Improved Fuzzy Load Models by Clustering Techniques in Distribution Network Control
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1 International Journal on Electrical Engineering and Informatics - Volume 3, Number 2, 20 Improved Fuzzy Load Models by Clustering Tecniques in Distribution Network Control George Grigoras, P.D. * and George Cartina, P.D. "George Asaci Tecnical University, Iasi, Romania * ggrigoras@yaoo.com Abstract: In operation and planning of te power systems, te analysis of te consumption trends depends on te evolution of economic activities and competition among several sources of energy, wic affect forecasts. Te loads estimation represents te basis of te system state estimation and influences various aspects of power system planning suc as: transformer and conductor sizing, capacitor bank placement and so on. Te main difficulties in modeling of te nodal loads result from te random nature of loads, te deficiency of measured data and te fragmentary and uncertain caracter of information on loads and customers. Tus, a modern metod for expressing te uncertainty in load models is fuzzy tecnique. A fuzzy set is a set containing elements tat ave varying degrees of membersip in te set. Tere are different ways to derive membersip functions. Subjective judgment, intuition and expert knowledge are commonly used in constructing membersip function. Because in many situations te coices of te membersip functions are subjective, in te paper te clustering tecniques are proposed for te improved of te defining of membersip functions corresponding to te load profiles and customers consumption categories. Results obtained demonstrate te ability of te fuzzy load models to overcome difficult aspects encountered in process control and operation problems. Keywords: clustering tecniques, fuzzy load models, distribution networks, load profiles.. Introduction Te electric load in distribution system varies wit time and place. Terefore electric companies need by te accurate load data of te supply customers for distribution network planning and operation, load management, customer service and billing. Tere are several factors tat influencing te customer s load [] - [3]: customer factor: type of consumption, type of electric eating, size of building etc; time factor: time of day, day of week, time of year; climate factor: temperature, umidity etc; oter electric loads correlated to te target load; previous load values; load curve patterns and so on. For an electric customer, te beavior is represented by a load profile figuring te electric power consumption for every period of time. Availability of suc data depends on te type of customer. Generally, te small customers (like residential ones) are poorly described since a communicating meter is too expensive regarding to teir consumption: for tese customers tere are only a few points of te curve every year. For larger customers, a communicating meter is often available for many reasons: te billing is done every mont, te consumption is ig and justifies te communicating meter investment, a detailed record of consumption is necessary because prices depend on te period. Te load profiles/curves may correspond to individual customer curves or to aggregates over an electric substation. 2. Clustering tecniques Cluster analysis is te organization of a collection of objects (usually represented as a vector of Received: Marc 0 t, 20. Accepted: June 20 t,
2 George Grigoras, et al. measurements) into clusters based on similarity. It is a wonderful exploratory tecnique to elp us understand te clumping structure of te data. Clustering is useful in several exploratory pattern-analysis, grouping, decision-making, and macine-learning situations, including data mining, documents retrieval, images segmentation, and patterns (objects) classification [] - [3], [6], [7], [8]. A pattern (object) (or feature vector, observation, or datum) x is a single data item used by te clustering algoritm. It consists, typically, of a vector of d measurements: x = (x, x 2, x d ). Te individual scalar components x i of a pattern x are called features (or attributes). A distance measure (a specialization of a proximity measure) is a metric (or quasi-metric) on te feature space, used to quantify te similarity of patterns. Tere are two major metods of clustering: ierarcical clustering and k-means clustering. Hierarcical clustering is subdivided into agglomerative metods, wic proceed by series of fusions of te n objects into groups, and divisive metods, wic separate n objects successively into finer groupings. Agglomerative tecniques are more commonly used. Hierarcical clustering may be represented by a two dimensional diagram, known as dendogram, wic illustrates te fusions or divisions made at eac successive stage of analysis. Hierarcical clustering is appropriate for small tables, up to several undred rows. An example of suc a dendogram is given below in Figure. Differences between agglomerative metods arise because of te different ways of defining distance (or similarity) between clusters. Several agglomerative tecniques will now be described in te following [4], [6] - [8]. Figure. Example of dendogram Single linkage clustering (connectedness or minimum metod). Te defining feature of te metod is tat distance between groups is defined as te distance between te closest pair of objects, were only pairs consisting of one object from eac group are considered: D(r,s) = min {d(i,j) : were object i is in cluster r and object j is in cluster s} Complete linkage clustering (diameter or maximum metod). Distance between groups is now defined as te distance between te most distant pair of objects, one from eac group: D(r,s) = max {d(i,j) : were object i is in cluster r and object j is in cluster s} 208
3 Improved Fuzzy Load Models by Clustering Tecniques in Average linkage clustering. Te distance between two clusters is defined as te average of distances between all pairs of objects, were eac pair is made up of one object from eac group: Trs D( r, s) = () Nr Ns were: T rs - te sum of all pair wise distances between cluster r and cluster s; N r, N s - te sizes of te clusters r and s. Centroid Metod. In te centroid metod te distance between two clusters is defined as te squared Euclidean distance between teir means. Te centroid metod is more robust to outliers tan most oter ierarcical metods. 2 D( r, s) = X r X s (2) K-means clustering. K-means is one of te simplest unsupervised learning algoritms tat solve te well-known clustering problem. Te main idea is to define k centroids, one for eac cluster so minimize an objective function, in tis case a squared error function: J were = k n j = i= ( j) 2 i c j ( j ) x i c (3) j 2 x is a cosen distance measure between a data point x i (j) and te cluster centre c j. 3. Membersip functions Matematical models and algoritms in electric power system teory aim to be as close to reality as possible. Modeling can be performed in numerous ways, inclusively using te Fuzzy Tecniques (FT). Te basic idea of FT is to model and to be able to calculate wit uncertainty. Uncertainty in fuzzy logic is a measure of nonspecifically tat is caracterized by possibility distributions. Linguistic terms used in our daily conversation can be easily captured by fuzzy sets for computer implementations. A fuzzy set is a set containing elements tat ave varying degrees of membersip in te set. Te membersip values of eac function are normalized between 0 and. Tere are different ways to derive membersip functions. Subjective judgment, intuition and expert knowledge are commonly used in constructing membersip function. Even toug te coices of membersip function are subjective, tere are some rules for membersip function selection tat can produce well te results [4]. A fuzzy number A can ave different forms but, generally, tis is represented as trapezoidal or triangular form, Figure 2, usually represented by its breaking points: A ( x, x2, x3) = [ m, a, b] (4) A x, x, x, x ) = [ m, n, a, ] (5) ( b μ(x) μ(x) a b a b 0 m x 0 m n x Figure 2. Triangular and trapezoidal membersip functions Using of te clustering tecniques, te steps for defining some trapezoidal membersip function in te case of a set of two dimensional objects, are presented in Figures 3, 4, and 5. Tis is an improved fuzzy models metod by clustering tecniques because te breaking points are calculated wit statistical caracteristics [3] - [5]: 209
4 George Grigoras, et al. x x 3 = m k σ ; = m + k σ ; 3 x x 2 4 = m k σ 2 = m + k σ 4 (6) were: m - average value, σ - standard deviation value, and k i (i =,,4) - coefficients wose value is determined by experts. x 2 x Figure 3. Example of ungrouped objects x 2 H 2 H 3 H x Figure 4. Grouping of te objects x 2 H 2 H 3 H x A, A 2, Figure 5. Defining of te membersip function 4. Fuzzy modeling of load profiles For an electric customer, te beavior is represented by a load profile figuring te electric power consumption for every period of time. Availability of suc data depends on te type of customer. Generally, te small customers (like residential ones) are poorly described since a communicating 20
5 Improved Fuzzy Load Models by Clustering Tecniques in meter is too expensive regarding to teir consumption: for tese customers tere are only a few points of te curve every year. For larger customers, a communicating meter is often available for many reasons: te billing is done every mont, te consumption is ig and justifies te communicating meter investment, a detailed record of consumption is necessary because prices depend on te period. Analyzed load curves may correspond to individual customer curves or to aggregates over an electric substation []. A. Load modeling In distribution networks, except te usual measurements from substations, tere is few information about te network state. Te feeders and te loads are not usually monitored. As a result, tere is a ig degree of uncertainty about te power demand and, consequently, about te network loading, voltage level and power losses. Terefore, te fuzzy approac may reflect better te real beavior of a distribution network under various loading conditions [4]. Tus, te ourly loading factor of a particular distribution transformer can be employed to approximate te nodal load. And, because te most utilities ave not istorical records of feeders, it is proposed to use linguistic terms, usually used by dispatcers, to describe te uncertain ourly loading factor. For modeling of te loads from distribution substations, two primary fuzzy variables can be considered: te loading factor ki (%) and power factor cosϕ, so tat te fuzzy representation of te active and reactive powers result from relations [4]: ki P = S n cos ϕ; Q = P tan ϕ (7) 00 were S n (kva) is te nominal power of te transformer from te distribution substations. Te fuzzy variables, ki and cosϕ, are associated to trapezoidal membersip functions, (4), (5), Figure 2. Te two fuzzy variables must be correlated, as indicated in Table. Linguistic Category VS (Very Small) S (Small) M (Medium) Table. Linguistic categories of te ki and cosφ x x Linguistic ki cosϕ Category ki (%) cosϕ (%) x M x x (Medium) x x x x H x x (Hig) x x x x x x VH x x (Very Hig) x x x Tese linguistic terms are defined in function by te loading of te transformers at te peak load. Eac loading level represented by a linguistic variable is described by a fuzzy variable and its associated membersip function. Te loading factor ki and te power factor cosϕ were divided into five linguistic categories wit te trapezoidal membersip function, Table. B. Load profile modeling In tis section, an approac to daily load profile determination for te nodes of an electric distribution network (20 kv) is presented. For tis purpose te use of ierarcic clustering metod conjunctively wit fuzzy models is applied to a distribution network, containing te data for 39 nodes, to classify load profiles of te 20 kv nodes into groups, representing typical load profiles. 2
6 George Grigoras, et al. Active power profiles corresponding to te considered nodes were normalized relatively to te proper energy consumption (during te day wen te load peak was recording) using te following relation was used: Pi p i = ; =,..., 24; i =,..., N (8) R W were: i i P - te active power (kw), demanded by te i node at our, W i - te active energy [kw], consumed by te i node during te day wen te peak load was recorded, and N R = te total number nodes tat were taken into consideration in te clustering process, for te active power. Tus, four groups were determined for te active power. For every obtained group, C P, C P2, C P3 and C P4, te average and te variance values ( m and of te distribution system, were calculated using relations: m = NC pi i= = j N C NC ; C,...,24;,..., N G σ, j =, 2, 3, 4), during te peak load day 2 ( pi m) i= σ = (0) N were: N G - te number of te resulted groups from te clustering process and number nodes from every group. Te signification of te coefficients NC (9) - te total m is following: tese coefficients transform te energy consumed by te medium member of te group in average active power demanded by it. Tese coefficients lead us to te typical load profiles corresponding to te active power, corresponding to te 20 kv nodes, Figure 6. m (kw/kw) Figure 6. Typical Load Profiles for te obtained groups Using tese load profiles corresponding to te active power and te factor cosϕ, te loading factor of te every transformer from te distribution substations can be determined. Also, using te statistical model (6) and values of te coefficients k i (i =,,4) from te a fuzzy load model is determined, Table 2 and Table 3. () 22
7 Improved Fuzzy Load Models by Clustering Tecniques in Table 2. Values of coefficients k -4 from te model (6) for eac group C P-P4 k C P C P2 C P3 C P4 k k k k Table 3. Fuzzy model for typical load profile C P m σ x x 2 x 3 x 4 m σ x x 2 x 3 (kw/kw) (kw/kw) (kw/kw) (kw/kw) x 4 0,0396 0,0044 0,0347 0,0352 0,0440 0, ,0467 0,0023 0,0442 0,0444 0,0490 0, ,0344 0,0030 0,03 0,034 0,0374 0, ,050 0,0027 0,0480 0,0483 0,0537 0, ,0298 0,0025 0,027 0,0273 0,0323 0, ,0535 0,0035 0,0497 0,0500 0,0569 0, ,027 0,0027 0,024 0,0244 0,0298 0, ,0528 0,0039 0,0485 0,0489 0,0566 0, ,026 0,0028 0,0230 0,0233 0,0290 0, ,059 0,0030 0,0486 0,0489 0,0550 0, ,0257 0,0027 0,0227 0,0230 0,0285 0, ,0509 0,0028 0,0478 0,048 0,0537 0, ,0277 0,0024 0,025 0,0253 0,030 0, ,0482 0,0046 0,0432 0,0437 0,0528 0, ,0324 0,0024 0,0298 0,030 0,0348 0, ,047 0,0047 0,0420 0,0424 0,058 0, ,0366 0,004 0,032 0,0325 0,0406 0, ,0509 0,0048 0,0456 0,046 0,0557 0, ,0387 0,0037 0,0346 0,0350 0,0424 0, ,05 0,0033 0,0474 0,0477 0,0544 0,0548 0,048 0,0032 0,0383 0,0386 0,045 0, ,0480 0,0053 0,0422 0,0427 0,0533 0, ,044 0,0037 0,0400 0,0404 0,0478 0, ,0439 0,0042 0,0393 0,0397 0,048 0,0485 C. Determination of te Consumption Categories In tis paragrap it is presented an algoritm for identification of te consumption categories of te consumers and grouping tem into te classes wit te similar caracteristics. Te customer classes must be obtained from istorical data and must be updated to follow te canges on consumer s beavior. Due to te large amount of data predicted to be available in te future and te necessity of easy actualization, te algoritm provides a clear separation of different steps based on te application of clustering tecniques. Te algoritm is based on te load profiling process. Te major steps identified are:. Load Researc: In tis step a representative sample of te set of consumers is identified, te most relevant attributes to be measured, te cadence for data collection is defined. Finally te collected data is gatered in a large database. 2. Data cleaning and pre-processing: In real problems, like tis, involving a large number of measurements, spread over a large geograpic area, collecting data during a considerable period of time different kind of problems will affect te quality of te database. Te most relevant and frequent are communication problems, outages, failure of equipment and irregular atypical beavior of some consumers. Te result will be a very large database wit problems like noise, missing values and outliers. Tis data (after being cleaned, pre-processed and reduced) is used to obtain te division of te initial data set in classes. 3. Partitioned into macro-categories: Te wole customer database is preliminary partitioned into consumption categories defined by te activity type: residential, commercial and industrial. 4. Classification: Inside eac consumption category a classification into 3 5 classes, in function by te daily energy consumption of te consumers is done. For realization of tis classification, te k-means metod is used. For every consumer from te class is determined te normalized load profile using a suitable normalizing factor (energy consumption by te period. Ten, using a ierarcical clustering metod, te normalized load profiles are refined so as to desist at te unrepresentative profiles. Te typical load profile for eac class is obtained by averaging te values for eac our. 5. Assignation: Finally, to te eac customer class is assigned a typical load profile according to teir consumption category. In our study, we ave considered a database described by 296 load curves of various consumers. Eac measurement is a curve of 24 ourly points describing te beavior of a consumer during a day. Te wole customer database was preliminary partitioned into consumption categories defined by te activity type: residential (47 consumers), commercial (97 consumers) and industrial (52 consumers). 23
8 George Grigoras, et al. In te next step te load profiles corresponding to te consumers from eac category were normalized relatively to te daily energy consumption. For tis purpose, te following relation was used: P ij p ij = ; =,..., 24; i =,..., N () Cj Wij were: Pij - te active power of te customer i from te class j, at te our, (kw), W ij - te daily energy consumption of te customer from te class j, (kw), and N Cj - te total number of te customers from te class j tat were taken into consideration in te clustering process. Ten, using te average linkage metod from te ierarcical clustering metods [2], [3], te normalized load profiles of te customers are refined so as to desist at te unrepresentative profiles. Tus, using tis refinement, te number of te load profiles of te customers from eac class is less. Te average values corresponding for eac consumer class were calculated using te relations: m = Nij pij i ; =,..., 24 ; i =,..., N (2) Cj N Cj Figure 7. TLPs of te consumption classes from te residential consumer s category Figure 8. TLPs of te consumption classes from te commercial consumer s category 24
9 Improved Fuzzy Load Models by Clustering Tecniques in Figure 9. Typical load profiles of te consumption classes from te industrial consumer s category Figure 0. TLPs of te consumer categories (residential, commercial and industrial) Tese coefficients transform te energy consumed by te medium member (customer) of te class in average active power demanded by it. Te typical load profile (TLP) for eac class from te consumer categories is obtained by representation of tese coefficients, Figures 7 0. For te residential consumer s category it obtained 5 consumption classes, te TLPs of tese are presented in Figure 7. In te case of te commercial consumer s category, 3 consumption classes were obtained. In Figure 8, te TLPs of tese classes are sown. Finally, tree consumption classes were obtained for industrial consumer s category, Figure 9. In Figure 0, te TLPs corresponding to te case in wic customers categories are not grouping in classes are represented. From te comparison of te typical load profiles it results tat a classification of te customer s categories (residential, commercial and industrial) in classes is useful in view of te building of te tariff structures, demand-side management, optimal operation and planning of distribution system and so on. 5. Conclusions In tis paper an algoritm, based on te clustering tecniques, is proposed for determination of te consumption profiles and categories of te customers. Clustering tecniques are extremely useful for assisting te distribution services providers in te process of electric customer 25
10 George Grigoras, et al. classification on te basis of load profile. Te results obtained on a database of residential customers demonstrate tat te metodology can be used wit te success in building of te tariff structures for te customers or in te optimal operation and planning of distribution system. 6. References [] S.Gasperic, D. Gerbek, and F.Gubina, Determination of te Consumers Load Profiles. Available online: /2002Sep/ 2-5_Gasperic.pdf. [2] G. Cicco,, R. Napoli, F. Piglione, P. Postolace, M. Scutariu, and C. Toader, A Review of Concepts and Tecniques for Emergent Customer Categorization. Available online: Sep/ 2-4_Cicco.pdf. [3] G. Cartina, G. Grigoras, E. C Bobric, and C. Lupaşcu, Improving of Fuzzy Models by Clustering Tecniques in Optimal Reconfiguration of te Distribution Networks, International World Energy System, Torino, Italy, [4] G. Cartina, G. Grigoras, and E. C. Bobric. Clustering Tecniques in Fuzzy Modeling. Power Systems Applications, Iasi, Romania: Venus Publising House, [5] G. Cartina, G. Grigoras, and V. Alexandrescu, Pilot pattern coosing by clustering and fuzzy tecniques, Proc. of International Conference Energy-Environment, Bucarest, Romania, 2005, pp [6] Hierarcical Clustering, Available online: ttp:// [7] A.K. Jain, M.N Murty, and P.J. Flynn. Data Clustering: A Review. Available online: articles. [8] JMP Statistics and Grapics Guide. Version 3, SAS Institute Inc., Cary, NC, USA, 995. George Grigoras was born in Vatra Dornei, Romania, on February 28, 976. He received te M. SC. and P. D. degrees in Electrical Engineering from George Asaci Tecnical University of Iasi, Romania, in 2000 and 2005 respectively. He is currently lecturer in te Department of Power Systems of Electrical Engineering Faculty at te same university. His main areas of interest are analysis, planning, and optimization of power systems. George Cartina received te M. SC. and P. D. degrees in Electrical Engineering from George Asaci Tecnical University of Iasi, Romania, in 964 and 972 respectively. He is currently professor in te Department of Power Systems of Electrical Engineering Faculty at te same university. His researc interests include especially to monitoring and optimal control of power systems. 26
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