of convergence tolerance, the JPEBD algorithms performance is close to the PEBD.10Fig.Sensitivity of execution time with respect to SP1 and SP2 tolerancesConclusionsIn this paper, a modified version of BD called PEBD is introduced with enhanced computational efficiency to solve NCUC problems. To speed up the process, the time periods that are satisfied are eliminated from next iterations of the PEBD. A convex model of NCUC with inter‐regional power constraints and generator stepwise bidding is used to evaluate the proposed PEBD algorithm. 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Journal

"IET Generation, Transmission & Distribution"Wiley

Published: May 1, 2019

Keywords: power generation dispatch; integer programming; optimisation; power system security; power generation scheduling; power system interconnection; power generation economics; power markets; iteration; SPs; time period; PEBD method; larger power systems; satisfied time periods; efficient period elimination BD; network‐constrained AC unit commitment; recent complexities; interconnected power system operation call; efficient approaches; fast approaches; operational problems including network‐constrained unit commitment; period‐elimination Benders decomposition; computational efficiency; AC NCUC problems; power balance; inter‐regional transactions; classical BD; solution procedure

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