This paper introduces a novel multiobjective approach for capacity benefit margin (CBM) assessment taking into account tie-line reliability of interconnected systems. CBM is the imperative information utilized as a reference by the load-serving entities (LSE) to estimate a certain margin of transfer capability so that a reliable access to generation through interconnected system could be attained. A new Pareto-based evolutionary programming (EP) technique is used to perform a simultaneous determination of CBM for all areas of the interconnected system. The selection of CBM at the Pareto optimal front is proposed to be performed by referring to a heuristic ranking index that takes into account system loss of load expectation (LOLE) in various conditions. Eventually, the power transfer based available transfer capability (ATC) is determined by considering the firm and nonfirm transfers of CBM. A comprehensive set of numerical studies are conducted on the modified IEEE-RTS79 and the performance of the proposed method is numerically investigated in detail. The main advantage of the proposed technique is in terms of flexibility offered to an independent system operator in selecting an appropriate solution of CBM simultaneously for all areas.
1. Introduction
In a deregulated power system environment, electricity is considered as a commodity that can be traded in a free market where the generators and loads participated. The transition to a new structure of electricity market is to ensure the quality and efficient production of electrical energy that can be offered at a lower electricity price as well as maximizing the utilization of generation and transmission facilities [1, 2]. Hence, it is important for the independent system operator (ISO) to calculate and provide the information of available transfer capability (ATC) associated with the transfer paths to the open access same-time information system (OASIS) so that electricity market could be conducted in an effective manner [3, 4]. ATC is defined as the maximum amount of power that can be transferred from a selling area to a buying area without jeopardizing a system security [5]. ATC can be calculated as the total transfer capability (TTC) reduced by the transmission reliability margin (TRM), capacity benefit margin (CBM), and existing transmission commitment (ETC). CBM is one of the main components considered in the ATC calculation and is defined as the amount of transfer capability reserved by load-serving entities, which is anticipated to be used in cases of generation deficiency [5–9]. Inaccurate determination of CBM may result in either underestimation or overestimation of the ATC. Underestimating the ATC value possibility will cause an ineffective use in the transmission facility, while overestimating the ATC value will threaten a power system security [3, 7].
So far, several methods have been proposed to determine CBM [10–19]. The basic method used to compute the CBM for each area of an interconnected system is based on trial and error [10], by prescribing 5% of the maximum transfer capability [11] or the CBM value is specified as zero [12, 13]. Reference [14] has proposed an analytic model used for multiarea generation reliability assessment and then applied into the sequential quadratic programming (SQP) for determining the CBM values considering the loss of load expectation (LOLE) as the system reliability criterion. Rajathy et al. [15] use the differential evolution and Monte Carlo techniques to determine the CBM. A method that has been proposed in [16] is used to determine the CBM for each area of an interconnected system using the evolutionary programming (EP) as an accelerated search technique. Furthermore, CBM determination is formulated as an optimization problem which is solved by using the particle swarm optimization (PSO) technique [17, 18]. In order to provide a set of choices for different cases, three methods have been proposed in [17, 18] which will provide different values of CBM. It is observed that the existing CBM calculations do not provide adequate flexibility for the ISO to select a CBM value in accordance with system requirements [10–19]. In addition, tie-line availability is an influential factor which has an effect on the reliability of an interconnected system followed by the value of CBM. This imperative factor has been taken into account for CBM calculation in [19].
A novel multiobjective based optimization approach is presented in this paper to determine several optimum values of CBM using the Pareto-based EP technique that takes into account the tie-line reliability of an interconnected system. The proposed Pareto-based EP technique has several advantages compared to the methodology previously presented in [16] and it provides the ISO with several choices of optimum CBM values. The multiobjective function of EP technique is referred to as the transfer capability margin of CBM for all areas with LOLE less than a specified value at initial condition. Moreover, the CBMs of all areas are obtained simultaneously at every execution of the proposed technique. The first order sensitivity with modified Gaussian formulation is used as a new mutation technique to enhance the EP performance in searching for a new population at global maximum domain with less computational time. Then, the Pareto optimal front approach is used to select several optimal solutions of CBM values using the ranking index of total LOLE and total difference of LOLE. A modified IEEE-RTS79 is used as the numerical test bed to verify effectiveness of the proposed method in providing the solutions of CBMs [17]. The robustness of the proposed method in CBM determination is compared with that of the basic methodology used for the CBM calculation [17]. Performance comparison has also been performed which investigates the effect of tie-line reliability included in the CBM determination. Finally, the significance of CBM considered as firm and nonfirm transfers can be observed through its impact on the ATC determination.
2. Multiobjective Functions of Capacity Benefit Margins Determination
A process involved in the Pareto-based EP technique used for determining the multiobjective function of CBMs is described as follows.
Step (a). Establish a solved base case power flow solution.
Step (b). Determine the LOLE for each area of the interconnected system at the base case condition.
Step (c). Identify the assisting areas with LOLE less than the specified value, ξ (e.g., 2.4 hrs/yr). It signifies that these areas conserve a certain amount of reserve generating capacity that could be used to compensate for the generation deficiency which may occur in the assisted area. LOLE associated with the assisted area is usually greater than ξ. It is important to mention that the assisting and assisted areas are the terms used to signify the direction of power transfer based CBM (CBMasgPareto) and this is different from the selling and buying areas which are the terms used to signify the direction of power transfer based ATC.
Step (d). Identify the assisted area with the largest LOLE above ξ.
Step (e). Determine the parent or initial population for each assisting area with LOLE below ξ. Equation (1) is used to generate the individuals xparm,asg, for parent or initial population using uniform random distribution. The determination of xparm,asg is based on either total rating of all tie-lines connecting between the assisting and assisted areas, PLIt_{asg}, or the total reserve generating capacity of the assisting area, DPGt_{asg}. The xparm,asg is determined based on the former condition when DPGt_{asg} exceeds the PLIt_{asg}. This means that tie-lines are the constraining factors for power transfer based CBM and, thus, xparm,asg are generated randomly based on PLIt_{asg}. The latter condition is used to determine xparm,asg when DPGt_{asg} is less than PLIt_{asg}. Each individual, xpar, is considered as an external generating capacity, PG_{Ext}, or CBM, which is provided by the assisting area to support generating capacity deficiency in the assisted area having the highest LOLE:(1)xparm,asg=randmDPGtasg,ifDPGtasg<PLItasg,randmPLItasg,ifDPGtasg>PLItasg,where(2)DPGtasg=PGtasg-PLtasg,PLItasg=∑l=1LPLIlasg.CBMm,asg or xparm,asg is the CBM in the case of transfer from assisting area to assisted area; PGt is the total generating capacity; PLt is the total peak load; PLI is the tie-line rating; L is the total number of tie-lines; m is 1,2,3,…,pop; asg is 1,2,3,…,Nasg; pop is the population size; and Nasg is the total number of assisting areas.
Step (f). Calculate a new total generation capacity, newPGtm,asg, for each assisting area according to CBM or xparm,asg as given in (3) and (4). The generating capacity of the assisting area is reduced as it is partially assigned to the assisted area. The new generating capacity for each bus g of the assisting area newPGtm,asg is obtained based on the ratio of generating capacity as(3)newPGtm,asg=∑g=1NGnewPGgm,asg,where(4)newPGgm,asg=PGgasg-PGgasg∑g=1NGPGgasg×xparm,asg.PG is the generating capacity and NG is the total number of generator buses.
Step (g). Determine the LOLE for each assisting area (LOLEm,asg) considering the newPGtm,asg, hourly peak load, and cumulative probability of generation capacity outage (PC(Cs)) as discussed in [19].
Step (h). Determine a new total generation capacity, newPGtm,asd=1, for an assisted area with the largest LOLE above ξ using (5) and (6). In (6), apportionment of the total xparm,asg or total CBMm,asg to each generator is performed based on the ratio of generating capacity and total generating capacity of an assisted area. For an assisted area, there are pop number of individuals for the size of new total generating capacity, newPGtm,asd=1,(5)newPGtm,asd=1=∑g=1NGnewPGgm,asd=1,where(6)newPGgm,asd=1=PGgasd=1+PGgasd=1∑g=1NGPGgasd=1∑asg=1Nasgxparm,asg,where asd is the number of assisted areas, 1.
Step (i). Calculate the fitness value (fm), that is, LOLEm,asd=1 as discussed in [19]. fm is an important parameter used to assist the determination of a new xparm,asg and the convergence criteria for the optimization process. This will be explained thoroughly in the following steps. fm or LOLEm,asd=1 is calculated by taking into account the increased amount of newPGtm,asd=1 obtained in Step (h).
Step (j). Perform the mutation to obtain an offspring for each assisting area with LOLE less than ξ. In the proposed mutation approach, the modified Gaussian technique is used to improve the capability of global maximum search of a new population with less computational time [16]. This technique is suitable in solving the optimization problems in which considerable discrepancy does exist among the individual values. Each offspring comprising new individuals, xoffm,asg, is originated from xparm,asg. The new individuals, xoffm,asg, are obtained using a new mutation technique that incorporates the first order sensitivity, ∂xparasg/∂Nf,ξ,σ, and the modified Gaussian formulation, Nfm,ξ,σ, as expressed in (7). The value of xoffm,asg is varied in accordance with the changes in fm to the estimated LOLE limit, ξ. Consider(7)xoffm,asg=xparm,asg+∂xparasg∂Nf,ξ·σ1-Nfm,ξ,σ,where(8)∂xparasg∂Nf,ξ,σ=maxxparasg-minxparasgmaxNf,ξ,σ-minNf,ξ,σ,Nfm,ξ,σ=e-fm-ξ2/2σ2,where maxxparasg and minxparasg are the maximum and minimum values of xparm,asg for every assisting area, respectively; maxN(f,ξ,σ) and minN(f,ξ,σ) are the maximum and minimum values of N(fm,ξ,σ), respectively; and σ or fmax is the maximum value of fitness, fm or LOLEm,asd=1.
The first order sensitivity is used to overcome the impediment of local maxima or minima which normally occurs in the case of large fm. Hence, robustness in searching for the global maxima or minima can easily be guaranteed by using the new mutation technique.
Step (k). Perform Steps (h) and (i) to determine fm or LOLEm,asd=1 in relation to a new value of newPGtm,asd=1 obtained according to (5) considering xoffm,asg. This implies that the xparm,asg in (6) has been replaced by xoffm,asg, yielding to a new value of newPGtm,asd=1. Apart from the newPGtm,asd=1 obtained based on xoffm,asg, determination of LOLEm,asd=1 also requires several other parameters such as the hourly peak load and new cumulative probability of the generation capacity outage (PC(Cs)) as discussed in [19].
Step (l). Perform pairwise comparison to determine the next generation of population comprising the best individuals selected from xoffm,asg and xparm,asg. For each assisting area, fm or LOLEm,asd=1 has been used as a reference for selecting the best individuals as the next generation of xparm,asg. In this case, fm for xparm,asg and xoffm,asg are obtained from Steps (h) and (j), respectively. The concept of selection is elucidated in terms of the formulation given in (9). Otherwise, when the total number of chosen individuals is not adequate for population size, pop, then the offspring, xoffm,asg, is selected as the next generation of xparm,asg as illustrated in(9)xselm,asg=xoffm=1,asgfm=1xoffm=1,asg<ξ⋮⋮xoffm,asgfmxoffm,asg<ξ⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯xparm=1,asgfm=1xparm=1,asg<ξ⋮⋮xparm,asgfmxparm,asg<ξ,(10)xparm,asg=xselm,asg,ifsizexselm,asg≥pop,xoffm,asg,ifsizexselm,asg<pop,where xselm,asg is the best individuals selected from xoffm,asg and xparm,asg having fm<ξ; fm(xoffm,asg) is the fm corresponding to the mth value of individual xoffm,asg; fm(xparm,asg) is the fm corresponding to the mth value of individual xparm,asg; and size(xselm,asg) is the size of xselm,asg.
Step (m). The convergence criteria for the EP optimization process is achieved when the mismatch between maximum fitness, fmax, and minimum fitness, fmin, is within a specified range, ε. fmax and fmin are the maximum and minimum values of fm, respectively, obtained based on the xparm,asg in Step (l):(11)fmax-fmin≤ε,where fmin is the minimum value of fm or LOLEm,asd=1 and ε is the desired accuracy, 0.1 for an example [16].
Go to Step (f) for the next generation of EP optimization process when the mismatch does not reach to the desired level and the new value of xparm,asg obtained in Step (l) will be used to calculate a newPGtm,asg in Step (f). Otherwise, proceed to Step (n) once the mismatch has reached the predetermined limit ε.
Step (n). Record the optimized multiobjective function of CBM_{asg} for the transfer case from assisting areas to an assisted area. The optimized multiobjective CBM_{asg} will be recorded at the last iteration of the optimization process. The CBM_{asg} is obtained as the average value of xparm,asg or CBMm,asg associated with the assisting area previously calculated in Step (l). This implies that the CBM_{asg} is calculated through (12). Hence, the multiobjective function (M.O.F) comprising several optimized CBM_{asg} for the case of power transferred from the assisting areas can be expressed by (13). Then, LOLE_{asg} is computed based on the CBM allocated for each assisting area, CBMm,asg, as discussed in [19]. Consider(12)CBMasg=μxparm,asg=μCBMm,asg,(13)M.O.F=CBMasg=1,CBMasg=2,…,CBMasg=Nasg.
Therefore, CBM_{asd} for an assisted area is calculated by summing the optimum amount of CBM_{asg} transferred from all the assisting areas as given in(14)CBMasd=1=∑asg=1NasgCBMasg.
Step (o). Repeat Steps (a)–(n) several times in order to obtain numerous optimal solutions of multiobjective CBM_{asg}. These results will be applied into the Pareto optimal concept in such a way to find several superior multiobjective CBM_{asg}. Figure 1 presents the flowchart of the proposed EP optimization technique used to determine several multiobjective functions of CBMs.
Proposed EP technique to determine several multiobjective functions of CBMs.
3. Ranking Index in the Pareto Optimality Concept for the Best Selection of Optimal Multiobjective Capacity Benefit Margins
Pareto optimality is a concept that has been commonly used to select several optimal solutions of the multiobjective CBM_{asg} designated as multiobjective CBMasgPareto. This implies that the concept of Pareto does not provide a single solution that can be considered as the global optima for a problem related to the multiobjective CBM_{asg}. This is important to the ISO since it will provide flexibility to select the optimal as well as the most inexpensive result of multiobjective CBMasgPareto. These inexpensive results usually fall under the cluster of the Pareto optimal front. However, it is not worthy to select an expensive optimal result of multiobjective CBM_{asg} and this type of solution is usually categorized under the cluster of non-Pareto optimal. Figure 2 shows an example elucidating two clusters of the Pareto optimal concept. In Figure 2, F1 represents the axis plane of CBMasg=1 solution for the transfer case from assisting area 3 to assisted area 1. F2 is the axis plane of CBMasg=2 solution for the transfer case from assisting area 2 to assisted area 1.
Pareto and non-Pareto optimal fronts for the multiobjective function CBM_{asg}.
The EP optimization technique is performed several times in order to provide numerous optimal solutions of CBM_{asg}. In addition, solution x is the intersection point for the two CBM_{asg} results. The solutions x marked with a circle represent the cluster of Pareto optimal front. Usually, the best optimal solution of CBM_{asg}, so-called CBMasgPareto, is selected from the cluster of Pareto optimal front. Solutions x marked with × represent the cluster of non-Pareto optimal front which do not have the best optimal solution of CBM_{asg} due to their expensive multiobjective function. For instance, this can be observed through the comparison between x1 and x3, which have the same CBMasg=1 value for the F1 axis, that is, the transfer case from assisting area 3 to assisted area 1. However, by referring to the F2 axis, that is, the transfer case from assisting area 2 to assisted area 1, x3 yields to an expensive CBMasg=2 value compared to x1. Thus, x3 and x1 are optimal solutions of multiobjective CBM_{asg} which can be categorized under the non-Pareto and Pareto optimal fronts, respectively.
Theoretically, the Pareto optimal front can be defined as the solution x that is not dominated by any other feasible solutions x [20]. If the domination operator is labeled “≻,” the Pareto optimal concept can be described through the following criteria and this is referring to Figure 2.
x1≻x3 and x2≻x3. Hence, the x3 solution is said to be dominated or a non-Pareto optimal front solution.
x1≻x2 and x2≻x1. Hence, the x1 and x2 solutions are said to be nondominated or Pareto optimal front solution.
The aforementioned criteria can also be used to determine the Pareto optimal front for a multiobjective function which has more than two transfer case solutions of CBM_{asg}.
Furthermore, the selection of CBMasgPareto will be performed by the ISO according to the ranking index of either total LOLE or total LOLE difference. The proposed method has the advantage of introducing CBMasgPareto which will also provide the optimum results of LOLE and LOLE difference located at the Pareto optimal front cluster. In the initial selection based on the ranking index of total LOLE, CBMasgPareto is arranged according to the ranking index of total LOLE sorted in an ascending order. Then, the CBMasgPareto is selected in accordance with the ranking index of total LOLE as shown in(15)CBMasgPareto∈RanktotalLOLE,where(16)totalLOLE=∑asg=1NasgLOLEasg.
Equation (15) shows that CBMasgPareto is selected based on the ranking index of reliability or total LOLE in the assisting areas.
In the subsequent selection based on the ranking index of total LOLE difference, CBMasgPareto is arranged according to the ranking index of total LOLE difference sorted in an ascending order. Then, the ranking index of total LOLE difference is used to select CBMasgPareto. This is illustrated in(17)CBMasgPareto∈RanktotalΔLOLE,where(18)totalΔLOLE=∑asg=1NasgLOLEasg-LOLEasgo,where LOLEasgo is the LOLE at the base case condition of each assisting area.
Finally, the selected CBMasgPareto will be taken into account as firm and nonfirm transfer margins in the ATC determination.
4. Firm and Nonfirm Available Transfer Capability Determination
This section discusses the ATC determination that takes into account each optimum CBMasgPareto value selected by referring to the ranking index of total LOLE and total LOLE difference. The proposed method uses the iterative power flow solutions to determine ATC by taking into account CBMasgPareto for the transfer case from an assisting area to an assisted area [21]. Basically, the determination of ATC considering CBMasgPareto requires an iterative power flow solution to be performed at every increase of generation capacity and load at the respective selling and buying areas until one of the system constraints is met. This method is used to determine ATC considering CBMasgPareto for the next case of power transfer. It is important to note that two approaches are available to calculate ATC taking into account CBMasgPareto as firm or nonfirm transfer. In the former approach, the assisting and assisted areas are experiencing changes in total generation capacity according to the firm transfer of CBMasgPareto, whereas, in the latter approach, ATC is determined as the total transfer capability, TTC, reduced by CBMasgPareto. The procedure for both approaches discussed in this paper are implemented as follows.
Step (a). Establish a solved base power flow solution.
Step (b). Specify the selling and buying areas for a power transfer.
Step (c). Proceed to Step (e) if CBMasgPareto is considered to be a nonfirm transfer. Otherwise, adjust the generation outputs according to CBMasgPareto for all areas. The modification of generation outputs in assisted area and assisting area is done by using (19) and (20), respectively,(19)newPGgasd=1=PGgasd=1-PGgasd=1∑g=1NGPGgasd=1∑asg=1NasgCBMasgPareto,(20)newPGgasg=PGgasg+PGgasg∑g=1NGPGgasgCBMasgPareto.Notice that (19) and (20) may cause the assisting area to transfer its reverse generation capacity (CBMasgPareto) for compensating the generation deficiency which may occur in the assisted area. This is different from what has been dealt previously with, with (4) and (6) whereby the generating capacity of an assisting area and assisted area is decreased and increased, respectively, in order to identify the amount of generation capacity reserved for the CBM so that LOLE will be less than ξ.
Step (d). Perform the power flow solution to allow an assisting area to transfer power based CBMasgPareto required for compensating the generation deficiency occurring in the assisted area.
Step (e). Simultaneously, increase the power injection and extraction at the selling and buying areas, respectively, until either one of the line flows or voltage constraints is met through the load flow solution. The lower and upper voltage limits are considered to be 0.90 and 1.10 p.u., respectively. The injected power is referring to the increase of generation capacity in a selling area resulting in a power transfer which will be extracted by the load increased in a buying area. The maximum power transfer so-called TTC is acquired once the increased power flow solution has met one of the system constraints as mentioned previously.
Step (f). Calculate the ATC at three different cases of TTC determined in Step (e). In conjunction with the TTCo for the first case, the ATC at base case condition is obtained by employing (21) which does not require the execution of Steps (c) and (d):(21)ATCo=TTCo-ETC,where TTCo is the total transfer capability or the maximum power transfer at base case condition obtained and ETC is the existing transmission commitment or base case load flow solution considering system components variations.
With regard to the TTCo and CBMasgPareto for the second case, (22) is used to calculate ATC taking into account nonfirm transfer of CBM:(22)ATCnonfirm=TTCo-CBMasgPareto-ETC.By referring to TTCCBMasgPareto given for the third case, the CBM is taken as a firm transfer for ATC determination and the associated formulation is introduced through(23)ATCfirm=TTCCBMasgPareto-ETC.By referring to (23), the modification of generation capacity is performed in Step (c) consecutively with the load flow solution performed in Step (d) so that the ATC is determined by considering the firm transfer of CBM.
Step (g). Repeat Steps (a)–(f) to determine ATC for the next transfer case between the selling and buying areas. The determination of ATC for the next transfer case will also consider the same CBMs determined for the assisting and assisted areas.
The flowchart of ATC determination that takes into account the firm and nonfirm transfer margins of CBMasgPareto is illustrated in Figure 3.
Flowchart of firm and nonfirm ATC determination technique.
5. Results and Discussion
A modified IEEE-RTS79 is used to demonstrate the effectiveness of the proposed method in determining the CBM for each area [19, 22]. The generating units and transmission line information are given in [19, 22]. In this paper, the specified value of LOLE limit, ξ, is assumed to be 2.4 hrs/yr.
5.1. Capacity Benefit Margin Considering Interconnected System Reliability
In the base case condition of a modified IEEE-RTS79, the total generation, total load, and LOLE associated with each area is presented in Table 1. Based on the predetermined LOLE, areas 2 and 3 are considered the assisting areas and area 1 is referred to as the assisted area.
Generation, load, and LOLE for the three areas.
Area
Generation [MW]
Load [MW]
LOLE [hrs/yr]
1
2035
1125
4.7756
2
1748
1141
0.6380
3
784
584
0.6917
Table 2 presents the results of CBM considering tie-line reliability and is determined using the basic methodology discussed in [19]. It is observed that 88 MW and 33 MW are the amount of CBM reserved for the transfer from assisting areas 2 and 3 to area 1, respectively, resulting in the LOLE value being below 2.4 hrs/yr. Hence, new generation capacities of 2156 MW, 1660 MW, and 751 MW are obtained for areas 1, 2, and 3, respectively.
CBM results considering interconnected system reliability using the method introduced in [19].
Area
Generation [MW]
CBM [MW]
LOLE [hrs/yr]
Assisted area 1
2156
121
2.3972
Assisting area 2
1660
88
1.3943
Assisting area 3
751
33
1.3569
5.2. Multiobjective Capacity Benefit Margins Result Determined by the Ranking Index in Pareto-Based Evolutionary Programming Technique
It is noteworthy that Table 1 has presented the total generation capacity and total load for every area at base case condition of IEEE-RTS79. In conjunction with this matter, the LOLE less than 2.4 hrs/yr implies that the assisting areas 2 and 3 have sufficient amount of total reserve generation capacity that can be used as a reference to estimate the amount of CBM for accommodating the generation deficiency which may occur in the assisted area 1 with LOLE above 2.4 hrs/yr. Hence, the EP optimization technique is used to perform simultaneous determination of CBM that can be transferred from the assisting areas 2 and 3 towards the assisted area 1.
In the EP optimization technique, there are 10 individuals in a population representing the xparm,asg=1 or CBMs for assisting area 2. The same situation goes to the next population representing the xparm,asg=2 or CBMs for assisting area 3. The initial process of EP optimization technique will randomly generate a uniform distribution of xparm,asg using (1) based on the reserve generating capacity available in the assisting area. In particular, the initial population, xparm,asg=1, for assisting area 2 is obtained through the randomly generated variables that are in the range of 1 MW and 607 MW. This signifies that 1748 MW − 1141 MW = 607 MW is the reserved generating capacity available in the assisting area 2. In the overleaf case, that is, referring to the assisting area 3, the initial population, xparm,asg=2, is obtained via the randomly generated variables which are within the range of 1 MW and 200 MW. Both of the xparm,asg representing the initial population for assisting area 2 and area 3 are tabulated in Table 3. Simultaneously, both of the initial populations are applied into the mutation in (7) and pairwise comparison process (10) to obtain xoffm,asg and a new xparm,asg, respectively, for the assisting areas 2 and 3. All of the optimization process embedded in the EP optimization technique is repeated until the difference between maximum fitness, fmax, and minimum fitness, fmin, for the assisted area 1 is equal or less than the specified ε = 0.1. In the last iteration of EP optimization process, the average value of xparm,asg for both populations represents the optimum value of CBM for assisting areas 2 and 3. The xparm,asg obtained at the final iteration of EP optimization process are shown in Table 4. In relation to each population of xparm,asg, it is obvious that a relatively similar value is obtained for all of the individuals, and the average value of xparm,asg in (12) may yield to CBM specified for the assisting areas 2 and 3. This result is obtained only for one optimization run of EP technique. The EP optimization technique is executed for several times so that the Pareto optimal fronts of CBMs (CBMasgPareto) are obtained which provides flexibility to the transmission provider in selecting optimum CBMs in tandem with the changes of economic, load-serving entity requirement or resource planner. The analysis of CBMasgPareto will be elucidated in the following discussion.
Initial population of EP technique for the assisting areas 2 and 3.
Number of individuals
Assisting area 2
Assisting area 3
xparm,asg=1 or CBM (MW)
LOLE (hrs/yr)
xparm,asg=2 or CBM (MW)
LOLE (hrs/yr)
1
474.61
52.67
47.96
1.89
2
237.57
5.22
71.63
3.44
3
147.71
2.29
165.24
37.63
4
246.18
5.46
4.08
0.70
5
59.55
1.03
9.60
0.80
6
81.11
1.25
34.80
1.43
7
572.83
137.47
130.82
15.57
8
581.37
148.06
147.35
23.20
9
350.15
15.24
130.55
15.57
10
37.29
0.85
91.19
5.62
Final population for the assisting areas 2 and 3 based on one run of EP optimization process.
Number of individuals
Assisting area 2
Assisting area 3
xparm,asg=1 or CBM (MW)
LOLE (hrs/yr)
xparm,asg=2 or CBM (MW)
LOLE (hrs/yr)
1
86
1.32
35
1.43
2
86
1.32
35
1.43
3
86
1.32
35
1.43
4
86
1.32
35
1.43
5
86
1.32
35
1.43
6
86
1.32
35
1.43
7
86
1.32
35
1.43
8
86
1.32
35
1.43
9
85
1.29
35
1.43
10
85
1.29
35
1.43
Figure 4 shows different optimized values of CBM obtained at every execution of the EP optimization process. The x-axis represents the CBM transferred from the assisting area 2 to assisted area 1, whereas the y-axis represents the CBM transferred from assisting area 3 to assisted area 1.
Pareto and non-Pareto optimal fronts of CBM for the two transfer cases.
It is observed that, with an increase in CBM associated with a particular assisting area, CBM at the other assisting area would decrease and vice versa. The best optimum values for the multiobjective function of CBMs are obtained based on the Pareto optimal front and the cluster for this case is illustrated in Figure 4. The other cluster represents the non-Pareto optimal front of CBMs with excessive value which may yield to an invidious violation of power system security and ineffective utilization of the existing network resources. Figure 5 represents the cluster of Pareto optimal front of CBMs extracted from Figure 4. In the Pareto optimal front, the results of CBM have less potential in violating system security compared with the excessive amount of CBMs obtained based on the non-Pareto optimal front.
Pareto optimal fronts of CBMs for the two transfer cases.
Furthermore, the Pareto optimal front approach used in the EP technique gives sufficient flexibility to the ISO in selecting the optimum value of CBM for every transfer case depending on the system requirements. This is obviously contradictory with CBM results tabulated in Table 2 which are obtained using a basic approach [19]. Based on the CBM results shown in Table 2, ISO does not have the flexibility to select other choices with suitable set of CBMs for compensating any generation deficiency at different system operating states. In relation to Figure 5, CBM results for each area yielding to the Pareto optimal front are also tabulated in Table 5. Every result of Pareto optimal front CBM will be used as a reference to estimate the power transferred from assisting areas 2 and 3 to accommodate possible generation deficiency in the assisted area 1.
Pareto optimal front of CBM values for each area.
EP run
Assisted area 1
Assisting area 2
Assisting area 3
CBM [MW]
LOLE [hrs/yr]
CBM [MW]
LOLE [hrs/yr]
CBM [MW]
LOLE [hrs/yr]
1
114
2.3997
63
1.0646
51
2.0485
2
117
2.3627
69
1.1475
48
1.8874
3
117
2.3794
73
1.1739
44
1.731
4
121
2.3967
85
1.2913
36
1.4705
5
117
2.3656
70
1.1458
47
1.9693
6
117
2.3723
71
1.1237
46
1.8714
7
116
2.3867
70
1.1458
46
1.8714
8
116
2.3941
71
1.1237
45
1.7887
9
118
2.3641
74
1.1847
44
1.731
10
118
2.3667
76
1.2076
42
1.6394
11
121
2.3996
86
1.3183
35
1.431
12
122
2.3785
86
1.3183
36
1.4705
13
125
2.3411
90
1.3775
35
1.431
14
121
2.3954
84
1.2902
37
1.539
15
122
2.3766
85
1.2913
37
1.539
16
124
2.3534
90
1.3775
34
1.3913
17
123
2.3695
90
1.3775
33
1.3569
18
123
2.3711
91
1.3914
32
1.2769
19
122
2.3931
93
1.428
29
1.2597
20
128
2.3248
102
1.5166
26
1.1351
21
125
2.3697
102
1.5166
23
1.0929
22
131
2.3091
110
1.6479
21
1.0247
23
132
2.3999
112
1.7002
20
1.0107
24
133
2.3934
113
1.7095
20
1.0107
25
132
2.3985
114
1.7198
18
0.9511
26
134
2.373
116
1.7116
18
0.9511
27
133
2.3841
116
1.7116
17
0.9235
28
136
2.3455
119
1.7702
17
0.9235
29
132
2.3932
121
1.8074
11
0.8044
It is observed that the Pareto optimal front of CBM values was obtained while fulfilling the LOLE criterion of less than 2.4 hrs/yr. The other advantage of the proposed method is that CBMasgPareto results also yield Pareto optimal front clusters of LOLE and difference in LOLE values. This can be verified in Figure 6 where LOLE located at the Pareto optimal front cluster refers to the CBMasgPareto results obtained for each case of power transfer depicted in Figure 4. Consequently, the results of total LOLE obtained through (15) are arranged in ascending order and the ranking index is assigned to every result to distinguish the reliability of the assisting areas shown in Table 6. With respect to each value of total LOLE, total CBMasgPareto was obtained based on the two transfer cases also shown in Table 6. The total CBMasgPareto is equivalent to CBM for an assisted area, CBMasd=1.
CBM results with ranking index of total LOLE and total LOLE difference.
CBMasdPareto received by area 1 [MW]
CBMasgPareto from area 2 to area 1 [MW]
CBMasgPareto from area 3 to area 1 [MW]
Total LOLE [hrs/yr]
Total difference of LOLE [hrs/yr]
Rank index
131
125
6
2.608
1.278
1
125
102
23
2.610
1.280
2
132
121
11
2.612
1.282
3
132
125
7
2.629
1.300
4
133
116
17
2.635
1.305
5
128
102
26
2.652
1.322
6
134
116
18
2.663
1.333
7
123
91
32
2.668
1.339
8
132
114
18
2.671
1.341
9
131
110
21
2.673
1.343
10
122
93
29
2.688
1.358
11
136
119
17
2.694
1.364
12
132
112
20
2.711
1.381
13
133
113
20
2.720
1.391
14
123
90
33
2.734
1.405
15
124
90
34
2.769
1.439
16
122
86
36
2.789
1.459
17
125
90
35
2.809
1.479
18
121
84
37
2.829
1.500
19
122
85
37
2.830
1.501
20
118
76
42
2.847
1.517
21
117
73
44
2.905
1.575
22
116
71
45
2.912
1.583
23
118
74
44
2.916
1.586
24
117
71
46
2.995
1.665
25
116
70
46
3.017
1.688
26
117
69
48
3.035
1.705
27
114
63
51
3.113
1.783
28
117
70
47
3.115
1.785
29
Pareto optimal fronts of LOLE for areas 2 and 3.
Figure 7 represents the difference between LOLE values located at the Pareto optimal front cluster. The results are referring to the CBMasgPareto obtained based on the power transfer cases shown in Figure 4. Then, the results of total LOLE difference calculated using (17) were arranged in ascending order and the ranking index is assigned to each result indicating level of reliability available for the assisting areas as shown in Table 6. Table 6 reveals that the total CBMasgPareto or CBMasdPareto values were arranged according to the total LOLE and total LOLE difference possessing the same ranking index. The results divulge that the Pareto-based EP method has the advantage of providing simultaneous optimum results of CBMasgPareto, LOLE, and difference of LOLE in which all are located at the Pareto optimal front cluster.
Pareto optimal fronts of LOLE difference for areas 2 and 3.
As noted earlier and clearly presented in Table 6, the proposed method has the advantage of providing several choices of CBM that can be selected by ISO based on the ranking index of total LOLE and/or total LOLE difference. For instance, the ISO shall set the CBMasgPareto, respectively, to 125 MW and 6 MW for the transfer case from assisting areas 2 and 3, respectively, to the assisted area 1 so that the assisting areas will operate in a highly reliable condition because the aforementioned power transfers are obtained based on the lowest total LOLE of 2.608 hrs/yr at the 1st ranking index. The combination of CBMasgPareto for both transfer cases will provide a total CBMasgPareto of 131 MW which results in the lowest total LOLE difference of 1.278 hrs/yr at the 1st ranking index as shown in Table 6. Due to a relatively large total CBMasgPareto of 131 MW, the tie-line capacity will not be fully utilized as a medium power transfer based ATC for electricity transfer. The total CBMasgPareto of 131 MW can also be obtained at the 10th ranking index as shown in Table 6. However, a total CBMasgPareto of 131 MW at the 10th ranking index will not be the best choice for the ISO since the assisting areas will operate in a less reliable condition due to the total LOLE of 2.673 hrs/yr and total LOLE difference of 1.343 hrs/yr. Furthermore, the 10th ranking index yields to a result that is close with the largest total CBMasgPareto of 136 MW located at the 12th ranking index. However, total LOLE of 2.694 hrs/yr and total LOLE difference of 1.364 hrs/yr signify a reasonable or moderately reliable operation of the assisting areas in conjunction with the largest total CBMasgPareto of 136 MW at the 12th ranking index.
In another situation whereby the ISO is not interested in a highly reliable condition of a power system, the CBMasgPareto of 70 MW can be selected for the transfer case from assisting area 2 to area 1 and the CBMasgPareto of 47 MW can be chosen for the transfer case from assisting area 3 to assisted area 1. This would be a less reliable choice prior to the largest value of total LOLE which is 3.115 hrs/yr at the 29th ranking index as tabulated in Table 6. Consequently, the total CBMasgPareto of 117 MW is obtained contributing to the largest total LOLE difference of 1.785 hrs/yr located at the 29th ranking index. For this case, a highly reliable condition incurred from a specific amount of CBM reserved through tie-line capacity is not the main intention for the ISO. Besides, ISO is more interested in the utilization of tie-line capacity for ATC in order to enhance and perform as an important role in the electricity market. Similar to the 29th ranking index, the CBMasgPareto of 117 MW at the 22nd ranking index can also be used in this case study. It has the advantage in providing total LOLE of 2.905 hrs/yr and total LOLE difference of 1.575 hrs/yr which is much better than the results obtained at the 29th ranking index. By comparing with the total CBM of 117 MW at the 22nd ranking index, ISO may choose the lowest value of total CBM, that is, 114 MW at the 28th ranking index, only when the objective is not solely on the reliability improvement of the assisting areas.
In a detailed analysis, ISO may select the CBMasgPareto of 90 MW and 33 MW for the transfer case from the assisting areas 2 and 3 to the assisted area 1, respectively, so that the assisting areas are operating at the mid ranking level (index 15) having the total LOLE of 2.734 hrs/yr and total LOLE difference of 1.405 hrs/yr. This indicates that ISO has chosen the value of CBMasgPareto for both transfer cases resulting in 50% priority on the reliability of assisting areas and 50% priority on the power transfer based ATC reserved for electricity market activities. The aforementioned discussion shows that the optimal value of CBM specified for each case of power transfer is actually dependent on similar ranking indices of total LOLE and total LOLE difference.
The previous results have well demonstrated that CBMasgPareto, LOLEs, and difference of LOLEs clustered in the Pareto optimal front are the criteria to be satisfied by the ISO before conducting the finest selection of CBMasgPareto. The performance of the Pareto optimal front embedded in the proposed optimization technique is not limited only to the CBMasgPareto value that provides the highest reliability of assisting areas due to the lowest total LOLE and total LOLE difference associated with the 1st ranking index. Nevertheless, it also is not confined to the CBMasgPareto value with large amount of ATC yielding to the largest total LOLE and total LOLE difference selected at the 29th ranking index. This implies that ISO has several choices for CBM for each case of power transfer depending on the ranking index selected based on the Pareto optimal front of total LOLE and total LOLE difference.
5.3. Performance Comparison with Existing Capacity Benefit Margin Calculation Methods
It is worthwhile to mention that the proposed method is robust in providing simultaneous optimum results of the CBMasgPareto, LOLE, and difference in LOLE, all of which are located at the Pareto optimal front cluster. In the proposed method, the ranking index has the advantage of providing a clearer depiction on the relationship between the three optimal results which will be a great help to the ISO in making the finest decision for selecting optimum CBM values. This is contradictory to other methods in [17], whereby the optimization process is performed separately to find the minimum total LOLE, minimum total LOLE difference, or minimum CBM considering weight of the tie-lines. As shown in Figure 8, the lowest total LOLE of 2.608 hrs/yr and lowest total LOLE difference of 1.278 hrs/yr, computed using the method presented in [17], will give a total CBM result of 131 MW which is quite large according to the Pareto optimal front tabulated in Table 6. Using both methods discussed in [17], ISO does not have a choice other than to utilize a large total CBM value of 131 MW to ensure a highly reliable operating condition of the assisting areas in accordance with the lowest total LOLE of 2.608 hrs/yr and lowest total LOLE difference of 1.278 hrs/yr. Thus, the proposed method of Pareto optimal front provides a solution to the abovementioned problem by providing the total CBMasgPareto of 125 MW at the 2nd ranking index considered as the other option that is smaller than the total CBM of 131 MW at the 1st ranking index in Table 6 and Figure 8. The result of total CBMasgPareto, that is, 125 MW at the 2nd ranking index, also provides a highly reliable operating condition for the assisting areas that are nearly identical with the lowest total LOLE and lowest total LOLE difference at the 1st ranking index. Consequently, the total CBMasgPareto of 125 MW at the 2nd ranking index provides a more conservative space to transfer the power based ATC compared with the total CBM of 131 MW at the 1st ranking index.
Total CBMasgPareto selection based on the ranking index in Pareto optimal front concept.
The method discussed in [17] provides some limited choices of CBM results as it considers the weight specified on each tie-line. However, the proposed method provides several CBMasgPareto values without considering the weight for the tie-lines. Once the tie-line weight is not considered in [17], the minimum total CBM of 114 MW is obtained. It is depicted in Figure 8 and Table 6 that the total CBM of 114 MW is obtained at the assisted area when the CBMs of 63 MW and 51 MW are transferred from the assisting areas 2 and 3, respectively. However, the total LOLE of 3.113 hrs/yr and total LOLE difference of 1.783 hrs/yr are relatively large at the 28th ranking index although the minimum total CBM of 114 MW is obtained using the abovementioned equation given in [17]. Therefore, the proposed Pareto optimal front concept is used to provide several choices of solution that are relatively similar to the minimum total CBM of 114 MW. For this case, the total CBMasgPareto of 116 MW is chosen from the 23rd ranking index of Pareto optimal front and it is the nearest value to the minimum total CBM of 114 MW. By referring to Figure 8 and Table 6, it can be observed that the total CBMasgPareto of 116 MW will improve the reliability of the assisting areas due to the total LOLE of 2.912 hrs/yr and total LOLE difference of 1.583 hrs/yr which are smaller than the LOLE results obtained from the minimum total CBM of 114 MW. For other cases of different weights assigned to each tie-line, the selection of CBM result is performed similarly by referring to the abovementioned explanation of Pareto optimal front concept.
5.4. Results of Available Transfer Capability Incorporating Capacity Benefit Margin
In this section, the ATC results are obtained based on the four cases of power transfer as shown in Tables 7, 8, 9, and 10. The results of ATCs are obtained by considering the firm and nonfirm transfers of CBMasgPareto located at the Pareto optimal front cluster as depicted in Table 6. Hence, for every case of power transfer, there are 29 results of ATC that give the flexibility to the ISO in choosing a suitable power transfer. It is noted that CBMasgPareto taken as a firm transfer contributes to slightly larger ATC values as compared to CBMasgPareto which is taken as a nonfirm transfer. It can be concluded that, by incorporating the nonfirm transfer of CBMasgPareto into ATC, there will be loss for certain amount of ATC in the power transfer contracts.
Results of ATC from area 1 to area 2.
EP run
ATC_{base}[MW]
ATC_{firm}[MW]
ATCnonfirm[MW]
1
586
542
523
2
586
538
517
3
586
535
513
4
586
527
501
5
586
537
516
6
586
537
515
7
586
537
516
8
586
537
515
9
586
535
512
10
586
533
510
11
586
526
500
12
586
526
500
13
586
524
496
14
586
528
502
15
586
527
501
16
586
524
496
17
586
524
496
18
586
523
495
19
586
522
493
20
586
515
484
21
586
515
484
22
586
510
476
23
586
508
474
24
586
508
473
25
586
507
472
26
586
506
470
27
586
506
470
28
586
504
467
29
586
502
465
Results of ATC from area 1 to area 3.
EP run
ATC_{base}[MW]
ATC_{firm}[MW]
ATCnonfirm[MW]
1
271
258
220
2
271
259
223
3
271
260
227
4
271
262
235
5
271
259
224
6
271
259
225
7
271
259
225
8
271
260
226
9
271
260
227
10
271
260
229
11
271
262
236
12
271
262
235
13
271
262
236
14
271
262
234
15
271
262
234
16
271
263
237
17
271
263
238
18
271
263
239
19
271
264
242
20
271
265
245
21
271
265
248
22
271
266
250
23
271
266
251
24
271
266
251
25
271
267
253
26
271
267
253
27
271
267
254
28
271
267
254
29
271
268
260
Results of ATC from area 2 to area 1.
EP run
ATC_{base}[MW]
ATC_{firm}[MW]
ATCnonfirm[MW]
1
1171
1115
1108
2
1171
1110
1102
3
1171
1106
1098
4
1171
1096
1086
5
1171
1109
1101
6
1171
1108
1100
7
1171
1109
1101
8
1171
1108
1100
9
1171
1106
1097
10
1171
1104
1095
11
1171
1095
1085
12
1171
1095
1085
13
1171
1091
1081
14
1171
1097
1087
15
1171
1096
1086
16
1171
1091
1081
17
1171
1091
1081
18
1171
1090
1080
19
1171
1088
1078
20
1171
1080
1069
21
1171
1080
1069
22
1171
1073
1061
23
1171
1071
1059
24
1171
1070
1058
25
1171
1069
1057
26
1171
1067
1055
27
1171
1067
1055
28
1171
1065
1052
29
1171
1063
1050
Results of ATC from area 3 to area 1.
EP run
ATC_{base}[MW]
ATC_{firm}[MW]
ATCnonfirm[MW]
1
71
21.5
20
2
71
24.5
23
3
71
28.5
27
4
71
36.5
35
5
71
25.5
24
6
71
26.5
25
7
71
26.5
25
8
71
27.5
26
9
71
28.5
27
10
71
30.5
29
11
71
37.5
36
12
71
36.5
35
13
71
37.5
36
14
71
35.5
34
15
71
35.5
34
16
71
38.5
37
17
71
39.5
38
18
71
40.5
39
19
71
43.5
42
20
71
46.5
45
21
71
49.5
48
22
71
51.5
50
23
71
52.5
51
24
71
52.5
51
25
71
54.5
53
26
71
54.5
53
27
71
55.5
54
28
71
55.5
54
29
71
21.5
20
6. Conclusion
This paper has presented a new approach for calculating CBM taking into account tie-line reliability in the interconnected system. The proposed approach employs the ranking index in a Pareto-based EP technique that provides several choices of optimum CBM values. The effectiveness of the proposed method in determining the CBM has been tested on the modified IEEE-RTS79. The results presented have shown that the Pareto optimal front of CBMs is an inexpensive solution compared to the CBMs located at the non-Pareto optimal front. The other advantage associated with the proposed method is due to its ability in providing simultaneous optimal results of CBM, LOLE, and LOLE difference whereby all are located at the Pareto optimal front cluster. Hence, selection of the result does not rely solely on the value of CBM, but it is also concurrently based on the impact of total LOLE and total LOLE difference included under the ranking of Pareto optimal front. In short, ISO has the flexibility to select the CBM at the Pareto optimal front referring to the ranking index of total LOLE and total difference of LOLE. Finally, CBM taken as a firm transfer yields to a relatively large value of ATC compared to CBM considered as nonfirm transfer.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported in part by the Research Management Institute (RMI), Universiti Teknologi MARA, Malaysia, under Grants 600-RMI/DANA 5/3/PSI (186/2013) and 600-RMI/DANA 5/3/CFI (56/2013); the Ministry of Higher Education (MOHE), Malaysia, under Grant 600-RMI/ERGS5/3 (18/2012); and the Ministry of Science, Technology and Innovation (MOSTI), Malaysia, under Grant 03-01-01-SF0476.
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