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220kv降压变电所电气一次系统设计275,220,kv,降压,变电所,电气,一次,系统,设计,275
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华 北 电 力 大 学 科 技 学 院毕 业 设 计（论 文）附 件外 文 文 献 翻 译学 号： 081901090217 姓 名： 庞斌 所在系别： 电力工程系 专业班级： 农电08k2 指导教师： 苏海锋 原文标题：A new optimization model for distribution substationsiting, sizing, and timing2012年 6 月 20 日一个新的配电变电站选址，容量和定时的优化模型一 摘要本文提出了一种新的配电变电站选址，定容和定时规划的优化模型。该模型使用线性函数来表达总成本函数。该模型包括不同的电气约束，像电压降落，变电站和变压器容量，潮流和径向流的限制.这种规划问题，被作为一个混合整数线性规划问题来进行制定，从而避免使用非线性规划，也同时避免了可能被困地的解决方案。一个数值例子来验证模型的有效性。关键词：配电系统规划;配电变电站;优化;径向分布系统二、主要内容1 .介绍 配电变电站规划被认为是在电力系统规划过程中迈出的重要一步。这是因为它代表电力传输和配电网之间的主要联系。可以选择的变电站选址和变电站的容量都会对变电站在传输和配电系统的规划过程都明确的限制，因此，变电所设计参数对馈线路有很大的影响。虽然变电站的成本对于配电系统规划总成本相对较小，但是其对配电系统规划的整体经济有一定的限制。，变电站是最复杂的，尤其是当它涉及到配电系统的规划时。60实用规划者认为这个一阶段发生在在传输阶段系统规划的过程中，20认为这是在配电系统规划中，而其余的（20）认为是一个单独的进程1,2。 变电站规划涉及变电站选址，变电站的规模和服务领域的决心，和变电站设备安装的时间。变电站的选址不完全是基于对电力的考虑。通常，城市规划和环境的限制在这个过程中的是主要决定因素。在最好的情况下，该市将提供一套规划可用的地址以供选择。然而这些地址很有可能都不是最佳的选择方案。这时，规划选址就不得不选择第二个好的方案。在一般情况下，变电站选址过程中被视为作为一个筛选过程，通过这个过程所有可能的地点被调查分为不合适，候选的，或者是将来需要评价的。通常，变电站的规模和生活区决定于电气因素和制约因素，如：设备容量和馈线的电压下降等制约因素1,2。 已开发几个布局规划模型可以划分为四个主要类别3,4：静态负载的总系统模型：确定最佳变电站的位置和规模，网络路由，负载和馈线大小之间的负荷传输。动态负载子系统模型：确定规模，变电站的位置，安装变电站其设备和“最佳馈线路线的时间。动态负载的总系统模型：确定规模，位置和安装配电变电站设别和主馈线的时间。 5中提出了确定变电站规模和时间的方案。在这个方案中通过使用伪动态的方法变电站的规模和时间确定会分开确定。这种方法需要连续应用单时间周期静态规划模型。此外，在这个算法中，是否兴建变电站是完全基于电压下降的考虑。大家都知道电压预测决定变电站的建设因素，然而这种方法的主要缺点是电压的预测主要基于在变电站的服务区内负荷密度均匀的假设，对于实际情况这不是可行的。此外，该模型没有考虑到设备的位置。在 6中，为解决变电站的位置，规模的确定交通运输方式方面的问题得到了发展。这种方法假定总需求等于总供给，它的目标是确定一个可行性模式，这种模式最大限度地减少了运输成本，同时满足所有的需求。这种方法不包括设备在目标函数方面的费用，而且将现有的和可能有的变电站作为蓝本，这种方法对正在运行的所有变电站，尽管他们中只承担一些小负荷，会产生一个最佳的解决方案。然而，这种方法没有考虑任何约束，如电压的限制。此外，也不包括电压下降方面的计算。 在7中，为了最优的变电站选址，提出了一种固定费用的中转模型。该模型的目标函数包括固定和可变成本的构成，通过使用一个整数的分支定界技术来进行解开。然而，这个模型是静态模型，不考虑任何随着时间的要求的变化因素。此外，它不包括任何限制例如电压限制。为了解决了多期的配电系统规划的问题，固定费用中转模型网络程序问题的建议在8中被提出来。这种技术被用于优化配电变电站和主馈线规划。然而，这种技术不包括任何限制如电压限制。 910的启发式组合优化算法来确定变电站的最佳容量，同时通过损失最小化降低在馈线损失，一种多源定位算法用于变电站的容量分配。这程序不需要选择候选变电站位置。 在11中为了解决最佳变电站的位置和规模一个自适应变异粒子群优化算法的被提出。这种方法不需要候选变电站的位置同时考虑到变电站建设的投资和地理信息系统（GIS）。一种基于最低馈线损耗的变电站综合服务区和馈线路方法在 12提出来。在此方法中，与分布数据的基础上，计算机图形学，地理信息系统最小的路径和负荷开关模式算法综合起来解决规划问题。最小路径算法被用来重新分配负载点。在远离主变的末端，开关的负荷进行集中和分配。负荷开关模式用于连接变电站馈线路径和分发加载点。许多约束，如潮流约束，功率流动和网络辐射都考虑其中。另一个变电站扩建规划程序被开发出来13。为了确定可行的候选地址，它提出了一个数学的聚类技术，同时考虑到变电站容量，馈线容量和电压限制。然后，为了解决现有的变电站和新的变电站分配和容量的扩展的最优解决方案问题一种遗传算法被提出来。这些上述程序9-13不包括电压范围内的任何约束。然而，问题的解决方案没有考虑随时间变化的需求。 在14中，配电变电所通过概率方法来选址。这种方法考虑到了每小时（或每天）负载周期。对于不同的逐时负载的情况，根据其负载大小，来进行负荷中心位置的确定和加权。然后用这些地点建立一个概率分布，来确定应位于变电站的概率的最大周长，过程中还考虑到，如土地供应和土地成本等因素。在15 中提出了一种非离散函数，这种模型考虑到变电站成本和各种约束条件等关于变电站选址，规模和时间的因素。此外，该模型考虑到随时间变化的需求。然而，在这个模型中的主要缺点是，每个规划间隔独立于以前的时间间隔所取得的成果。这导致了在被安装在较早时期的一些馈线，然后移除后与新的馈线安装在以后时期，这实际上是不可行的。此外，这个问题导致了一个混合整数非线性规划（MINLP），由于非线性，这可能影响最佳解决方案。本文组织如下：第二节介绍了正在研究的系统。在第3节提出了对配变电站选址，容量，和时序的选择的优化模型.使用建模和解决提出的题制定,通用代数建模软件（GAMS）求解。在第4节，从提出的优化模型产生出结果。一个修改的拟定，来避免低效的电力传输的问题在第5节讨论. 本节还提出了修改制定生成的结果。2.研究制度 根据调查的服务区由9个部门构成如图2。每个部门的面积为0.44平方公里，假设变电站安装部门，由城市部门2，4和6构成。规划期被设定为10年， 每2年的时间间隔共5次。表115给出的行业需求的增长超过10年的规划期内（每隔5）的增长。每个变电站的额定功率为40兆伏安，可配备了两个变压器，每个最高额定为20兆伏安。每个变压器的效率，地点，并分配给变量在表2中的细节被提出来。变电站固定成本假设是$ 200,000，而变压器的单位安装成本假定为150美元，能源成本假定为0.17元/千瓦时。利息，税收，通货膨胀，保险费率被认为是10，10，6和1。变压器允许被加载到其额定值的75。它也被认为沿着每条馈线的最大允许压降为275 V（下被设定为11千伏系统的额定电压的2.5）。在额定功率的变压器铜损假设是127千瓦15。每个变电站九馈线（总可用馈线27馈线），并假定每个馈线直接供应相应部门的需求的（表3）。此表还介绍了为每个可用馈线和现有的路线分配的变量。3问题制定 当规划安装配电系统变电站及其部件，其主要目标是尽量减少设备的安装和能量损失的整体成本。这个费用取决于一些因素，如变电站选址，设备安装时间，设备（变压器）负荷。对于变电站的选址，增加安装变电站或不当的选址，会大大增加整个系统的成本。利率，通货膨胀率，税款，保险费率的影响设备安装的时间，从而整体成本也受到影响。由于能量损失的大小是取决于设备的负荷，负荷水平的增加将导致整体成本增加。此外，以一个间隔期为基础的以往的规划模型上，可能会导致不切实际的解决方案，如安装在同期馈线，然后在后期消除15。在这种情况下，就需要人们用专业知识来消除这些不切实际的解决方法。考虑到所有这些因素，提出了一个新的方案来最大限度地减少整体成本。为了避免人类专业知识的必要性，以实现以下目标：1确定设备安装的最佳时机。2充分确定变电站的选址和容量。在整个规划地平线的主要目标写成如下：其中q是在10年的规划期内，设计间隔数等于设置等于5。因此，两年内选择每个设计间隔提供足够的时间进行设备安装。CS1，CS2和CS3是分别为固定成本变电站1，2，3,，CT11，CT12是变压器的两个单位的成本将安装在变电站1. CT21，CT22是变压器的两个单位的成本将安装在变电站2. CT31，CT32是变压器的两个单位的成本将安装在变电站3. C是能源成本（元/千瓦时），Htr是变压器单元（MVA）的评级，Xin是从单位交付的电力.I在给定的周期n安装单元（变压器）。如果未安装单元（变压器）它设置为零。Rn和BN是固定的收费率和现值因素，对于一个给定的时间间隔n分别计算.如下：其中，R，T，F，分别为利润，税收，通货膨胀和保险费率。 在此方案中，变压器的功率损耗依据变压器负载的百分比来进行计算。假设这一比例相当于在额定功率（PCU）时变压器的铜损变压器单元评级（HTR）的比率。3.1 额外的限制，以克服非线性决策变数yin是用来确定是否变压器提供电力。通过乘以变量xxin（由变压器提供电源）二进制变量yin。然而，这将强制非线性问题。为了避免这种情况，变量xin被推出来取代双方的产品一些变量和约束被添加如下：其中M是一个大数，并选择是等于10000，以保证约束式显示。（4）和（5）将表明是否使用或变压器，XXin是从单元（变压器）i在一定时期内N（MVA）的电力功率，Yin是一个二进制变量表示在给定的周期n的功率耗散。一个它的价值等于1表示电流通过变压器传输，而零表示没有电流传出。3.2 固定成本约束如前所述，Sij表示二进制的决策变量，决定了一个单位的安装. 一个单位的成本取决于今年由于安装引起的变化。二进制变量F增加，其中影响的发电量为在连续两年的单位提供决策变量。3.3 容量限制每个变电站，每个变压器容量分别40兆伏安和20兆伏安。然而，他们被允许加载到其额定容量的75从而最高效率运作。这个导致了对于每个变电站，每个变压器有30兆伏安和15兆伏安容量的限制. 此外，变电站和变压器负荷的下限设置为0.这可以表示如下： 3.4 功率流的限制这些约束代表的能量守恒定律，各变电站的总负载在一个给定的时间间隔n内等价于个人变压器单位的负荷总和，同时，在同一时间内等于本部门要求提供的总变电站和子变电站单位的总铜损。这些约束先解释如下：3.5 径向流约束 假定每条馈线提供每块地区的电能需求，直接从某变电站提取。而且，假定每块区域从一个变电站得到电能供给，主要是为了满足径向流动的限制。这些限制可以表示如下：3.6电压限制 电压的限制是为了确保在每个节点上的电压保持在允许的水平。这些约束以每条馈线压降的最大值来表示。这些约束可以表示如下： 是在被给定时间N内沿馈线j 的电压降，Vnominalis表示系统的额定电压（kV）。ZJ是馈线J的阻抗，XJ，n是一个二进制的变量，表示在给定的一段时间内存在馈线的存在。表示最大电压降落。4.优化结果在第2节提出的问题解决了，在GAMS很容易使用的方式解式。（1） - （22）的问题解决了而且最优的系统配置如图 3。这种配置造成在的10年规划期内，总成本为2,839,253.817美元。表4给出了馈线的变量的状态，可以得出的结论是没有必要安装变电站3。然而，此表显示，最佳的解决方案包括安装馈线6来为区域6供电，这是一种低效馈线路由（2变电站已安装）。尽管电力要输送到较远的区域，但是所有涉及的限制，包括压降限制，均已得到满足，如表5和表6。这些制约因素还没有显示出来，因为区域6在研究期间的的需求最小。结果表明，所需的负荷变电站1和2分别为28.179兆伏安和25.16兆伏安。变电站1和2的服务区域在表3中显示。调查结果显示，在这个配置规划中需要四台变压器同时两台变压器要在第一规划期内安装，另外两台在第二规划期内安装。同时可以看出这个问题的制定是通过延时一些变压器的安装来考虑整体变压器的时限问题的 。比先前15中提出的问题的主要优点有：1 这些问题不包含任何非线性问题2 在整个规划期内这些问题得到了优化，从而避免了任何不切合实际的方案，像在第一段时间内安装一条馈线，可是在第二个规划时期就拆除了。这种问题的制定包括了三个变电站可能的位置，而不仅仅是为了满足负荷的需要，从而考虑用算式来确定在满足负荷条件下变电站的数目。三、 结论本文提出了一种新的配电变电站选址，规模和时序规的划优化模型。这个模型使在这个规划时期内的总成本得到了优化。通过最大限度地减少整个规划期内的成本，从而得到准确，实际的成果。 根据研究，只考虑成本优化可能会导致一些问题，在规划期内特别低效，对未来负荷的增长也是行不通的。所得的结果显示，电压降落不足以防止无效的功率传输，尤其是负荷水平在规划期内不那么高。所以，修改后的规划方案，包括同时最大限度的降低成本和压降，才能保证这种做法。在这种情况下得出的结论能够提供一种更好的和高效的供电装置，从而满足将来负荷的增长。原文作者及出处：T.H.M. El-Fouly a,*, H.H. Zeineldin b, E.F. El-Saadany a, M.M.A. Salama aA new optimization model for distribution substationsiting, sizing, and timingT.H.M. El-Foulya,*, H.H. Zeineldinb, E.F. El-Saadanya, M.M.A. SalamaaaDepartment of Electrical and Computer Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1bMasdar Institute of Science and Technology, P.O. Box 45005, Abu Dhabi, United Arab EmiratesReceived 24 August 2006; received in revised form 4 September 2007; accepted 22 October 2007AbstractThis paper presents a new planning optimization model for distribution substation siting, sizing, and timing. The proposed modelinvolves using linear functions to express the total cost function. The developed model includes different electrical constraints such asvoltage drops, substation and transformer capacities, power flow, and radial flow constraints. The proposed planning problem is formu-lated as a Mixed Integer Linear Programming (MILP) problem to avoid the use of nonlinear programming and thus avoiding the pos-sibility of getting trapped in local solutions. A numerical example is presented to validate the effectiveness of the developed model.? 2007 Elsevier Ltd. All rights reserved.Keywords: Distribution system planning; Distribution substation; Optimization; Radial distribution systems1. IntroductionDistribution substation planning is considered the mostimportant step in the power system planning process. Thisis because it represents the main link between transmissionand distribution system. The available sites and sizes ofsubstations result in definite constraints on both transmis-sion and distribution systems planning process and thus,substations design parameters have a great impact on feed-ers routing. Substations usually set limits on the overalleconomics of the distribution system planning althoughtheir cost represents a relatively small part of the total costof distribution system. Substations are the most involvedcomponent of distribution system when it comes to theplanning of the system. This stage is considered by 60%of utility planners as one of the stages in the transmissionsystem planning process, 20% consider it a stage in distri-bution system planning, while the rest (20%) deals with itas a separate process 1,2.Substation planning involves substation site selection,substation size and service areas determination, and sub-stations equipment installation timing. The site selectionof the substation is not entirely based on electrical consid-eration. City plans and environmental restrictions are usu-ally the main determining factors in this process. In the bestscenario, the city will provide the planner with a set ofavailable sites (candidate sites) to choose from. It is possi-ble that none of these sites will meet the optimal solution.However, the planner has to select the second best site. Ingeneral, the substation site selection process is consideredas a screening process, through which all-possible site loca-tions are investigated and classified into unsuitable, candi-date, and future evaluation sites. This process is illustratedin Fig. 1. The substation size and the service area are usu-ally determined based on electrical considerations and con-straints, such as: equipment capacities and feeders voltagedrop constraints 1,2.Several distribution planning models have been devel-oped that can be divided into four main categories 3,4:0142-0615/$ - see front matter ? 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijepes.2007.10.002*Corresponding author. Tel.: +519 888 4567x7061; fax: +519 746 3077.E-mail address: telfoulyhivolt.uwaterloo.ca (T.H.M. El-Fouly)./locate/ijepesAvailable online at Electrical Power and Energy Systems 30 (2008) 308315 Static Load Subsystem Models: Determine the size andthe location of either the distribution substation or pri-mary feeders. Static Load Total System Models: Determine the opti-mal substation location and size, network routing, loadtransfer among stations and feeders sizes. Dynamic Load Subsystem Models: Determine the size,the location, and timing of installing substations andits equipment or the optimal feeders routings. Dynamic Load Total System Models: Determine thesize, the location, and the timing of installation of thedistribution substations and primary feeders.In 5, an approach to determine the sizing and timing ofsubstations was proposed. In this approach sizing and tim-ing were effectively decoupled by using the Pseudo-Dynamic approach. This approach requires sequentialapplicationsofthesingle-time-periodstaticplanningmodel. Moreover, in the proposed algorithm, the questionof whether or not to construct a substation is based com-pletely on voltage drop considerations. The major draw-back of this method is that voltage forecasts, whichdecide the possibility of constructing a substation, arebased on the assumption that load densities are uniformwithin a substation service area. This is not true for mostpractical cases. Moreover, this model did not take into con-sideration the equipment locations. In 6, a transportationapproach for solving the substation location, sizing, andserviceareaproblemwasdeveloped.Thisapproachassumed that the total demand is equal to the total supplyand the objective was to determine a feasible flow patternthat minimizes the total transportation cost, while satisfy-ing all demands. This approach did not include the equip-ment costs in its objective function and also modeled allexisting and potential substations as source nodes, whichleads to an optimal solution with all substations being uti-lized, even though some of them serve only a small amountof load. Moreover, this method did not consider any con-straints, such as voltage constraints, in its solution. In addi-tion, no voltage drop calculations were included. In 7, afixed charge transshipment model for the problem ofchoosing an optimal substation location was developed.The objective function of the developed model includedboth the fixed and the variable cost components and wassolved using an integer branch-and-bound technique.However, this developed model was a static model; it didnot consider any variation in the demands with time.Moreover, it did not include any constraints for voltagelimits. A fixed charge transshipment network procedureto solve the multi-period distribution system-planningproblem was suggested 8. This technique was used foroptimal distribution substation and primary feeders plan-ning. However, this technique did not include any con-straints for voltage limits.In 9 and 10 a Heuristic Combinational Optimizationalgorithmwasproposedtodeterminetheoptimumrequired substations capacities and then a Multi sourceLocating algorithm is used to allocate the substations byminimizing the cost of energy losses on the feeders. Thisprocedure does not require the selection of candidate sub-station locations. In 11 an adaptive mutation particleswarm optimization algorithm was developed to solve forthe optimal substation location and sizing. This approachdoes not require candidate substation location and it takesinto account both the substation construction investmentand the geographic information system (GIS). An optimalsubstation service area and feeder routing method based onminimum feeder loss was developed 12. In this method, aGIS with distribution data base, computer graphics, andthe minimal-path and the load switch pattern algorithmwere integrated to solve the planning problem. The mini-mal-path algorithm was used to redistribute the loadpoints. Load between two switched were lumped andassigned to the switch at the farther end from the maintransformer. The load switch pattern was used to connectthe feeder paths for the substation and to distribute theload points. Constraints such as flow limits, power flow,and network radiations were taken into account. Anothersubstation expansion planning procedure was developed13. It proposed a mathematical clustering technique todetermine the feasible candidates while considering thesubstation capacities, feeder capacities, and voltage regula-tions limitations. After that, a genetic algorithm is used tosolve the optimization problem for expansion requirementsfor existing substations and new substation allocations andcapacities determination. These aforementioned proce-dures 913 did not include any constraints for voltage lim-its. Moreover, the problem formulations did not consider atime varying demand.In 14, a probabilistic methodology for distribution sub-station location selection was presented. This methodologytook into account the hourly (or daily) load cycle. For dif-ferent hourly load scenarios, the load center locations aredetermined and weighted according to their load magni-tude. These locations are then used to develop a probabilitydistribution that is used in determining the maximum prob-Fig. 1. Substation siting selection process.T.H.M. El-Fouly et al. / Electrical Power and Energy Systems 30 (2008) 308315309ability perimeter of the area where the substation should belocated. The process also takes into account factors such asland availability and the cost of land. A model developednon-discrete functions for distribution substation sizing,sitting, and timing taking into account the different compo-nents for the substations cost function and various con-straints including, voltage, power flow, radial flow, andcapacities constraints was presented 15. Moreover, themodel considered a time varying demand for the sectorsunder investigation. However, the main drawback in thismodel is that the optimization process is carried out foreach planning interval independent on the results obtainedfor previous intervals. This results in some feeders beinginstalled at earlier periods then removed latter with newfeeders installed in the next periods, which is practicallyinfeasible. Moreover, the problem was formulated as aMixed Integer Nonlinear Programming (MINLP), whichcould result in local optimal solutions due to nonlinearity.This paper addresses these drawbacks.This paper is organized as follows: Section 2 presents thesystem under study. The proposed problem formulationfor the distribution substation siting, sizing, and timingoptimization model is presented in Section 3. The proposedproblem formulation was modeled and solved using theGeneral Algebraic Modeling Software (GAMS) solvers16. In Section 4, the results generated from the proposedoptimization model are presented. A modified problem for-mulation to ensure preventing inefficient transmission ofpower is discussed in Section 5. This section also presentsthe generated results from the modified formulation.Finally, in Section 6, conclusions are presented.2. System under studyThe service area under investigation consists of 9 sectorsas shown in Fig. 2. The area of each sector is 0.44 km2andit is assumed that the proposed sectors for substationinstallation by the city are sectors 2, 4, and 6. The planningperiod is set to be 10 years divided into 5 time intervals of 2years each. The sectors demand growth over the 10-yearplanning period (5 intervals) is given in Table 1 15. Eachsubstation is rated at 40 MVA and can be equipped with amaximum of two transformer units each rated at 20 MVA.The units ratings, proposed locations, and assigned vari-ables are given in details in Table 2.The substation fixed cost is assumed to be $200,000while the transformer unit installation cost is assumed tobe $150 and the cost of energy is assumed to be 0.17 $/kWh. The interest, tax, inflation, and insurance rates areconsidered to be equal to 10%, 10%, 6%, and 1%, respec-tively. Transformer units are allowed to be loaded to 75%of its rated value. It is also assumed that the maximumallowable voltage drop along each feeder is 275 V (2.5%of the system nominal voltage which is set to be 11 kV).The transformer copper loss at rated power is assumed tobe 127 kW 15. Nine feeders are available for each substa-tion (total available feeders are 27 feeders), and each feederis assumed to supply the sector demand directly withoutintermediate points as given in Table 3. This table alsoFig. 2. Area under study indicating the proposed substation sites by thecity.Table 1Sector demands over studied periodsDp,n(MVA)Interval (n)Sector # (p)1234567891221.51414162344352.5547355545375746664.5537675767553767Table 2Proposed units capacities and variablesUnit number (m)TypeRating (MVA)LocationVariable1Substation40Sector # 4xx1,n2Substation40Sector # 6xx2,n3Substation40Sector # 2xx3,n4Transformer20Sector # 4xx4,n5Transformer20Sector # 4xx5,n6Transformer20Sector # 6xx6,n7Transformer20Sector # 6xx7,n8Transformer20Sector # 2xx8,n9Transformer20Sector # 2xx9,nTable 3Feeders variables and their available routesVariableRoutesectornumberVariableRoutesectornumberVariableRoutesectornumberFromToFromToFromToX141X1061X1921X242X1162X2022X343X1263X2123X444X1364X2224X545X1465X2325X646X1566X2426X747X1667X2527X848X1768X2628X949X1869X2729310T.H.M. El-Fouly et al. / Electrical Power and Energy Systems 30 (2008) 308315presents the assigned variable for each available feeder andits available route.3. Problem formulationWhen planning to install a distribution system substa-tion and its components, the main objective is to minimizethe overall cost of equipment installation and energy losses.This cost depends on factors such as substation siting, tim-ing of equipment installation, and equipment (transformer)loading. Regarding the substation siting, increasing thenumber of installed substations or improper proposed siteselection could greatly increase the overall system cost.Interest rates, inflation rates, taxes, and insurance ratesimpact the timing of installation of equipment, and hencethe overall cost is affected. Since the amount of energy lossis dependent on the equipment loading, an increase in load-ing level will result in an increase in the overall cost. More-over, previous planning models based on a one-intervalperiod could result in an impractical solution such asinstalling a feeder in an earlier period and then removingit in a latter period 15. In such cases, human expertise isrequired to eliminate these impractical solutions. To takeinto account all these factors, the paper proposes a newproblem formulation that minimizes the overall cost. Theproblem was formulated over the whole planning periodto avoid the necessity of human expertise and to accom-plish the following targets:1. Determining the optimal time of equipment installation.2. Adequately determine the siting and sizing of thesubstations.The main objective over the whole planning horizon canbe written as follows:cost Xqn1RnbnCS1? S1;n CS2? S2;n CS3? S3;n CT11? S4;n CT12? S5;n CT21? S6;n CT22? S7;n CT31? S8;n CT32? S9;n? 8760 ?Pcu? CHTr?X9i4xi;n1where q is equal to the number of design intervals withinthe 10-year planning period and is set equal to 5. Thus, atwo-year period is chosen for each design interval to pro-vide sufficient time for equipment installation. CS1, CS2,and CS3are the fixed cost for substations 1, 2, and 3,respectively, CT11, CT12are the cost of the two transformerunits to be installed at substation 1 (including the cost ofthe iron losses), CT21, CT22are the cost of the two trans-former units to be installed at substation 2 (including thecost of the iron losses), CT31, CT32are the cost of the twotransformer units to be installed at substation 3 (includingthe cost of the iron losses), Si,nis a binary variable indicat-ing the installation of unit i at a given period n, Pcuis thetransformer copper loss at rated power (kW), C is the costof energy ($/kWh), HTris the transformer unit rating(MVA), xi,nis the power delivered from unit (transformer)i at a given period n if this unit (transformer) is installedand it is set to zero if the unit (transformer) is not installed,Rnand bnare the fixed charge rate and the present worthfactor for a given interval n respectively, and are calculatedas follows:Rn i t r r1 r2q1?n? 18 n;wheren 1;.;q2bnf ? r1f1r?2q1?n? 1?8;wheren 1;.;q3where r, t, f, i are the interest, tax, inflation, and insurancerates, respectively.In this formulation, transformers power losses are calcu-lated as a percentage of the transformer loading. This per-centage is assumed equal to the ratio of the transformercopper loss at rated power (Pcu) to the transformer unit rat-ing (HTr). The objective function is minimized subject tothe following constraints:3.1. Additional constraints to overcome nonlinearityA decision variable yi,nis used to determine whether ornot a transformer is delivering power. This was done bymultiplying the variable xxi,n(power delivered by trans-former) by the binary variable yi,n. Unfortunately, this willenforce nonlinearity in the problem. In order to avoid this,a variable xi,nis introduced to replace the product of bothvariables and constraints are added as follows:0 6 xi;n6 xxi;n8 i;where i 4;.;9 and 8 n;where n 1;.;q4xxi;n? M1 ? yi;n 6 xi;n6 M ? yi;n8 i;where i 4;.;9 and 8 n;where n 1;.;q5where M is a big number and was chosen to be equal to10,000 to guarantee that the constraint shown in Eqs. (4)and (5) will converge to indicate whether a transformer isused or not, xxi,nis the power delivered from unit (trans-former) i at a given period n (MVA), and yi,nis a binaryvariable indicating the dissipated power from unit i at a gi-ven period n. A value of yi,nequal to 1 indicates that poweris transferred through the transformer while a value of zeromeans that no power is transferred.3.2. Fixed cost constraintsAs mentioned earlier, Sijrepresents a binary decisionvariable that determines the installation of a unit i in a cer-tain year j. The cost of a unit will vary depending on theyear it is installed due to the change in both R and b. A bin-ary variable F was added which relates the amount ofT.H.M. El-Fouly et al. / Electrical Power and Energy Systems 30 (2008) 308315311power supplied by a unit in two consecutive years to thedecision variable Sij.Si;16 Mxi;18 i 1;.96Si;1Pxi;1308 i 1;.97Fi;j6 Mxi;j8 i 1;.9 and 8 j 2;.58Fi;jPxi;j309Si;jP ?Mxi;j?1 Fi;j8 i 1;.9 and j 2;.510Si;j6 ?xi;j?130 Fi;j8 i 1;.9 and j 2;.511The first two constraints focus on the installation of a unitin the first year. If power is being delivered by a unit on thefirst year, Si1will equal 1 and thus indicating the operationof this unit. The variable Fijindicates whether a unit hasbeen installed starting from the second period as high-lighted in Eqs. (8) and (9). Eqs. (10) and (11) have been for-mulated to force variableSijto equal 1 once a unit i hasbeen installed in period j and above.3.3. Capacity constraintsEach substation and each transformer has a capacity of40 MVA and 20 MVA, respectively. However, they areallowed to be loaded to 75% of their rated capacity formaximum efficiency operation. This results in capacity lim-its of 30 MVA and 15 MVA for each substation and eachtransformer, respectively. Moreover, the lower limits forthe substations and transformers loading are set to zero.This could be expressed as follows:0 6 xxi;n6 15yi;n8 i;where i 4;.;9 and 8 n;where n 1;.;q120 6 xxl;n6 308 n and where l 1;2; and 3133.4. Power flow constraintsThese constraints represent the law of conservation ofenergy, where the total loading of each substation at agiven time interval n is equal to the sum of loading of itsindividual transformers units and at the same time equalsto the sum of the demands of the sectors supplied by thissubstation and the total copper loss of the substationsunits. These constraints are expressed as follows:xx1;nX9z1X9p1Dp;nxzhiPcu1000xHTr? x4;n x5;n8 n; where n 1;.;q14xx2;nX18z10X9p1Dp;nxzhiPcu1000xHTr? x6;n x7;n8 n; where n 1;.;q15xx3;nX27z19X9p1Dp;nxzhiPcu1000xHTr? x8;n x9;n8 n; where n 1;.;q16xx1;n xx4;n xx5;n8 n; where n 1;.;q17xx2;n xx6;n xx7;n8 n; where n 1;.;q18xx3;n xx8;n xx9;n8 n; where n 1;.;q19where Dp,nis the demand of sector p at a given period n(MVA), and xzis a binary variable indicating whether acertain sector is being supplied from a given substation ata given period n or not.3.5. Radial flow constraintEach feeder is assumed to supply a sector demanddirectly from a certain substation. Moreover, it is assumedthat each sector is supplied from one substation only to sat-isfy the radial flow constraints. These constraints can beexpressed as follows:xz xz9 xz18 18 z; where z 1;.;9203.6. Voltage constraintsThe voltage constraints are used to ensure that the volt-age levels at all nodes are within the standard permissiblelevels. These constraints are expressed in the form of limit-ing the voltage drop along each feeder to a specified max-imum limit. These constraints can be expressed as follows:DVj;n1000Dp;nVnominalZj? xj;n8 j and 8 p;where j 1;.;27 and where p 1;.;9210 6 DVj;n6 DVmax8 j and 8 n;where j 1;.;27 and where n 1;.;522where DVj,nis the voltage drop along the feeder j at a givenperiod n (V), Vnominalis the system nominal voltage (kV),Zjis the electrical impedance of the feeder j (X), xj,nis a binaryvariable indicating the existence of feeder j at a given per-iod n, and DVmaxis the maximum allowable voltage droplimit (V).4. Optimization resultsThe problem formulated in Section 2 was solved usingthe OSL solver in GAMS. Eqs. (1)(22) were solved andthe optimal system configuration is shown in Fig. 3. Thisconfiguration results in a total optimized cost over the10-year planning period of $2,839,253.817. Table 4 pre-sents the status of the feeders variables, and it can be con-cluded that there is no need to install Substation 3.However, this table indicates that the optimal solutioninvolves installing Feeder X6that feeds sector 6 (whereSubstation 2 is installed) from Substation 1 installed in sec-tor 4, which is an inefficient routing for a feeder. Despite oftransferring the power to far sectors, all involved con-straints, including the voltage drop constraints, have beensatisfied as presented in Tables 5 and 6. These constraintshave not been violated because sector 6 has the smallest312T.H.M. El-Fouly et al. / Electrical Power and Energy Systems 30 (2008) 308315demand over the studied period. The presented resultsreveal that the required loading for substations 1 and 2are 28.179 MVA and 25.16 MVA, respectively. The servicearea of substation 1 and 2 are presented in Fig. 3. Theresults also revealed that four transformers will be neededin this configuration and that two transformers will beinstalled in the first period while the other two will beinstalled in the second period. It can be seen that the pro-posed problem formulation takes into account the timingof installation of the transformers by delaying the installa-tion of some of the transformers. The main advantages ofthe proposed formulation over the previously proposedformulation in 15 are:1. The problem does not include any nonlinear functions.2. The planning problem is optimized over the entire plan-ning period and thus, avoiding any impractical solutionssuch as installing a feeder at one period and removing itin the next coming period.The formulation included three possible sitings for thesubstation (more than the loading requirements), thusallowing for the algorithm to choose the optimal numberof substations based on the installed loading levels.5. Modified formulationTo ensure the prevention of inefficient transmission ofpower (i.e. to a far sector), the same problem formulationwas solved, but the sum of voltage drops is added in themain objective function. The reason behind this is to min-imize both the overall cost and the voltage drops simulta-neously. Both objective functions were normalized andmultiplied by weights where the sum of these weights isequal to one. The new objective function can be writtenas follows:minw1?costcostmax w2Pqn1Phi1DVj;nhiPqn1Phi1DVj;nhimax26437523where w1and w2are the assigned weights for the normal-ized functions and they have been assumed to be 0.8 and0.2, respectively, costmaxis the maximum permissible costfor the system (calculated assuming that two substationsare equipped with four transformers being installed fromthefirstperiod)andisequalto$2,839,313.817,Pqn1Phi1DVj;nhiis the summation of all voltage dropsvalues obtained in the problem,Pqn1Phi1DVj;nhimaxisthe maximum permissible summation for the voltage dropsand is given by:Pqn1Phi1DVj;n?max p ? number of feeders ? DVmax 9x9x275 22275V24Eqs. (2)(22) were solved for the optimal system configura-tion with the objective taking the form given in (23) and theresults obtained are presented in Fig. 4. This configurationresulted in a total cost of $2,839,253.817 over the 10-yearplanning period, which is equal to that obtained from theFig. 3. Feeders status for the main formulation.Table 4Feeders variables status (main formulation)VariableX1X2X3X4X5X6X7X8X9Status100101110VariableX10X11X12X13X14X15X16X17X18Status011010001VariableX19X20X21X22X23X24X25X26X27Status000000000Table 5Substation and transformers loading over the studied period (mainformulation)UnitInterval number1234519.05817.61224.15326.66928.179213.58620.12822.14124.15325.16030000049.05815.00015.00015.00015.000502.6129.15311.66913.179613.58615.00015.00015.00015.000705.1287.1419.15310.160800000900000Table 6Calculated voltage drops over the selected feeders (main formulation)SectorInterval number12345X160.78291.174151.956182.347212.738X660.782151.956182.347182.347182.347X7121.565151.956212.738212.738212.738X842.980171.919214.898257.878257.878X1185.959171.919214.898257.878257.878X1245.587121.565151.956182.347212.738X14121.565151.956151.956151.956151.956X18182.347212.738212.738212.738212.738Note. The voltage drop for X4is zero.T.H.M. El-Fouly et al. / Electrical Power and Energy Systems 30 (2008) 308315313previous formulation. However, the data in Table 7 (pre-senting the status of the feeders variables) reveals thatthe new optimal solution does not include the installationof Feeder X6and sector 6 (where Substation 2 is installed)is supplied from Substation 2, which is much favorablethan the previously obtained results. Moreover, the resultsreveal that there is no need to install Substation 3 and thatall involved constraints have been satisfied as presented inTables 8 and 9. Fig. 4 presents the service area of substa-tions 1 and 2. As in the previous formulation, four trans-formers would be required to be installed to satisfy theloading levels during the 10-year planning period. Two ofthem are to be installed in the first interval, while the othersare to be installed in the second interval. Thus, the totalcost over the 10-year planning period from both proposedformulations is the same. The main distinguishing differ-ence of this formulation than the previous one is the pre-vention of inefficient transmission of power.Results using the proposed modified problem formula-tion are compared with the corresponding results generatedfor the same system by Temarz and Salama 15 (where theproblem was formulated as a Mixed Integer NonlinearProgramming). This comparison revealed the following: In Temraz and Salamas results for units loading condi-tions presented in Table 10, there are non-logic changingin the substations and transformers loading conditionsfrom one time interval to the next interval. For example,a decrease in Substation 1 (SS1) loading condition (from14.09 MVA at interval 1 to 10.016 MVA at interval 2) isobserved. This is non-logic solution with the continu-ously increasing demand. Such variations are notobserved in the proposed problem formulations Table 8. Temraz and Salamas results revealed that some feedersbeing installed at a specific interval and then removed inthe next intervals, which is practically infeasible. Theproposed formulations overcome such deficiency byforcing any installed feeder at earlier intervals to remaininstalled during later intervals. Temraz and Salama formulated the planning problem asa Mixed Integer Nonlinear Programming (MINLP),which could result in local optimal solutions due to non-linearity. This paper addresses this drawback by formu-lating the planning problem as a Mixed Integer LinearProgramming (MILP). The planning problem formulation by Temraz and Sala-ma did not actually solve the substation location prob-lem as it proposed two candidate sites only for thesubstations and the demand ratings exceeds the capacityFig. 4. Feeders status for the modified formulation.Table 7Feeders variables status (modified formulation)VariableX1X2X3X4X5X6X7X8X9Status100110100VariableX10X11X12X13X14X15X16X17X18Status011001011VariableX19X20X21X22X23X24X25X26X27Status000000000Table 8Substation and transformers loading over the studied period (modifiedformulation)UnitInterval number12345111.07016.10221.13422.64424.153211.57321.63725.16028.17929.185300000411.07015.00015.00015.00015.000501.1026.1347.6449.153611.57315.00015.00015.00015.000706.63710.16013.17914.185800000900000Table 9Calculated voltage drops over the selected feeders (modified formulation)SectorInterval number12345X160.78291.174151.956182.347212.738X5121.565151.956151.956151.956151.956X7121.565151.956212.738212.738212.738X1185.959171.919214.898257.878257.878X1245.587121.565151.956182.347212.738X1742.980171.919214.898257.878257.878X18182.347212.738212.738212.738212.738Table 10Substation and transformers loading over the studied peri
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