A Fast Computational Technique to Assess Total Transfer Capability Using Broyden Shamanski Method

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1 Global Journal of researches in engineering Electrical and electronical engineering Volue 11 Issue 5 Version 1. July 211 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: A Fast Coputational Technique to Assess Total Transfer Capability Using Broyden Shaanski Method By K. Chandrasekar, N. V. Raana Tagore Engineering College, Chennai, Tailnadu, India Abstracts - In the deregulated power syste assessent of Total Transfer Capability (TTC) is a coplex task which has to be done at periodic intervals for each source sink pairs. Though there are any ethods available to assess TTC, the ost accurate ethod is Repeated Power Flow using Newton Rap son (RPFNR). This ethod suffers fro the drawback of high coputational tie due to the presence of ultiple Jacobian inverses. In this paper a novel ethod, Repeated Power Flow using Broyden Shaanski ethod with Sheran Morrison forula (RPFBSS) is eployed which eliinates the drawback of RPFNR ethod without coproising accuracy. The proposed approach is tested with WSCC 9 bus, New England 39 bus and IEEE 118 bus test syste and the results are copared with the conventional RPFNR ethod. Keywords: Electric power deregulation, Total Transfer Capability, Repeated Power Flow, Broyden Shaanski, Newton Rap son. GJCST Classification : 967 A Fast Coputational Technique to Assess Total Transfer Capability Using Broyden Shaanski Method Strictly as per the copliance and regulations of : 211 K. Chandrasekar, N. V. Raana This is a research/review paper, distributed under the ters of the Creative Coons Attribution-Noncoercial 3. Unported License peritting all non coercial use, distribution, and reproduction in any ediu, provided the original work is properly cited.

2 A Fast Coputational Technique to Assess Total Transfer Capability Using Broyden Shaanski Method K. Chandrasekar α, N. V. Raana Ω July 211 Abstract - In the deregulated power syste assessent of Total Transfer Capability (TTC) is a coplex task which has to be done at periodic intervals for each source sink pairs. Though there are any ethods available to assess TTC, the ost accurate ethod is Repeated Power Flow using Newton Rap son (RPFNR). This ethod suffers fro the drawback of high coputational tie due to the presence of ultiple Jacobian inverses. In this paper a novel ethod, Repeated Power Flow using Broyden Shaanski ethod with Sheran Morrison forula (RPFBSS) is eployed which eliinates the drawback of RPFNR ethod without coproising accuracy. The proposed approach is tested with WSCC 9 bus, New England 39 bus and IEEE 118 bus test syste and the results are copared with the conventional RPFNR ethod. Keywords : Electric power deregulation, Total Transfer Capability, Repeated Power Flow, Broyden Shaanski, Newton Rap son. I. INTRODUCTION A ccording to NERC report [1], Total Transfer Capability (TTC) is defined as the aount of electric power that can be transferred over the interconnected transission network in a reliable anner while eeting all defined pre and post contingencies. Available Transfer Capability (ATC) is a easure of transfer capability reaining in the physical transission network for further coercial activity over and above already coitted uses. Therefore ATC = TTC Coitted Uses and Coitted Uses = TRM + Existing Transission Coitents (including CBM) where TRM is Transission Reliability Margin and CBM is Capacity Benefit Margin. Deterination of TTC is the key coponent in ATC calculation. There are nuber of ethods reported till date in literature to copute TTC. DC load flow [2] for transfer capability calculation is faster but does not consider losses in the network, voltage liits etc. Methods which use DC Power Author α : Assoc. Professor, EEE Dept, Tagore Engineering College, Chennai, Tailnadu, India. Currently he is pursuing Ph. D in JNTUH, College of Engineering, Hyderabad. Telephone: E- ail : chaandru74@gail.co. Ω Author : professor and Head, EEE Dept, JNTUH, College of Engineering, Nachupally, Karinagar Dist. A.P., India.Telephone: E- ail : nvrjntu@gail.co. Transfer Distribution s [3] or AC Power Transfer Distribution s [4-5] can fetch accurate results only to those cases which are too close to the base case fro which those distribution factors are derived. Artificial Neural Network [6-7] and Fuzzy logic [8] based ethods are uch faster in coputing TTC but it requires a clear understanding of the coplex network topology so that these intelligent systes can be trained for accurate results. Optial Power Flow based ethods [9 12] assess TTC with other factors such as Econoic Dispatch, can get better results but at the cost of higher coputational tie. Two other well known ethods which solve full AC load flow repeatedly to find TTC is, Continuation Power Flow (CPF) ethod [13] and Repeated Power Flow ethod (RPF) [14]. CPF ethod involves predictor, step length control, corrector and paraeterization which ake the procedure uch coplicated. When copared to CPF, RPF ipleentation is uch easier and it also provides a part of P-V curve if voltage stability has to be taken into account. Fro the literature available it is understood that the effectiveness of ajority of ethods in assessent of TTC has been proved by coparing it with the benchark ethod RPF [8 9], [15] since the results obtained using this ethod is very uch accurate. RPF ethod norally uses Newton Raphson (RPFNR) for power flow which suffers fro the drawback of high coputational tie. Coputational tie plays a vital role in TTC assessent since a typical assessent frequency [16] is Hourly TTC for the next 168 Hours : Once per day Daily TTC for the next 3 days : Once per week onthly TTC for onths 2 through 13 : Once per onth In RPFNR the priary task is to find the loading factor for TTC assessent in which ajor portion of CPU tie is spent in functional evaluations (coputation of Jacobian eleents) and arithetic operations (inverting the Jacobian atrix) which is a coon procedure in Newton Raphson ethod. In this paper a novel ethod RPF using Broyden Shaanski ethod with Sheran Morrison 211 Global Journals Inc. (US) 13 Global Journal of Researches in Engineering ( f ) Volue XI Issue V vv Version I

3 211 July 14 Global Journal of Researches in Engineering ( f ) Volue XI Issue V Version I forula (RPFBSS) [17-19] is used for coputation of TTC which eliinates the drawback of RPFNR ethod. This ethod is basically a generalization of the Secant ethod for solving Non Linear equations hence it reduces the nuber of functional evaluations when copared to NR ethod. Also presence of Sheran Morrison forula helps us to reduce the nuber of arithetic operations by taking inverse of Jacobian only once for a given topology of network and for the reaining iterations a rank one update is done to copute the inverse (an approxiate Jacobian inverse) irrespective of any transfer directions and source/sink pairs which reduces the tie required for coputing TTC. II. MATHEMATICAL FORMULATION OF TTC TTC deals with the transfer of iu possible power flow in a transission network subject to the satisfaction of certain constraints like network theral liits, voltage liits, generation liits etc. The atheatical forulation for TTC [2] is: n -PDi - Plossij = (1) j= 1 n QGi -QDi - Qlossij = (2) j= 1 Subject to Vi in Vi Vi (3) S P ij Sij (4) Gi (5) Where is the real power generation at bus i PDi is the real load in bus i Plossij is the active power loss in the line ij QGi is the reactive power generation at bus i QDi is the reactive load in bus i Qlossij is the reactive power loss in the line ij Vi is the voltage at bus i Vi in and Vi are the iniu and iu voltage liits at bus i Sij is the apparent power flow in the line ij Sij is the theral liit of line ij is the iu real power generation available at bus i ' In RPF ethod, power flow equations are solved repeatedly by increasing the coplex load with unifor load distribution factor and power factor at every load bus in the sink area and increasing the injected real power at generator bus in the source area until liits are incurred. P Gi (real power in source area), P Di (real power in sink area) and Q Di (reactive power in sink area) are changed in the following way. λttc = (1 + ) (6) PDi PDi λttc = (1 + ) (7) QDi QDi λttc = (1 + ) (8) Where is the original real power generation at bus i in source area. P Di is the original active load in bus i in sink area. is the original reactive load in bus i in sink QDi area. λttc is the scalar paraeter representing the increase in bus load or generation. λ ttc = correspond to no transfer (base case) and λttc = λttc correspond to iu transfer. The TTC level in (noral or contingency state) is given by: TTC P λ = Di ( ) (9) And ATC neglecting TRM, ETC is given by Di ( λ ) Di (1) ATC = P P Where P λ Di ( ) λ = λ. is the su of load in sink area when PDi is the su of load in sink area when λ =. III. POWER FLOW USING BSS METHOD In general, an NR ethod finds the value of ' x ' iteratively such that F( x ) = (11) In the iterative process, say in is updated as given below 1 x x x th iteration ' x ' + = (12) And 1 x = ( J ) F( x ) (13) Where J is the Jacobian atrix. In the assessent of TTC the power flow equations are solved repeatedly, for every step increent of λttc there are ore than one iteration and for every iteration a Jacobian atrix of size n n is coputed and then inverted. For n non linear equations, coputation of Jacobian atrix eleents includes coputation of n 2 partial derivatives and n 2 nuber of coponent functions. Therefore n + n functional evaluations need to be done. Again inversion 211 Global Journals Inc. (US)

4 T A Fast Coputational Technique to Assess Total Transfer Capability Using Broyden Shaanski Method of an n n Jacobian atrix using Gauss Jordan eliination ethod requires n 3 arithetic operations or if sparsity technique is used to copute Jacobian inverse with soe for of Gauss eliination technique then the total tie taken for the inversion is k n, where k is the average non zero entries in a row or colun of the sparse LU factors and n is the size of the Jacobian atrix. This procedure takes ore coputational tie. The Quasi Newton BSS ethod [17-19] belongs to the class of two step iteration which differentiates it fro the conventional Broyden s ethod. Let us consider the expression (11) which has to be solved iteratively using BSS ethod. In the first iteration x is chosen as in the case of NR ethod, then w is calculated as given below 1 w = ( J ) F( x ) (14) Using (14) v is updated as v = x + w (15) this is the first step iteration. Using (15) as 1 s is coputed s = ( J ) Fv ( ) (16) Then with the value of 1 α s and v, 1 x is updated using x = v + ( M C. s ) s (17) Which is the second step iteration. Here M, C and α are the real variables defined in [17], where the role of M is to increase the rate of convergence, C and α keeps the new iteration in the convergence region. Fro the second iteration the above procedure is repeated by replacing the Jacobian atrix J with an equivalent atrix A which is defined at the th iteration as given below ( 1) 1 A = A + [ F( x) A ( x)] (18) where 1 F( x) = F( x ) F( x ) (19) 1 x x x = (2) This reduces the nuber of functional evaluations to n fro n 2 + n when copared to the case of NR ethod but akes the convergence of BSS as super linear when copared to quadratic convergence of NR ethod. Further the n 3 arithetic operation for coputing the inverse of A atrix can be reduced to operations using the Sheran Morrison forula as n 2 1 ( A ) Where ( 1) 1 [ A ] + U = (21) T 1 1 x [ A ] F( x) Unlike NR ethod, here the Jacobian inverse is coputed only once during the first iteration and for the reaining iterations a rank one update is done to copute the inverse. In a noral power flow, the quadratic convergence and the advantage of ipleenting sparsity technique in NR ethod proves to be superior to the super linear convergence of BSS ethod which has Sheran Morrison forula for Jacobian inverse. When it coes to the proble of TTC assessent power flow is solved repeatedly, which involves ultiple Jacobian coputations and inverses, in this process coputation using BSS ethod is faster when copared to NR ethod [21]. IV. ALGORITHM TO ASSESS TTC USING RPFNR AND RPFBSS METHOD The algorith to assess TTC using RPFNR and RPFBSS ethod differs only in the power flow technique used as given below a) RPFNR ethod Step 1 : Read Bus, line, generator data etc. Step 2 : Solve power flow using NR ethod. Check equations (3), (4), and (5), if there is liit violations go to step 4 else go to step 3. Step 3 : Make a step increase in λ ttca λttc = λttc + λttc. Copute equation (6), (7) and (8). Go to step 2. Step 4 : Copute TTC using (9) at λttc b) RPFBSS ethod U = { x [ A ] F( x)}*{ x[ A ] } (22) = λttc. Step 1 : Read Bus, line, generator data etc. Step 2 : solve the first iteration of power flow using NR Method. Step 3 : Use (14) to (22) for second iteration which replaces NR by BSS ethod. Solve power flow copletely. Check equations (3), (4), and (5), if there is liit violations go to step 5 else go to step 4. Step 4 : Make a step increase in λ ttc as, λttc = λttc + λttc. Copute equation (6), (7) and (8). Go to step 2. Step 5 : Copute TTC using (9) at λttc = λttc. V. RESULTS AND DISCUSSION The effectiveness of the proposed ethodology is illustrated using the WSCC 9 bus, New England 39 bus and IEEE 118 bus test syste. The power flow data for the test syste are considered fro [22-23]. Load flow progras are executed in MATLAB using odified 211 Global Journal of Researches in Engineering ( f ) Volue XI Issue V vv Version I July Global Journals Inc. (US)

5 211 July 16 Global Journal of Researches in Engineering ( f ) Volue XI Issue V Version I MATPOWER [24] coding in INTEL core 2 Duo CPU T55@ 1.66 GHz processor under Windows XP professional operating syste. a) WSCC 9 bus test syste WSCC 9 bus test syste consists of 3 generators with 9 transission lines. This syste has been divided into two areas for TTC coputation. Area 1 includes buses 3,6,8,9 and Area 2 has buses 1, 2,4,5,7. The base load in Area 1 is 125 MW and that of Area 2 is 19MW. The TTC value, MW loss and the liit condition obtained using NR and BSS are identical, hence a coon entry has been ade in Table 1. The TTC value for transfer of power for Area 1-2 under base case, selected line outage and with generator outage is 41.4 MW, MW and MW respectively as shown in Table 1. The total tie required to coplete the coputation of TTC for Area 1-2 with and without contingencies using BSS and NR ethod is (s) and (s). Siilarly the transfer of power for Area 2-1 with and without contingencies is also shown in Table 1. The overall tie to copute both transfer directions including contingencies using BSS ethod is (s) and for NR ethod is (s) which shows that the CPU tie for BSS ethod is 1.7 % less when copared to that of NR ethod. b) New England 39 Bus test syste New England 39 bus test syste has 1 generators and 46 transission lines. For coputing TTC this syste has been divided into three areas. The buses in each area are as shown in Table 2. The base loads in Area 1, Area 2 and in Area 3 is 1124 MW, MW and 2649 MW respectively. The transfer power for all transfer directions i.e., Area 1-2, Area 2-3 Area 3-1 and vice versa is shown in Table 3. The TTC value, liit condition and MW loss for power transfers in between areas using NR and BSS ethods are identical whereas the CPU tie for coputing these results for both these ethods differs as shown in Table 3. The coputational tie for all transfer directions for base case, line outage and generator outage case using BSS ethod is (s), (s) and (s) respectively. Siilarly using NR ethod, the CPU tie are (s), (s) and (s) respectively. Hence the overall tie to copute TTC value with and without contingencies for all transfer directions for BSS and NR ethod is (s) and (s) respectively which shows that the CPU tie for BSS ethod is % less when copared to that of NR ethod. c) IEEE 118 bus test syste This test syste has 54 generators and 186 transission lines. It has been divided into three areas with the area wise classification of buses as shown in Table 2. The base loads in Area 1, Area 2 and in Area 3 is 963 MW, 1937 MW and 1342 MW respectively. The transfer power for all transfer directions are coputed and furnished in Table 4. The TTC value, liit condition and MW loss for power transfers in between areas using NR and BSS ethods are identical. On the other hand the CPU tie for coputing these results for both these ethods differs as shown in Table 4. The coputational tie for all transfer directions for base case, selected line outage and generator outage case using BSS ethod is (s), (s) and 4.71 (s) respectively. Siilarly using NR ethod, the CPU tie are (s), (s) and (s) respectively. Hence the overall tie to copute TTC value with and without contingencies for all transfer directions for BSS and NR ethod is (s) and (s) respectively which shows that the CPU tie for BSS ethod is % less when copared to that of NR ethod. VI. CONCLUSION A fast coputational technique to assess TTC using BSS ethod is presented and tested on WSCC 9 bus, New England 39 bus and IEEE 118 bus test syste. Results indicate that the coputational tie to assess TTC using the proposed BSS ethod is far less when copared to the conventional NR ethod without losing accuracy. Further fro the results it is also evident that the percentage reduction in CPU tie for the proposed approach increases with the increase in size of the power syste when copared to that of conventional approach. REFERENCES RÉFÉRENCES REFERENCIAS 1. Available Transfer Capability Definitions and Deterination, NERC report, (1996). 2. G. Haoud, Assessent of Available Transfer Capability of Transission systes, IEEE Trans. Power Syst., 15/1, (2), Gabriel C. Ejebe, Jaes G. Waight, Manuel Santos- Nieto, and Willia F. Tinney, Fast Calculation of Linear Available Transfer Capability, IEEE Trans. Power Syst., 15/3, (2) Ashwani Kuar, S.C. Srivastava, AC Power Transfer Distribution s for Allocating Power Transactions in Deregulated Market, IEEE Power Engineering Review, (22), P.Venkatesh, R. Gnanadass, N. P. Padhy, Available transission capability deterination using power transfer distribution factors, International Journal of Eerging Electric P ower Systes, 1/2, (24), Luo, A. D. Patton, and C. Singh, Real power transfer capability calculations using ulti-layer feed-forward neural networks, IEEE Trans. Power Syst., 15/ 2, (2), T.Jain, S.N.Singh, and S.C. Srivastava, A Neural Network Based Method for Fast ATC Estiation in Electricity Markets, IEEE PES General eeting, Tapa, USA 7/GM746, (27). 211 Global Journals Inc. (US)

6 8. Azhar B. Khairuddin, S. Shahnawaz Ahed, M. Wazir Mustafa, Abdullah A. Mohd. Zin and Hussein Ahad, A Novel Method for ATC Coputations in Large Scale Power Syste, IEEE Trans. Power Syst., 19/2, (24), Yan Ou and Chanan Singh, Assessent of available transfer capability and argins, IEEE Trans. Power Syst., 17/2, (22), Mohaed Shaaban, Yixin Ni and Felix F. Wu, Transfer Capability Coputations in Deregulated Power Systes, 33 rd IEEE International Conference on syste sciences, Hawaii, (2), P. Bresesti, D. Lucarella, P. Marannino, R. Vailati and F. Zanellini, An OPF Based Procedure for fast TTC Analyses, IEEE PES Suer Meeting, Chicago, IL, USA, 3, (22), R. Ganadass, K. Manivannan, T G Palanivelu, N P Padhy, Assesent of Total and Econoic Transfer Capability for Practical Power Systes with Bilateral Transactions, JIE (India), 86, (25), Mark H. Gravener and Chika Nwankpa, Available transfer capability and first order sensitivity, IEEE Trans. Power Syst., 14/2, (1999), G. C. Ejebe, J. Tong, J. G. Waight, J. G. Frae, X. Wang, and W. F. Tinney, Available transfer capability calculations, IEEE Trans. Power Syst., 13/4, (1998) Manish Patel, Adly A. Girigis, Review of Available Transission Capacity Calculation Methods, Power Syste Conference, Cleson University, South Carolina, (29), Deterination of ATC within the Western Interconnection, WECC RRO Docuent MOD -3-, (21), S. Buhiler, N. Krejic and Z. Luzanin, Practical Qausi Newton algoriths for singular non linear systes, Journal on Nuerical Algoriths, Springer, vol. 55, n. 4, January 21, pp C. G. Broyden, A class of ethods for solving Non Linear Siultaneous Equations, Matheatics of Coputation, 19/92, (1965), Asif Seli, An Investigation of Broyden s Method in Load Flow Analysis, MS thesis report, Ohio University, (1994). 2. Yan Ou; Chanan Singh, Iproveent of Total Transfer Capacity Using TCSC and SVC, IEEE PES Suer Meeting, 2, (21), Vancouver, BC, K. Chandrasekar, N. V. Raana, A fast coputational technique to trace V- Q curve using Broyden Shaanski ethod, International Review on Modelling and Siulations (I.RE.MO.S), 4/1, (211), Dobson, S. Greene, R. Rajaraan, F.L. Alvarado, C.L. De-Marco, M. Glavic, A. DeSouza, R. Zieran, R.J. Thoaset al., Transfer capability calculator and tutorial, web site at Nov Power systes test case archive.http :// D. Zierann and Carlos E. Murillo-Sánchez, Matpower a Matlab power syste siulation package, User s Manual, Version 3.2, (27). July Global Journal of Researches in Engineering ( f ) Volue XI Issue V vv Version I 211 Global Journals Inc. (US)

7 July Global Journal of Researches in Engineering ( f ) Volue XI Issue V Version I Table 1 : TTC for WSCC 9 bus test syste Paraeters Area 1 2 Area 2 1 TTC (MW) MW Loss Base case Liiting V 5 V 9 CPU NR Tie(s) BSS Line outage Line 7-8 Line 7-8 TTC (MW) Contingency MW Loss (Line outage) Liiting V 7 V 9 CPU NR Tie(s) BSS Contingency ( Generator outage) Generator outage Bus 3 Bus 2 TTC (MW) MW Loss Liiting Line 1-4 Line 1-4 CPU Tie(s) NR BSS Table 2 : Area wise classification of Test syste Area 1 New England 39 Bus 1 3,17,18,25 27, 3, , 31,32, ,16,19 24, 28,29,33 36, 38 IEEE 118 Bus 1 23, 25 32, , ,38,65 112, 116, Global Journals Inc. (US)

8 Table 3 : TTC value with area wise power transfer of New England 39 bus test syste Paraeters Area Area Area Area Area Area TTC (MW) MW Loss Base case Liiting V 8 Line 6-11 Line 2-3 V 8 Line 4-5 Line CPU NR tie(s) BSS Line outage Line 3-4 Line 3-4 Line Line Line Line TTC (MW) Contingency MW Loss (Line outage) Liiting V 8 Line 6-7 Line 2-3 V 8 Line 4-14 Line CPU NR tie(s) BSS Generator outage Bus 37 Bus 32 Bus 32 Bus 38 Bus 38 Bus 37 Contingency ( Generator outage) TTC (MW) MW Loss Liiting Line Line 2-3 Line 2-3 Line 4-5 Line 4-5 Line CPU tie(s) NR BSS Table 4 : TTC value with area wise power transfer of IEEE 118 bus test syste Paraeters Area Area Area Area Area Area TTC (MW) MW Loss Base case Liiting Line 89- Line P G69 P G69 92 Line P G1 CPU NR tie(s) BSS Line outage Line Line Line Line Line Line TTC (MW) Line MW Loss Contingency Liiting Line 9- Line P G69 Line Line 9-91 P 91 G1 CPU NR tie(s) BSS Generator outage Bus 25 Bus 13 Bus 13 Bus 49 Bus 49 Bus 25 TTC (MW) Generator MW Loss Contingency Liiting Line 89- Line P G69 P G69 92 Line Line CPU NR tie(s) BSS July Global Journal of Researches in Engineering ( f ) Volue XI Issue V vv Version I 211 Global Journals Inc. (US)

9 July Global Journal of Researches in Engineering ( f ) Volue XI Issue V Version I This page is intentionally left blank 211 Global Journals Inc. (US)