220kv降壓變電所電氣一次系統(tǒng)設計275

220kv降壓變電所電氣一次系統(tǒng)設計275,kv,降壓,變電所,電氣,一次,系統(tǒng),設計 華 北 電 力 大 學 科 技 學 院 畢 業(yè) 設 計（論 文）附 件 外 文 文 獻 翻 譯 學 號： 081901090217 姓 名： 龐斌 所在系別： 電力工程系 專業(yè)班級： 農電08k2 指導教師： 蘇海鋒 原文標題：A new optimization model for distribution substation siting, sizing, and timing 2012年 6 月 20 日 一個新的配電變電站選址，容量和定時的優(yōu)化模型 一﹑ 摘要 本文提出了一種新的配電變電站選址，定容和定時規(guī)劃的優(yōu)化模型。該模型使用線性函數來表達總成本函數。該模型包括不同的電氣約束，像電壓降落，變電站和變壓器容??量，潮流和徑向流的限制.這種規(guī)劃問題，被作為一個混合整數線性規(guī)劃問題來進行制定，從而避免使用非線性規(guī)劃，也同時避免了可能被困 地的解決方案。一個數值例子來驗證模型的有效性。 關鍵詞：配電系統(tǒng)規(guī)劃;配電變電站;優(yōu)化;徑向分布系統(tǒng) 二、主要內容 1 .介紹 配電變電站規(guī)劃被認為是在電力系統(tǒng)規(guī)劃過程中邁出的重要一步。這是因為它代表電力傳輸和配電網之間的主要聯(lián)系?？梢赃x擇的變電站選址和變電站的容量都會對變電站在傳輸和配電系統(tǒng)的規(guī)劃過程都明確的限制，因此，變電所設計參數對饋線路有很大的影響。雖然變電站的成本對于配電系統(tǒng)規(guī)劃總成本相對較小，但是其對配電系統(tǒng)規(guī)劃的整體經濟有一定的限制。，變電站是最復雜的，尤其是當它涉及到配電系統(tǒng)的規(guī)劃時。60％實用規(guī)劃者認為這個一階段發(fā)生在在傳輸階段系統(tǒng)規(guī)劃的過程中，20％認為這是在配電系統(tǒng)規(guī)劃中，而其余的（20％）認為是一個單獨的進程[1,2]。 變電站規(guī)劃涉及變電站選址，變電站的規(guī)模和服務領域的決心，和變電站'設備安裝的時間。變電站的選址不完全是基于對電力的考慮。通常，城市規(guī)劃和環(huán)境的限制在這個過程中的是主要決定因素。在最好的情況下，該市將提供一套規(guī)劃可用的地址以供選擇。然而這些地址很有可能都不是最佳的選擇方案。這時，規(guī)劃選址就不得不選擇第二個好的方案。在一般情況下，變電站選址過程中被視為作為一個篩選過程，通過這個過程所有可能的地點被調查分為不合適，候選的，或者是將來需要評價的。通常，變電站的規(guī)模和生活區(qū)決定于電氣因素和制約因素，如：設備容量和饋線的電壓下降等制約因素[1,2]。 已開發(fā)幾個布局規(guī)劃模型可以劃分為四個主要類別[3,4]： ?靜態(tài)負載的總系統(tǒng)模型：確定最佳變電站的位置和規(guī)模，網絡路由，負載和饋線大小之間的負荷傳輸。 ?動態(tài)負載子系統(tǒng)模型：確定規(guī)模，變電站的位置，安裝變電站其設備和“最佳饋線路線的時間。 ?動態(tài)負載的總系統(tǒng)模型：確定規(guī)模，位置和安裝配電變電站設別和主饋線的時間。 [5]中提出了確定變電站規(guī)模和時間的方案。在這個方案中通過使用偽動態(tài)的方法變電站的規(guī)模和時間確定會分開確定。這種方法需要連續(xù)應用單時間周期靜態(tài)規(guī)劃模型。此外，在這個算法中，是否興建變電站是完全基于電壓下降的考慮。大家都知道電壓預測決定變電站的建設因素，然而這種方法的主要缺點是電壓的預測主要基于在變電站的服務區(qū)內負荷密度均勻的假設，對于實際情況這不是可行的。此外，該模型沒有考慮到設備的位置。在 [6]中，為解決變電站的位置，規(guī)模的確定交通運輸方式方面的問題得到了發(fā)展。這種方法假定總需求等于總供給，它的目標是確定一個可行性模式，這種模式最大限度地減少了運輸成本，同時滿足所有的需求。這種方法不包括設備在目標函數方面的費用，而且將現有的和可能有的變電站作為藍本，這種方法對正在運行的所有變電站，盡管他們中只承擔一些小負荷，會產生一個最佳的解決方案。然而，這種方法沒有考慮任何約束，如電壓的限制。此外，也不包括電壓下降方面的計算。 在[7]中，為了最優(yōu)的變電站選址，提出了一種固定費用的中轉模型。該模型的目標函數包括固定和可變成本的構成，通過使用一個整數的分支定界技術來進行解開。然而，這個模型是靜態(tài)模型，不考慮任何隨著時間的要求的變化因素。此外，它不包括任何限制例如電壓限制。為了解決了多期的配電系統(tǒng)規(guī)劃的問題，固定費用中轉模型網絡程序問題的建議在[8]中被提出來。這種技術被用于優(yōu)化配電變電站和主饋線規(guī)劃。然而，這種技術不包括任何限制如電壓限制。 [9][10]的啟發(fā)式組合優(yōu)化算法來確定變電站的最佳容量，同時通過損失最小化降低在饋線損失，一種多源定位算法用于變電站的容量分配。這程序不需要選擇候選變電站位置。 在[11]中為了解決最佳變電站的位置和規(guī)模一個自適應變異粒子群優(yōu)化算法的被提出。這種方法不需要候選變電站的位置同時考慮到變電站建設的投資和地理信息系統(tǒng)（GIS）。一種基于最低饋線損耗的變電站綜合服務區(qū)和饋線路方法在 [12]提出來。在此方法中，與分布數據的基礎上，計算機圖形學，地理信息系統(tǒng)最小的路徑和負荷開關模式算法綜合起來解決規(guī)劃問題。最小路徑算法被用來重新分配負載點。在遠離主變的末端，開關的負荷進行集中和分配。負荷開關模式用于連接變電站饋線路徑和分發(fā)加載點。許多約束，如潮流約束，功率流動和網絡輻射都考慮其中。另一個變電站擴建規(guī)劃程序被開發(fā)出來[13]。為了確定可行的候選地址，它提出了一個數學的聚類技術，同時考慮到變電站容量，饋線容量和電壓限制。然后，為了解決現有的變電站和新的變電站分配和容量的擴展的最優(yōu)解決方案問題一種遺傳算法被提出來。這些上述程序[9-13]不包括電壓范圍內的任何約束。然而，問題的解決方案沒有考慮隨時間變化的需求。 在[14]中，配電變電所通過概率方法來選址。這種方法考慮到了每小時（或每天）負載周期。對于不同的逐時負載的情況，根據其負載大小，來進行負荷中心位置的確定和加權。然后用這些地點建立一個概率分布，來確定應位于變電站的概率的最大周長，過程中還考慮到，如土地供應和土地成本等因素。在15 中提出了一種非離散函數，這種模型考慮到變電站成本和各種約束條件等關于變電站選址，規(guī)模和時間的因素。此外，該模型考慮到隨時間變化的需求。然而，在這個模型中的主要缺點是，每個規(guī)劃間隔獨立于以前的時間間隔所取得的成果。這導致了在被安裝在較早時期的一些饋線，然后移除后與新的饋線安裝在以后時期，這實際上是不可行的。此外，這個問題導致了一個混合整數非線性規(guī)劃（MINLP），由于非線性，這可能影響最佳解決方案。 本文組織如下：第二節(jié)介紹了正在研究的系統(tǒng)。在第3節(jié)提出了對配 變電站選址，容量，和時序的選擇的優(yōu)化模型.使用建模和解決提出的題制定,通用代數建模軟件（GAMS）求解。在第4節(jié)，從提出的優(yōu)化模型產生出結果。一個修改的擬定，來避免低效的電力傳輸的問題在第5節(jié)討論. 本節(jié)還提出了修改制定生成的結果。 2.研究制度 根據調查的服務區(qū)由9個部門構成如圖2。每個部門的面積為0.44平方公里，假設變電站安裝部門，由城市部門2，4和6構成。規(guī)劃期被設定為10年， 每2年的時間間隔共5次。表1[15]給出的行業(yè)需求的增長超過10年的規(guī)劃期內（每隔5）的增長。 每個變電站的額定功率為40兆伏安，可配備了兩個變壓器，每個最高額定為20兆伏安。每個變壓器的效率，地點，并分配給變量在表2中的細節(jié)被提出來。 變電站固定成本假設是$ 200,000，而變壓器的單位安裝成本假定為150美元，能源成本假定為0.17元/千瓦時。利息，稅收，通貨膨脹，保險費率被認為是10％，10％，6％和1％。變壓器允許被加載到其額定值的75％。它也被認為沿著每條饋線的最大允許壓降為275 V（下被設定為11千伏系統(tǒng)的額定電壓的2.5％）。在額定功率的變壓器銅損假設是127千瓦[15]。每個變電站九饋線（總可用饋線27饋線），并假定每個饋線直接供應相應部門的需求的（表3）。此表還介紹了為每個可用饋線和現有的路線分配的變量。 3．問題制定 當規(guī)劃安裝配電系統(tǒng)變電站及其部件，其主要目標是盡量減少設備的安裝和能量損失的整體成本。這個費用取決于一些因素，如變電站選址，設備安裝時間，設備（變壓器）負荷。對于變電站的選址，增加安裝變電站或不當的選址，會大大增加整個系統(tǒng)的成本。利率，通貨膨脹率，稅款，保險費率的影響設備安裝的時間，從而整體成本也受到影響。由于能量損失的大小是取決于設備的負荷，負荷水平的增加將導致整體成本增加。此外，以一個間隔期為基礎的以往的規(guī)劃模型上，可能會導致不切實際的解決方案，如安裝在同期饋線，然后在后期消除[15]。在這種情況下，就需要人們用專業(yè)知識來消除這些不切實際的解決方法?？紤]到所有這些因素，提出了一個新的方案來最大限度地減少整體成本。為了避免人類專業(yè)知識的必要性，以實現以下目標： 1確定設備安裝的最佳時機。 2充分確定變電站的選址和容量。 在整個規(guī)劃地平線的主要目標 寫成如下： 其中q是在10年的規(guī)劃期內，設計間隔數等于設置等于5。因此，兩年內選擇每個設計間隔提供足夠的時間進行設備安裝。CS1，CS2和CS3是分別為固定成本變電站1，2，3,，CT11，CT12是變壓器的兩個單位的成本將安裝在變電站1. CT21，CT22是變壓器的兩個單位的成本將安裝在變電站2. CT31，CT32是變壓器的兩個單位的成本將安裝在變電站3. C是能源成本（元/千瓦時），Htr是變壓器單元（MVA）的評級，Xin是從單位交付的電力.I在給定的周期n安裝單元（變壓器）。如果未安裝單元（變壓器）它設置為零。Rn和BN是固定的收費率和現值因素，對于一個給定的時間間隔n分別計算. 如下： 其中，R，T，F，分別為利潤，稅收，通貨膨脹和保險費率。 在此方案中，變壓器的功率損耗依據變壓器負載的百分比來進行計算。假設這一比例相當于在額定功率（PCU）時變壓器的銅損變壓器單元評級（HTR）的比率。 3.1 額外的限制，以克服非線性 決策變數yin是用來確定是否變壓器提供電力。通過乘以變量xxin（由變壓器提供電源）二進制變量yin。然而，這將強制非線性問題。為了避免這種情況，變量xin被推出來取代雙方的產品 一些變量和約束被添加如下： 其中M是一個大數，并選擇是等于10000，以保證約束式顯示。 （4）和（5）將表明是否使用或變壓器，XXin是從單元（變壓器）i在一定時期內N（MVA）的電力功率，Yin是一個二進制變量表示在給定的周期n的功率耗散。一個它的價值等于1表示電流通過變壓器傳輸，而零表示沒有電流傳出。 3.2 固定成本約束 如前所述，Sij表示二進制的決策變量，決定了一個單位的安裝. 一個單位的成本取決于今年由于安裝引起的變化。二進制變量F增加，其中影響的發(fā)電量為在連續(xù)兩年的單位提供決策變量。 3.3 容量限制 每個變電站，每個變壓器容量分別40兆伏安和20兆伏安。然而，他們被允許加載到其額定容量的75％從而最高效率運作。這個導致了對于每個變電站，每個變壓器有30兆伏安和15兆伏安容量的限制. 此外，變電站和變壓器負荷的下限設置為0.這可以表示如下： 3.4 功率流的限制 這些約束代表的能量守恒定律，各變電站的總負載在一個給定的時間間隔n內等價于個人變壓器單位的負荷總和，同時，在同一時間內等于本部門要求提供的總變電站和子變電站單位的總銅損。這些約束先解釋如下： 3.5 徑向流約束 假定每條饋線提供每塊地區(qū)的電能需求，直接從某變電站提取。而且，假定每塊區(qū)域從一個變電站得到電能供給，主要是為了滿足徑向流動的限制。這些限制可以表示如下： 3.6電壓限制 電壓的限制是為了確保在每個節(jié)點上的電壓保持在允許的水平。這些約束以每條饋線壓降的最大值來表示。這些約束可以表示如下： 是在被給定時間N內沿饋線j 的電壓降，Vnominalis表示系統(tǒng)的額定電壓（kV）。ZJ是饋線J的阻抗，XJ，n是一個二進制的變量，表示在給定的一段時間內存在饋線的存在。表示最大電壓降落。 4.優(yōu)化結果 在第2節(jié)提出的問題解決了，在GAMS很容易使用的方式解式。（1） - （22）的問題解決了而且最優(yōu)的系統(tǒng)配置如圖 3。這種配置造成在的10年規(guī)劃期內，總成本為2,839,253.817美元。表4給出了饋線的變量的狀態(tài)，可以得出的結論是沒有必要安裝變電站3。然而，此表顯示，最佳的解決方案包括安裝饋線6來為區(qū)域6供電，這是一種低效饋線路由（2變電站已安裝）。盡管電力要輸送到較遠的區(qū)域，但是所有涉及的限制，包括壓降限制，均已得到滿足，如表5和表6。這些制約因素還沒有顯示出來，因為區(qū)域6在研究期間的的需求最小。結果表明，所需的負荷變電站1和2分別為28.179兆伏安和25.16兆伏安。變電站1和2的服務區(qū)域在表3中顯示。調查結果顯示，在這個配置規(guī)劃中需要四臺變壓器同時兩臺變壓器要在第一規(guī)劃期內安裝，另外兩臺在第二規(guī)劃期內安裝。同時可以看出這個問題的制定是通過延時一些變壓器的安裝來考慮整體變壓器的時限問題的 。比先前15中提出的問題的主要優(yōu)點有： 1 這些問題不包含任何非線性問題 2 在整個規(guī)劃期內這些問題得到了優(yōu)化，從而避免了任何不切合實際的方案，像在第一段時間內安裝一條饋線，可是在第二個規(guī)劃時期就拆除了。 這種問題的制定包括了三個變電站可能的位置，而不僅僅是為了滿足負荷的需要，從而考慮用算式來確定在滿足負荷條件下變電站的數目。 三、 結論 本文提出了一種新的配電變電站選址，規(guī)模和時序規(guī)的劃優(yōu)化模型。這個模型使在這個規(guī)劃時期內的總成本得到了優(yōu)化。通過最大限度地減少整個規(guī)劃期內的成本，從而得到準確，實際的成果。 根據研究，只考慮成本優(yōu)化可能會導致一些問題，在規(guī)劃期內特別低效，對未來負荷的增長也是行不通的。所得的結果顯示，電壓降落不足以防止無效的功率傳輸，尤其是負荷水平在規(guī)劃期內不那么高。所以，修改后的規(guī)劃方案，包括同時最大限度的降低成本和壓降，才能保證這種做法。在這種情況下得出的結論能夠提供一種更好的和高效的供電裝置，從而滿足將來負荷的增長。 原文作者及出處：T.H.M. El-Fouly a,*, H.H. Zeineldin b, E.F. El-Saadany a, M.M.A. Salama a for involves using linear functions to express the total cost function The developed model includes di erent electrical constraints such as voltage 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 is because it represents the main link between transmission system planning process 20 consider it a stage in distri stations equipment installation timing The site selection in 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 voltage drop constraints 1 2 Several distribution planning models have been devel oped that can be divided into four main categories 3 4 Corresponding author Tel 519 888 4567x7061 fax 519 746 3077 E mail address telfouly hivolt uwaterloo ca T H M El Fouly Available online at Electrical Power and Energy Systems and distribution system The available sites and sizes of substations 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 overall economics of the distribution system planning although their cost represents a relatively small part of the total cost of distribution system Substations are the most involved component of distribution system when it comes to the planning of the system This stage is considered by 60 of utility planners as one of the stages in the transmission of 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 best scenario the city will provide the planner with a set of available 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 In general the substation site selection process is considered as 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 illustrated sibility of getting trapped in local solutions A numerical example is presented to validate the e ectiveness of the developed model C211 2007 Elsevier Ltd All rights reserved Keywords Distribution system planning Distribution substation Optimization Radial distribution systems 1 Introduction Distribution substation planning is considered the most important step in the power system planning process This bution system planning while the rest 20 deals with it as a separate process 1 2 Substation planning involves substation site selection substation size and service areas determination and sub A new optimization model siting sizing T H M El Fouly a H H Zeineldin b a Department of Electrical and Computer Engineering University of Waterloo b Masdar Institute of Science and Technology P O Received 24 August 2006 received in revised form Abstract This paper presents a new planning optimization model for distribution 0142 0615 see front matter C211 2007 Elsevier Ltd All rights reserved doi 10 1016 j ijepes 2007 10 002 distribution substation and timing E F El Saadany a M M A Salama a 200 University Avenue West Waterloo Ontario Canada N2L 3G1 Box 45005 Abu Dhabi United Arab Emirates 4 September 2007 accepted 22 October 2007 substation siting sizing and timing The proposed model 30 2008 308 315 and Static Load Subsystem Models Determine the size and the 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 load transfer among stations and feeders sizes Dynamic Load Subsystem Models Determine the size the location and timing of installing substations and its equipment or the optimal feeders routings Dynamic Load Total System Models Determine the size the location and the timing of installation of the distribution substations and primary feeders In 5 an approach to determine the sizing and timing of substations was proposed In this approach sizing and tim ing were e ectively decoupled by using the Pseudo Dynamic approach This approach requires sequential applications of the single time period static planning model Moreover in the proposed algorithm the question Fig 1 Substation siting selection process T H M El Fouly et al Electrical Power of 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 which decide the possibility of constructing a substation are based on the assumption that load densities are uniform within a substation service area This is not true for most practical cases Moreover this model did not take into con sideration the equipment locations In 6 a transportation approach for solving the substation location sizing and service area problem was developed This approach assumed that the total demand is equal to the total supply and the objective was to determine a feasible flow pattern that 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 all existing and potential substations as source nodes which leads to an optimal solution with all substations being uti lized even though some of them serve only a small amount of 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 a fixed charge transshipment model for the problem of choosing an optimal substation location was developed The objective function of the developed model included both the fixed and the variable cost components and was solved using an integer branch and bound technique However this developed model was a static model it did not consider any variation in the demands with time Moreover it did not include any constraints for voltage limits A fixed charge transshipment network procedure to solve the multi period distribution system planning problem was suggested 8 This technique was used for optimal 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 Optimization algorithm was proposed to determine the optimum required substations capacities and then a Multi source Locating algorithm is used to allocate the substations by minimizing the cost of energy losses on the feeders This procedure does not require the selection of candidate sub station locations In 11 an adaptive mutation particle swarm optimization algorithm was developed to solve for the optimal substation location and sizing This approach does not require candidate substation location and it takes into account both the substation construction investment and the geographic information system GIS An optimal substation service area and feeder routing method based on minimum feeder loss was developed 12 In this method a GIS with distribution data base computer graphics and the minimal path and the load switch pattern algorithm were integrated to solve the planning problem The mini mal path algorithm was used to redistribute the load points Load between two switched were lumped and assigned to the switch at the farther end from the main transformer The load switch pattern was used to connect the feeder paths for the substation and to distribute the load points Constraints such as flow limits power flow and network radiations were taken into account Another substation expansion planning procedure was developed 13 It proposed a mathematical clustering technique to determine the feasible candidates while considering the substation capacities feeder capacities and voltage regula tions limitations After that a genetic algorithm is used to solve the optimization problem for expansion requirements for existing substations and new substation allocations and capacities determination These aforementioned proce dures 9 13 did not include any constraints for voltage lim its Moreover the problem formulations did not consider a time varying demand In 14 a probabilistic methodology for distribution sub station location selection was presented This methodology took into account the hourly or daily load cycle For dif ferent hourly load scenarios the load center locations are determined and weighted according to their load magni Energy Systems 30 2008 308 315 309 tude These locations are then used to develop a probability distribution that is used in determining the maximum prob ability perimeter of the area where the substation should be located The process also takes into account factors such as land availability and the cost of land A model developed non discrete functions for distribution substation sizing sitting and timing taking into account the di erent compo nents for the substations cost function and various con straints including voltage power flow radial flow and capacities constraints was presented 15 Moreover the model considered a time varying demand for the sectors under investigation However the main drawback in this model is that the optimization process is carried out for each planning interval independent on the results obtained for previous intervals This results in some feeders being installed at earlier periods then removed latter with new kWh The interest tax inflation and insurance rates are considered 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 maximum allowable 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 to be 127 kW 15 Nine feeders are available for each substa tion total available feeders are 27 feeders and each feeder is assumed to supply the sector demand directly without intermediate points as given in Table 3 This table also Table 1 Sector demands over studied periods D p n MVA Interval n Sector p 123456789 1 2 21 51 41 416 2 3 44 3 52 5547 3 5 55 4 53 757 4 6 66 4 553 767 5 7 67 5 53 767 Table 2 Table 3 Feeders variables and their available routes Variable Route sector number Variable Route sector number Variable Route sector number From To From To From To X 1 41X 10 61X 19 21 X 2 42X 11 62X 20 22 X 3 43X 12 63X 21 23 X 4 44X 13 64X 22 24 X 5 45X 14 65X 23 25 X 6 46X 15 66X 24 26 X 7 47X 16 67X 25 27 310 T H M El Fouly et al Electrical Power and Energy Systems 30 2008 308 315 feeders installed in the next periods which is practically infeasible Moreover the problem was formulated as a Mixed Integer Nonlinear Programming MINLP which could result in local optimal solutions due to nonlinearity This paper addresses these drawbacks This paper is organized as follows Section 2 presents the system under study The proposed problem formulation for the distribution substation siting sizing and timing optimization model is presented in Section 3 The proposed problem formulation was modeled and solved using the General Algebraic Modeling Software GAMS solvers 16 In Section 4 the results generated from the proposed optimization model are presented A modified problem for mulation to ensure preventing ine cient transmission of power is discussed in Section 5 This section also presents the generated results from the modified formulation Finally in Section 6 conclusions are presented 2 System under study The service area under investigation consists of 9 sectors as shown in Fig 2 The area of each sector is 0 44 km 2 and it is assumed that the proposed sectors for substation installation by the city are sectors 2 4 and 6 The planning period is set to be 10 years divided into 5 time intervals of 2 years each The sectors demand growth over the 10 year planning period 5 intervals is given in Table 1 15 Each Fig 2 Area under study indicating the proposed substation sites by the city substation is rated at 40 MVA and can be equipped with a maximum 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 200 000 while the transformer unit installation cost is assumed to be 150 and the cost of energy is assumed to be 0 17 Proposed units capacities and variables Unit number m Type Rating MVA Location Variable 1 Substation 40 Sector 4 xx 1 n 2 Substation 40 Sector 6 xx 2 n 3 Substation 40 Sector 2 xx 3 n 4 Transformer 20 Sector 4 xx 4 n 5 Transformer 20 Sector 4 xx 5 n 6 Transformer 20 Sector 6 xx 6 n 7 Transformer 20 Sector 6 xx 7 n 8 Transformer 20 Sector 2 xx 8 n 9 Transformer 20 Sector 2 xx 9 n X 8 48X 17 68X 26 28 X 9 49 18 69X 27 29 and presents the assigned variable for each available feeder and its available route 3 Problem formulation When planning to install a distribution system substa tion and its components the main objective is to minimize the 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 the number of installed substations or improper proposed site selection could greatly increase the overall system cost Interest rates inflation rates taxes and insurance rates impact the timing of installation of equipment and hence the overall cost is a ected Since the amount of energy loss is 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 interval period could result in an impractical solution such as installing a feeder in an earlier period and then removing it in a latter period 15 In such cases human expertise is required to eliminate these impractical solutions To take into account all these factors the paper proposes a new problem formulation that minimizes the overall cost The problem was formulated over the whole planning period to 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 the substations The main objective over the whole planning horizon can be written as follows cost X q n 1 R n b n C S1 C1 S 1 n C S2 C1 S 2 n C S3 C1 S 3 n C T11 C1 S 4 n C T12 C1 S 5 n C T21 C1 S 6 n C T22 C1 S 7 n C T31 C1 S 8 n C T32 C1 S 9 n C138 8760 C1 P cu C1 C H Tr C1 X 9 i 4 x i n 1 where q is equal to the number of design intervals within the 10 year planning period and is set equal to 5 Thus a two year period is chosen for each design interval to pro vide su cient time for equipment installation C S1 C S2 and C S3 are the fixed cost for substations 1 2 and 3 respectively C T11 C T12 are the cost of the two transformer units to be installed at substation 1 including the cost of the iron losses C T21 C T22 are the cost of the two trans former units to be installed at substation 2 including the cost of the iron losses C T31 C T32 are the cost of the two transformer units to be installed at substation 3 including T H M El Fouly et al Electrical Power the cost of the iron losses S i n is a binary variable indicat ing the installation of unit i at a given period n P cu is the transformer copper loss at rated power kW C is the cost of energy kWh H Tr is the transformer unit rating MVA x i n is the power delivered from unit transformer i at a given period n if this unit transformer is installed and it is set to zero if the unit transformer is not installed R n and b n are the fixed charge rate and the present worth factor for a given interval n respectively and are calculated as follows R n i t r r 1 r 2 q 1C0n C0 1 8 n where n 1 q 2 b n f C0 r 1 f 1 r C16C17 2 q 1C0n C0 1 C18C198 where n 1 q 3 where r t f i are the interest tax inflation and insurance rates 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 transformer copper loss at rated power P cu to the transformer unit rat ing H Tr The objective function is minimized subject to the following constraints 3 1 Additional constraints to overcome nonlinearity A decision variable y i n is used to determine whether or not a transformer is delivering power This was done by multiplying the variable xx i n power delivered by trans former by the binary variable y i n Unfortunately this will enforce nonlinearity in the problem In order to avoid this a variable x i n is introduced to replace the product of both variables and constraints are added as follows 0 6 x i n 6 xx i n 8 i where i 4 9and8 n where n 1 q 4 xx i n C0 M 1 C0 y i n 6 x i n 6 M C1 y i n 8 i where i 4 9 and 8 n where n 1 q 5 where M is a big number and was chosen to be equal to 10 000 to guarantee that the constraint shown in Eqs 4 and 5 will converge to indicate whether a transformer is used or not xx i n is the power delivered from unit trans former i at a given period n MVA and y i n is a binary variable indicating the dissipated power from unit i at a gi ven period n A value of y i n equal to 1 indicates that power is transferred through the transformer while a value of zero means that no power is transferred 3 2 Fixed cost constraints As mentioned earlier S ij represents a binary decision variable that determines the installation of a unit i in a cer tain year j The cost of a unit will vary depending on the Energy Systems 30 2008 308 315 311 year it is installed due to the change in both R and b A bin ary variable F was added which relates the amount of lighted in Eqs 8 and 9 Eqs 10 and 11 have been for mulated to force variableS to equal 1 once a unit i has variable indicating the existence of feeder j at a given per and ij been installed in period j and above 3 3 Capacity constraints Each substation and each transformer has a capacity of 40 MVA and 20 MVA respectively However they are allowed to be loaded to 75 of their rated capacity for maximum e ciency operation This results in capacity lim its of 30 MVA and 15 MVA for each substation and each transformer respectively Moreover the lower limits for the substations and transformers loading are set to zero This could be expressed as follows 0 6 xx i n 6 15y i n 8 i where i 4 9 and 8 n where n 1 q 12 0 6 xx l n 6 30 8 n and where l 1 2 and 3 13 3 4 Power flow constraints These constraints represent the law of conservation of energy where the total loading of each substation at a given time interval n is equal to the sum of loading of its individual transformers units and at the same time equals to the sum of the demands of the sectors supplied by this substation and the total copper loss of the substations units These constraints are expressed as follows xx 1 n X 9 z 1 X 9 p 1 D p n x z hi P cu 1000 xH Tr C1 x 4 n x 5 n 8 n where n 1 q 14 xx 2 n X 18 z 10 X 9 p 1 D p n x z hi P cu 1000 xH Tr C1 x 6 n x 7 n 8 n where n 1 q 15 xx 3 n X 27 z 19 X 9 p 1 D p n x z hi P cu 1000 xH C1 x 8 n x 9 n power supplied by a unit in two consecutive years to the decision variable S ij S i 1 6 Mx i 1 8 i 1 9 6 S i 1 P x i 1 30 8 i 1 9 7 F i j 6 Mx i j 8 i 1 9 and 8 j 2 5 8 F i j P x i j 30 9 S i j PC0Mx i jC01 F i j 8 i 1 9andj 2 5 10 S i j 6 C0 x i jC01 30 F i j 8 i 1 9 and j 2 5 11 The first two constraints focus on the installation of a unit in the first year If power is being delivered by a unit on the first year S i1 will equal 1 and thus indicating the operation of this unit The variable F ij indicates whether a unit has been installed starting from the second period as