Approximate Loading Margin Methods Using Artificial Neural Networks in Power Systems

Transcription

1 Approximate Loadig Margi Methods Usig Artificial Neural Networks i Power Systems Arthit Sode-Yome, Member, IEEE ad Kwag Y. Lee, Fellow, IEEE Abstract This paper proposes approximate loadig margi methods usig Artificial Neural Networks (NN) for static voltage stability i power systems. Two methodologies, amely Actual LM with NN (ALM-NN) ad Maximum Loadig Margi with NN (MLM-NN), are proposed for fidig NN traiig data sets. Artificial Neural Network is used to approximate the loadig margi at particular geeratio directio. The proposed methods are validated ad compared with the Maximum Loadig Margi method i the modified IEEE 4-bus test system. The methods will help system operators to approximate voltage stability margi or loadig margi of the system i a short period of time. Idex Terms Loadig margi, maximum loadig margi method, eural etworks, geeratio directio. V I. INTRODUCTION oltage istability has bee a major cocer i power systems, especially i plaig ad operatio, as there have bee several major power iterruptios associated with this pheomeo, i the past []-[2]. Voltage istability due to the lack of the ability to foresee the impact of cotigecies is oe of the mai reasos for the worst North America power iterruptios o August 4th, 2003 [2]. Hece, may electric power utilities have bee devotig a great deal of efforts i voltage stability assessmet ad margi ehacemet. Major cotributory factors to voltage istability are power system cofiguratio, geeratio patter ad load patter [], [3]-[6]. eeratio patter is easier to cotrol by system operators compared to other factors, as log as there is eough margi left i the geerators [4]-[5]. Covetioally, i typical voltage stability studies, geeratio of participatig geerators are raised at the same rate or predefied rate. Icreasig geeratio at this rate may ot lead to highest voltage stability margi. Maximum Loadig Margi (MLM) approach, which provides the maximum Loadig Margi (LM) or static voltage stability margi is proposed i referece [5]. A approximate ad simple model represetig the relatioship betwee the This work was supported by Thailad Research Fud, office of the higher Educatio Commissio of Thailad ad the Siam Uiversity, Thailad uder grat No. MR A. Sode-Yome is with Siam Uiversity, Bagkok, Thailad ( arthit@ieee.org). K. Y. Lee is with Departmet of Electrical & Computer Egieerig, Baylor Uiversity, Waco, TX , USA ( Kwag_Y_Lee@baylor.edu). geeratio directio (D) ad the loadig margi is used to obtai the maximum loadig-margi poit. Although, MLM method ca provide maximum LM i the geeratio directio space, it approximate LM based o curve fittig methods. I additio, reactive power of all geerators except the swig geerator is at the limits for all cases at the LM poit [5]. Alteratively, oe may be iterested i approximatig the LM directly from the actual LM with correspodig geeratio directio from exhaustive simulatio. This ca be doe by usig a heuristic approach such as Artificial Neural Networks (NN) [7]. Based o the above observatio, attetio draw i this paper is to propose a simple simulatio approach that provides a approximatio of LM i the geeratio directio space usig Artificial Neural Networks. If the LMs at various geeratio directios are traied usig Neural Networks, operator ca approximate LM of the system at a particular geeratio directio i a fast ad simple way. It may be useful i the real-time Eergy Maagemet System at the load dispatchig ceter or System Operator. The rest of the paper is orgaized as follows: Sectio II presets static voltage stability. Existig eeratio Directio methods are summarized i Sectio III. New simulatio approaches are proposed i Sectio IV. I Sectio V, umerical results are preseted. Fially, i Sectio VI, major cotributios ad coclusios are give. II. STATIC VOLTAE STABILITY Voltage stability is the ability of power system to maitai adequate voltage magitude so that whe the system omial load is icreased, the actual power trasferred to that load will icrease []. It is maily associated with reactive power imbalace. I static voltage stability study, Cotiuatio Power Flow (CPF) ad optimizatio methods are the mai aalysis techiques that are used to fid voltage stability margi or loadig margi of the system [], [4]-[6], [8]. Major cotributory factors to voltage istability are power system cofiguratio, geeratio patter, ad load patter. The geeratio patter is easier to cotrol by system operators compared to load patter. Customarily, the geeratio of each participatig geerator is raised at the same rate, at a predefied rate, or accordig to their spiig reserves. I the followig sectio, existig methods of geeratio directios are summarized. III. ENERATION DIRECTION eeratio patter or directio is defied as the portios of geeratio icrease i each participatig geerator to serve

2 2 the desired load icrease ad losses i the system. Let K i be the factor for active power icrease at geerator i ad P i,o be the geeratio at the base case, the, the geeratio P i at a higher loadig poit ca be writte as ( ) P = P + K () i i, o i where i =, 2,, for all participatig geerators. The factor K i ca be viewed as the geeratio directio (D i ) ad is very crucial to voltage stability. eeratio Directio ca be worked out by fidig the slope of geeratio icrease for idividual geerator. Existig methods to idetify geeratio directios i voltage stability study are summarized below. A. Covetioal Approach Covetioally, the geeratio for a system is icreased by fixed percetage as pre-specified i the plaig stage, e.g., accordig to the spiig reserve [], [4]. The power geeratio of geerator i after the load icrease ca be writte as ad ( ) P = P + K = P + P (2) i i, o i i, o i Pi = PD + P (3) loss N where P i is the power geeratio of geerator i, P i,o is the geeratio of geerator i at base load, P i is the icrease of power geeratio at geerator i, P D is the total load icrease, P Loss is the total loss icrease, N is the umber of geerators. B. Optimal Power Flow Approach Traditioal Optimal Power Flow (OPF) ca be formulated to iclude voltage stability criteria as follows [8]: (4) Miimize C ( P ) ( 2 i = ai Pi + bi Pi + ci ) subject to ( λ ), ( δ δ ) i Di o i j ij ij ij ij N P + P U U cos + B si = 0 (5) Q + Q U U si B cos = 0 (6) ( λ ), ( δ δ ) i Di o i j ij ij ij ij P P P i mi i i max U U U i mi i i max S = P + Q S (9) 2 2 ij ij ij ij,max Where C is total operatig cost of the system, a i, b i, c i are cost coefficiets of geerator i, (7) (8) λ is load icremetal parameter or loadig factor (L.F.), P i, Q i are real ad reactive power geeratio at bus i, P Di,o, Q Di,o are real ad reactive power demad at bus i at base load, is umber of buses i the system, P i mi, P i max are lower ad upper power limits of geerator i, U i mi, U i max are lower ad upper limits of voltage magitude at bus i, P ij, Q ij,, S ij are the real, reactive ad apparet power i lie ij, S ij, max is the MVA (Thermal) limit of lie ij. eeratio directio, i this approach, ca be worked out by subtractig the ew dispatch from the old dispatch for idividual geerators. C. Cost Participatio Factor Approach Cost participatio factor is viewed as the easiest method to idetify amout of power geeratio with ecoomic load dispatch cosideratio. It is calculated based o geerators icremetal cost [8]: i '' ( / Ci ) N '' ( / C j ) P = P D (0) where C i is the cost fuctio of geerator i, C i is the secod derivative of the cost fuctio i, P i is the icrease i power geeratio for geerator i, P D is the total load icrease. Amog the existig methods, very few of them ca provide the highest LM of the system. Hece, i the followig sectio, the Maximum Loadig Margi method is preseted to maximize the LM by searchig i the geeratio directio space. D. Maximum Loadig Margi Method The MLM method idetifies a vector of the Ds of geerators that gives maximum LM by approximatig the surface of the LM as a fuctio of the geeratio directios. If oe ca approximate the LM surface as a fuctio of all geeratio directio variables (K i ), optimizatio techique ca be used to provide the highest LM poit. Mathematically, the method ca be formulated as follows: p LM = B0 + Bi, pki p= subject to N () Ki = i= (2) 0 K i (3) where K i is the geeratio directio for geerator i, B i,p are the coefficiets terms, B o is a costat term, p is the power term ad is the umber of terms of the polyomial

3 3 approximatio. If geeratio is icreased accordig to this directio, the system will have the maximum loadig margi. The MLM method provides a good approximatio of the D, which would give the maximum LM for a give case. Sice LM surface may have multiple maximum, as it ca be approximated by polyomial equatios, MLM method may be required to fid the global maximum. E. Other Existig Methods Liear ad quadratic estimates of the LM with respect to system parameters, icludig power geeratio, are computed by usig sesitivity method to locally predict the ew locatio of the maximum loadig margi poits [9]. From the existig methods, oly MLM method ca provide maximum loadig margi i the geeratio directio space. This method requires a approximated LM surface equatio based o the two-dimesioal LM curves i each geeratio directio. This may be useful whe oe would like to fid the maximum LM poit based o a optimizatio techique. At the load dispatchig ceter or System operator, however, oe may be iterested i fidig the LM of the system i a short period of time whe some geeratio directios are cosidered. Istead of approximatig LM i a equatio as proposed i the MLM method, oe may suggest to use some heuristic methods such as Artificial Neural Networks to approximate the LM surface. I the followig, ew LM approximatio methods are proposed based o the actual LM, MLM method ad Neural Networks. IV. PROPOSED METHODOLOIES A. Artificial Neural Networks [7] Neural etwork is a collectio of itercoected euros that icremetally lear from their data to capture essetial liear ad oliear treds i complex data. Neuro etworks perform a variety of tasks such as approximatio, predictio, etc. There are several NNs suitable for oliear aalysis, icludig multilayer perceptro (MLP) etworks, radial basis fuctio (RBF) etworks, etc. MLP is the most popular ad widely used oliear etworks for solvig may practical problems. The reaso for the popularity is that it is flexible ad ca be traied to assume the shape of the patters. The MLP ca be called uiversal approximators due to their ability to approximate ay oliear relatioship betwee iputs ad outputs to ay degree of accuracy. The power comes from the hidde layer of euro located betwee the iput layer ad output layer of euros. The hidde layer may cosist of oe or may oliear euros ad it performs cotiuous, oliear trasformatios of the weighted iput. Noliear activatio fuctio trasforms the weighted iput of a euro oliearly to a output. Sigmoid activatio fuctio is the popular oe. The most widely used sigmoid activatio fuctios are logistic fuctio ad hyperbolic taget fuctio. Output of each euro is give i Equatio (4), where the weighted sum of iputs is passed through a sigmoid fuctio as σ ω jx j + b (4) B. Proposed Methodologies The simulatio method is preseted i two steps: Step I- Obtaiig Traiig Data Set ad Step II-Approximatio of LM Usig NN. Step : Obtaiig Traiig Data Set Two methodologies for fidig traiig data set, amely Actual LM Method with NN (ALM-NN) ad Maximum LM Method with NN (MLM-NN) are proposed. The methodologies are described as follows: Actual LM Method with NN (ALM-NN): The actual loadig margis ad correspodig geeratio directios are foud from exhaustive simulatio usig ay voltage stability software to calculate loadig margi of the power system for a give geeratio directio [5]. The actual LM ad correspodig D are used as a traiig data set. Maximum LM Method with NN (MLM-NN): The relatioship of LM with respect to D of each geerator is foud from a sigle dimesioal space as oly oe geerator is cosidered, except the swig bus. The LM surface is approximated for multi-dimesioal case based o the separability coditio ad MLM method preseted i Sectio III.D. Step 2: Approximatio of LM usig NN After LM ad all possible Ds are foud, the traiig data set is the used to trai the NN. From the NN, the approximated LM ca be foud from ay geeratio directio value. Fig. summarizes the process of the proposed approximated LM methods. I the begiig, geeratio directio is firstly set for the CPF process. The the traiig data set is foud. For ALM-NN method, the completed PV curve is plotted to obtai the LM of the particular geeratio directio. The process is the repeated util eough traiig data are obtaied. For MLM-NN, the LM surface is approximated for all possible Ds. The traiig data is the used i NN process by itroducig Ds as iputs ad LM as a output. After the NN is traied to fid the appropriate weights ad other NN parameters, oe ca use the traied NN to approximate the LM at the Ds of iterest. The proposed method is validated i the followig sectio. A. Test Power Systems V. NUMERICAL RESULTS The modified IEEE 4-bus test system [5] is used to validate the proposed method. Its sigle lie diagram is depicted i Fig. 2, which cosists of five sychroous machies, icludig oe sychroous compesator used oly

4 4 for reactive power support ad four geerators located at buses, 2, 6 ad 8. The modificatio from the origial IEEE 4-bus test system is that geerators located at buses 6 ad 8 were chaged from sychroous compesators to geerators. I the system, there are twety braches ad fourtee buses with eleve loads totalig 259 MW ad 8.4 Mvar. The value of 259 MW is used for the base MVA of the IEEE 4-bus system. c ENERATORS SYNCHRONOUS COMPENSATORS THREE WINDIN TRANSFORMER EQUIVALENT 9 7 c c 3 Fig. 2. Sigle lie diagram of the modified IEEE 4-bus test system. Fig.. LM Approximatio Usig Artificial Neural Networks. Results preseted i the paper were produced with the help of UWPFLOW [0] ad aother Neural Network Software. The UWPFLOW is a research tool that has bee desiged to calculate loadig margi of the power system for a give load ad geeratio directio. I the followig sectio, simulatio results are preseted. B. Traiig Data Set The size of geeratio directio space is i proportio to the umber of dispatchable geerators cosidered i the study. To limit the umber of geerators i this study, a total of four geerators are used for the IEEE 4-bus test system. Three cases of three geerators are used throughout the paper.. ALM-NN Three cases of geerator locatios are cosidered: geerators are located at buses, 2 ad 6 (Case 26); at buses, 2 ad 8 (Case 28); ad at buses, 6 ad 8 (Case 68). The actual LM is foud with the help of UWPLOW program based o the methodology preseted i Sectio IV. The actual data of LM ad Ds are plotted i Figs. 3, 4 ad 5 for the 26, 28 ad 68 cases, respectively. These LM plots are obtaied from PV at all possible Ds i the D space with 0. D step. The maximum LM ad correspodig D of each case are show i Table I. From Table I, the maximum LM of 26, 28 ad 68 cases are.655,.0286 ad.0686, respectively. 2. MLM-NN The MLM method is based o the loadig margi of the system at various possible geeratio directios i the geeratio directio space. The approximate plots obtaied from MLM method for 26, 28 ad 68 cases are show i Figs. 6, 7 ad 8, respectively. Clearly, the plots obtaied from the MLM approach are almost the same as those obtaied from the actual LM plot. The LM obtaied from actual LM ad MLM methods are used to trai NN i the followig subsectio. C. Approximatio of LM usig NN From LM ad D data obtaied i previous subsectio, oe may use Ds as iputs ad LM as a output to trai the NN. Multilayer perceptro with activatio fuctios ad black propagatio are used to capture oliearity of the traiig data. There are 66 traiig data used i the traiig process. Table II shows the summary of the traiig data ad results for each case. The result is obtaied from the best solutio of 30 traiig results that provide the miimum traiig error. From the table, about 7-47 euros are used i the hidde layer. Number of euros ad types of activatio fuctio are obtaied from the best solutio usig NN software. The traiig times are oly few miutes ad testig time for each xyz case is less tha secod.

5 5 Fig. 3. Actual LMs i case 26. Fig. 6. Approximated LMs i case 26 usig the MLM approach. Fig. 4. Actual LMs i case 28. Fig. 7. Approximated LMs i case 28 usig the MLM approach. Fig. 5. Actual LMs i case 68. Fig. 8. Approximated LMs i case 68 usig the MLM approach. TABLE I. COMPARISON OF ENERATION DIRECTIONS AND LM. Solutios obtaied from the Actual LM Cases plot LM (p.u.) Ds (0,0.7,0.3) (0,,0) (0.6,0.4,0) The umerical solutios of each method are give i Table III for all the cases. The D data used for testig has 0.25 iterval where those for learig has 0. iterval. From the table it ca be see that eural etworks ca capture oliearity of LM surface. The approximated LM solutios are very close to the actual LM value as show i Figs. 3-5 ad to approximated LM from MLM method [5]. The proposed method provides a good approximatio of LM, which may help System Operator to obtai LM at some geeratio directio poits i a short time. The methodology

6 6 may be applied to EMS system at load dispatchig ceter or System Operator. TABLE II. TRAININ RESULT INCLUDIN HIDDEN LAYER, OUTPUT LAYER, ACTIVATION FUNCTION AND NUMBER OF NEURONS. Case Activatio Fuctio Hidde Layer No. of Neuros Activatio Fuctio For ALM-NN/ MLM-NN Output Layer No. of Neuros 26 Tah/Tah 5/7 Sie/Sie / 28 Tah/Sie 5/37 Tah/Logistic / 68 Tah/Exp 2/47 Exp/Idetity / TABLE III. NEURAL NETWORK OUTPUT AT DIFFERENT DS D for xyz case ALM-NN for xyz MLM-NN for xyz x y z VI. CONCLUSION This paper proposes ew methods, amely ALM-NN ad MLM-NN, for approximatig loadig margi or voltage stability margi of the system usig Artificial Neural Networks. Loadig margis i geeratio directio space are used to trai eural etworks. Multilayer perceptro with activatio fuctios ad black propagatio are used to capture oliear relatioship of the traiig data. The proposed methods are validated ad compared with Actual LM ad MLM methods i the three geerator cases i the modified IEEE 4-bus test system. The results show that the proposed method is able to fid the result i a short period of time. It ca be used i the EMS system to help System Operator to approximate voltage stability margi based o geeratio directios i a simple ad fast way. REFERENCES [] IEEE/PES Power System Stability Subcommittee, Voltage Stability Assessmet: Cocepts, Practices ad Tools, special publicatio, fial draft, Aug [2] Blackout of 2003: Descriptio ad Resposes, Available: [3] B. H. Lee ad K. Y. Lee, "Dyamic ad Static Voltage Stability Ehacemet of Power Systems," IEEE Trasactios o Power Systems, Vol. 8, No., pp , 993. [4] A. Sode-Yome ad N. Mithulaatha, Compariso of shut capacitor, SVC ad STATCOM i static voltage stability margi ehacemet, Iteratioal Joural of Electrical Egieerig Educatio, Vol. 4, No. 3. pp.58-7, [5] A. Sode-Yome, N. Mithulaatha, ad K. Y. Lee, A Maximum Loadig Margi Method for Static Voltage Stability i Power Systems, IEEE Tras. Power Syst., Vol. 2, pp , [6] C. A. Caizares, A. C. Z. De Souza, ad V. H. Quitaa, Compariso of performace idices for detectio of proximity to voltage collapse, IEEE Tras. Power Syst., Vol., No. 3, pp , Aug [7] S. Samarasighe, Neural Networks for Applied Scieces ad Egieerig, Auerbach Publicatios, Tylors ad Fracis roup, New York, [8] A. Sode-Yome ad N. Mithulaatha, eeratio Directio Based o Optimizatio Techique for Power System Static Voltage Stability, Australasia Uiversities Power Egieerig Coferece, Hobart, Australia, Sep , [9] S. ree, I. Dobso ad F. L. Alvarado, Sesitivity of Loadig Margi to Voltage Collapse with respect to Arbitrary Parameters, IEEE Tras. Power Syst, Vol. 2, No., pp , Feb [0] C. A. Cañizares, et al., UWPFLOW: Cotiuatio ad Direct Methods to Locate Fold Bifurcatios i AC/DC/FACTS Power Systems, Uiversity of Waterloo, available at Jue Arthit Sode-Yome (M 04) received the B.Eg degree i Electrical Egieerig from Price of Sogkla Uiversity, Thailad, i 993, M.S. degree i Electrical Egieerig from the Pesylvaia State Uiversity, USA, i 996 ad Doctoral Degree i Eergy from Asia Istitute of Techology, Thailad, i Dr. Arthit is a lecturer at Departmet of Electrical Egieerig, Siam Uiversity, Thailad. His mai research iterests are voltage stability, FACTS, eural etworks, optimizatio techiques ad HVDC. Kwag Y. Lee (F 0) received his B.S. degree i Electrical Egieerig from Seoul Natioal Uiversity, Korea, i 964, M.S. degree i Electrical Egieerig from North Dakota State Uiversity, Fargo, i 968, ad Ph.D. degree i system sciece from Michiga State Uiversity, East Lasig, i 97. He has bee o the faculties of Michiga State, Orego State, Housto, the Pesylvaia State Uiversity, ad Baylor Uiversity, where he is curretly Professor ad Chair of Electrical ad Computer Egieerig ad Director of Power ad Eergy Systems Laboratory. His iterests are power systems cotrol, operatio ad plaig, ad itelliget systems applicatios to power plats ad power systems cotrol. Dr. Lee is a Fellow of IEEE, Editor of IEEE Trasactios o Eergy Coversio, ad Associate Editor of IEEE Trasactios o Neural Networks.