Analysis and Comparison of Risk and Load Point Indices of Power System Model HLI and HLII

Transcription

1 Analysis and Comparison of Risk and Load Point Indices of Power System Model HLI and HLII L.B. Rana and N.Karki Abstract Aim of this work is to evaluate risk analysis of the Electrical Power Network with Discrete generation, load Model and transmission line constraints. The system under considered is analyzed using conditional probability approach and discrete Markov model. The generation load model and transmission model are convolved to form the appropriate risk model. Load point indices LOLP, LOLE, Expected Energy Not served EENS or Loss of Energy Expectation are computed with and without transmission facilities constraints. Also effect of transmission capacity and peak load are investigated. The basic generation model is capacity outage probability table(copt) and load model is daily peak load variation curve (DPLVC) or hourly variation of load (LDC).The calculated indices do not normally include transmission constraints. Basic modeling approach for HLI is shown in Fig.No.2. Keywords LOLP, LOLE EENS HLI and HLII S I. INTRODUCTION INCE the modern power system is composed by three subsystems Generation, Transmission and Distribution. Reliability and risk evaluation of the whole network is complex so it is divided in different hierarchical levels (HL) as shown in Fig.No.1. The concept of hierarchical levels (HL) has been developed in order to establish a consistent means of identifying and grouping these functional zones [1]. HLI represents generation and Load Model only. Whereas HLII includes transmission facilities also. And HLIII represents whole network. In this analysis first of all HLI is only considered. The generation and load models are convolved to compute probabilistic risks and adequacy indices [1]. Fig..2 Modeling approach for HLI [1] Then transmissions facilities constrain are imposed to form HLII. Then conditional probability approach is implied to get load point or delivery point indices seen by each individual network node. Scheme for calculation of reliability indices for HLII is shown in flow chart of Fig.3. The system is initializing with initial outage and line flow limit is imposed. Fig. 1 HL of Electrical Power Network [1] Lalit. Rana is PhD Scholar in Lappeernrata University of Technology, Lappeenranata Finland. rana.lalit@gmail.com N.Karki is Lecturer in Electrical Engineering Department, Institute of Engineering Thribhuvan University Nepal Fig. 3 Scheme for Reliability indices Computation 83

2 If there is no line flow limit then generation is reschedule and again same line flow limit is imposed to obtain the minimum load point power curtailment, otherwise it gives reliability index of load point without rescheduling. The steady state value of the availability and unavailability for any system is obtained from equation (1) when time approaches to infinity. μ λ -(λ + µ)t Availability, A(t) = + e (1.a) λ + μ λ + μ λ λ -(λ + µ)t Unavailability, U(t) = + e (1.b) λ + μ λ + μ μ λ Hence Availability, A= Unavailability, U= λ + μ λ + μ Also the when the system fulfill requirements needed for binominal distribution i.e. trails are independent and probability of failure and success for each trail are constant can be applied.generalized expression is given in equation (2). (p + q) n = p n + n p n-1 n(n 1) q + p n-2 q 2 n(n 1)(n - 2) + 2! 3! p n-3 q 3 n(n 1)..[n - (r -1)] p n-r q r +. + r! q n..(2) n(n 1)..[n - (r -1)] n! Where p n-r q r = = ncr r! r!(n - r)! II. LOAD POINT INDICES Departure Rate is defined as the rate by which system n departed from state j to all other states ν = a kj k = 1, j Frequency of occurrence/ annum is defined as summation of the product of departure rate from j state to all other state and probability of system being in state j. Frequency of n occurrence/ annum f = P * a j kj k = 1, j Loss of load probability (LOLP) is defined as the probability of the generating capacity is less than daily peak load over a space of time usually one year n LOLP = P( C ). P( L > C ) i i i i = 1, Where P (Ci) is individual probability of the state i which is obtained from the capacity outage probability table. Where P(Li>Ci) is percentage of time for which outage capacity will cause loss of loads The Loss of Load Expectance (LOLE) is risk index and the most widely accepted and used probabilistic method in system reliability evaluation for generating systems. Two models are required and employed. One is the previously 84 studied Load Duration Curve (LDC), and the other is the COPT. These two models are convolved in the process. The units of the LOLE are in days per year (d/y). The LOLE evaluation method is expressed in the following mathematical formula. LOLE = P. t. Hence LOLE is k k. defined as summation of the product of probability of outage and the percentage of time that cause load loss Expected Energy Not supplied (EENS) or Loss of Energy Expectance (LOEE), EENS or LOEE in load point in per unit is computed t j P D j j j 100 EENS = where. D = L. dt., and, D = ( L C ). dt D j j 0 0 Where Dj is energy curtailed due to capacity outage Qk, Pj is probability of capacity outage Qk. Where D is total energy under load curve.eens is computed from DLPVC as shown in Fig.4 shaded area represents EENS. III. SYSTEM DESCRIPTION The single line diagram of the system under consideration is shown Fig. 5. It consist six generators of equal capacity connected in common bus, two parallel transmission line and load model. TABLE I SYSTEM DATA [1] Generator data 6 x 40 MW units λ=0.01 f/day = 3.65 f/yr µ = 0.49 r/day = r/yr Dj Load Capacity outage 27.77% ti % % of Time Fig. 4 DLPVC Transmission elements Individual line-cap. =160 MW 2 lines λ= 0.5 f/yr r=7.5 hours/repair Load Peak load = 180MW load-duration curve varies from the 100% to the 60% Load (MW) %of Time Generating System Transmission System Daily Load Curve Fig. 5 System considered P0 240 P1 200 P Max 180 P2 160 P3 120 P4 80

3 IV. MODEL FOR STATISTICAL ANALYSIS 4.1. Generation Model The Discrete Markov model for the system is developed as shown in Fig.6 assuming there is no common cause failure and all failure is independent. Also repair actions are carried out independent of each other (i.e. maintenance crew is thus not a limiting factor). The system consist seven dependent equations hence the steady state probability of system being in any state can be obtained from any of six dependent and another equation i.e. is sum of probabilities of being all possible states is equal to 1 [1-2]. Hence limiting state probability of the generating station computed from equation (1) of appendix I which is illustrated in Table II. TABLE II STEADY STATE PROBABILITY OF THE GENERATION SYSTEM P0 P1 P2 P P4 P5 P All Six Up 4.2. Transmission Line Model For this analysis two parallel TL is considered each of capacity 160MW. Failure rate and repair rate of TL is shown in table I. Fig.No.7 represents Discrete Markov model of TL. The steady state Markov Model of TL comprises three dependent equations and solved by considering another equation which gives sum of being any states is equal to unity [1-2] as shown in Appendix I. Both Up 2λt µt 6λ µ One Up 5 Up 1DN Fig. 6 Generation Markov Model Since the Generation and the transmission outage are independent and no common cause failure, so the two independent models is merged and load model is imposed to obtain load point indices. λt 2µt Fig. 7 TL Markov Model λ 6µ All Six down Both down Probability of failure failure k K V. ANALYSIS Q = P. P and Frequency of k F = F. P, Where, P I, F and P K are Outage condition of the Network, Probability of existence of outage, Frequency of occurrence of outage and probability of the load at bus exceeding the I K maximum load that can be supplied at that bus during the outage respectively Expected number of load curtailments n =. F x Where j belongs to number of states which includes all contingencies resulting in line overloads which are alleviated by load curtailment at bus K. Expected load curtailed= F. x = L Where L is the load not supplied at an load point due all contingencies to alleviate overloading and F is state frequency Expected Load Curtailed = Load Curtailed x frequency by which that amount of load is curtailed. Expected energy not supplied EENS = F. L D x Where Dj is duration in hours of the load curtailment arising due to the outage j. EENS = ELL x hours associated to that ELL 5.1 Load Point Indices with and without TL The system is analyzed with and without transmission line is computed and result are compared and illustrated in Table IV and Fig.No.8. It is observed that system indices are affected with TL constraints and performance becomes poorer. Here eighteen possible states are obtained after merging two models. Indices for each state are computed as below. Let consider the system is in state 2 i.e. All Generators are functioning and but only one line is functioning capacity Available = 160 MW State Probability =Probability of all Gen. functioning x Probability of one TL functioning Total frequency occurrence per year =Prob. of being that state x depart rate from the state = P(0G1L)x(6λ+λt+µt) Similarly state probability and frequency per annum is computed and listed in table P kj = probability of load at bus K exceeding the maximum load that can be supplied at that bus without failure. It s computed as shown For state 2 when capacity available is 160 MW 85

4 P = = k =,Failure Probability = x0.2777= and Failure Frequency = x0.2777= The result for all possible states without and with TL are shown Appendix II and III TABLE.III TRANSMISSION LINE OUTAGE Outage TLin Prob Depart Rate Freq. Occ/year 0 2X 0, , X 0, ,5 1, ,89E , MW. The results are presented in Table VI and Fig.No 10. TABLE V INDICES WITH DIFFERENT TL CAPACITY Indices TL of 170MW T.L of 160 MW TL of 150 MW Failure Probability days per year Failure Fequency EENS(MWhr) per Year Expected Load Curtailed MW TABLE IV LOAD POINT INDICES WITH AND WITHOUT TL Indices Failure Probability days per year Failure Fequency EENS(MWhr) per Year Expected Load Curtailed (MW) Without TL constraints With T.L constraints Fig. 9 Indices with different TL Capacity TABLE VI Fig. 8 Indices with and without TL 5.2. Load Point Indices with different TL capacity Here two cases are considered one is when Transmission Capacity is increased to 170 MW and other is decreased to 150 MW. The Results are compared in Table V and Fig. No Indices for Different System Peak To perform this analysis transmission capacity per line is considered as 160MW and the Load Model is also constant ie Linear variation from 100% to 60%,only the Peak varies. Here again two situations are considered one is Peak load increased to 200MW and other is decreased to INDICES WITH DIFFERENT PEAK Indices 200MW Peak 180 MW Peak 160 MW Peak Failure Probability days per year Failure Frequency EENS(MWhr) per Year Expected Load Curtailed CONCLUSION Fig. 10 Indices with different System Peak 86

5 It is observed that the risk will be more when the TL constraints will imposed and it will be increased with increase the value of peak load of system.also it will affected the Transmission line capacity for the given load model i.e. decreases with increases the TL capacity for given load model and same failure and repair rate.also it is further influenced from the size of the generators. REFERENCE [1] Reliability Evaluation of Power Systems Second Edition by Roy Billinton University of Saskatchewan College of Engineering Saskatoon, Saskatchewan, Canada and Ronald N. Allan University of Manchester Institute of Science and Technology Manchester, England [2] System Reliability Theory: Models, Statistical Methods, and Applications, Second Edition by Marvin Rausand and Arnljot Høyland [3] Introduction to Reliability Engineering Second edition by E. E. Lewis Department of Mechanical Engineering, Northwestern University, Euanston, Illinois uiy, 1994 Appendix I Steady state Equation for the Generation Steady state Equation for the Transmission Appendix II Pin A 0.98 U 0.02 F R Indi. Cum Dept % Load En Pout Prob Prob Rate F/year time LOLP Curt Curt E E E E E E E E E-05 LOLE ELL

6 Appendix III Cond. Cap (Avai) State State Probabilities and Frequencies and other indices Failure Prob. Frequency Pij Prob Freq Hrs Outage Load Curt ELL EENS 0G0L G1L G2L E E G0L G1L E E G2L E E E E G0L G1L E E G2L E E E E E E-05 3G0L G1L E E G2L E E E E E E-05 4G0L E E G1L E E E E G2L E E E E E E-07 5G0L E E E E G1L E E E E E-06 2E-05 5G2L E E E E E E-09 6G0L E E E E E G1L E E E E E E-08 6G2L E E E E E E