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Reliability Studies in Power Generation: Adequacy Planning and Analysis Seminar

Learn about reliability indices calculation methodology, terms in reliability studies, and input data interpretation for single/multi-area studies. Define LOLE, LOLP, EUE, FOR, PFOR, and more. Explore power generation adequacy objectives and test study parameters sensitivity. Understand probabilistic density functions and generator states representation for improved planning accuracy.

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Reliability Studies in Power Generation: Adequacy Planning and Analysis Seminar

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  1. Generation Adequacy PlanningLOLE/LOLP Study SeminarbyGene Prestong.preston@ieee.orgDecember 6, 2002

  2. Seminar Purpose • Show the methodology for calculating the reliability indices using graphics and examples • Define terms used in reliability studies such as LOLE, LOLP, EUE, FOR, PFOR, pdf, etc. • Provide information to stakeholders concerning input data and interpretation of study results for single area and multi-area studies

  3. Generation Adequacy Study Objectives • Ensure installed generation reserve is sufficient • Test the sensitivity of study parameters

  4. f(x) ~ Pmax time at each MW level f(x) total area = 1 0 x - megawatts (MW) Pmax forced out of service See notes for each slide for more information. pdf = probabilistic density functionof a typical generator

  5. random variables f(x) F(x)   1 .5 x 1- area = value expected value or mean value F(x) = 1 - Pr[generation MW ≤x] = Pr[generation MW >x] Cumulative distribution function of the pdf

  6.   Combination of two generator pdfs using convolution

  7. x1 gen 1 Pr x2 gen 2 Pr .125 .25 .125 .1653 .375 .25 .375 .2123 .625 .25 .625 .2725 .875 .25 .875 .3499 Representation of pdfs with discrete states

  8. X Pr 0.25 0.0413250 0.50 0.0944000 0.75 0.1625250 1.00 0.2500000 1.25 0.2086750 1.50 0.1556000 1.75 0.0874750 Take all combinations of Pr’s and MW’s

  9. .8 f(x) .1 PFOR .1 FOR 0 x - MW Derated Pmax Representation of pdfs with discrete states (one generator with states: 0, Derated, Pmax)

  10. 1 probability of failure = t = 0 1/λ mean time to failure 1 – exp(–λt) probability of failure = Generator failure as an exponential function of time

  11. up Pr[unit is up] = P1 µ λ Pr[unit is down] = P0 µP0 = λP1 down and P0 + P1 = 1 λ = failure rate gives P0 = λ / (λ+µ) µ = repair rate also FOR = P0 = Pr[down] FOR = per unit down time Steady state FOR (forced outage rate)derived from λandµ

  12. P1 Pmax MW λ1+λ2 –µ2 –µ1 –λ2 µ2+λ3 –µ3 1 1 1 P1 P2 P3 0 0 1 λ2 µ2 P2 µ1 Pder MW λ1 λ3 PFOR = per unit derated time (partial forced outage rate) µ3 0 MW P3 FOR = per unit down time Markov representation of a 3-state generator

  13. U U U 9 1 4 1 2.33 1 D D D unit 1 FOR=.1 10 MW unit 2 FOR=.2 15 MW unit 3 FOR=.3 20 MW Use of the Markov process to represent three two-state generators

  14. P1 UUU 9 2.33 4 1 1 1 P2 UUD P3 UDU P5 DUU 2.33 2.33 9 9 4 4 1 1 1 1 P4 UDD P6 DUD P7 DDU 1 9 4 2.33 1 P8 DDD 1 1 Markov representation of three generators

  15. P1 P2 P3 P4 P5 P6 P7 P8 0 0 0 0 0 0 0 1 3 -2.3 -4 0 -9 0 0 0 -1 4.3 0 -4 0 -9 0 0 -1 0 6 -2.3 0 0 -9 0 0 -1 -1 7.3 0 0 0 -9 -1 0 0 0 11 -2.3 -4 0 0 -1 0 0 -1 12.3 0 -4 0 0 -1 0 -1 0 14 -2.3 1 1 1 1 1 1 1 1 Markov representation of three generators

  16. .504 45 MW .216 25 MW .126 30 MW .054 10 MW .056 35 MW .024 15 MW .014 20 MW .006 0 MW P1 P2 P3 P4 P5 P6 P7 P8 Markov representation of three generators

  17. Individual States Cumulative Gen 1 Gen 2 Gen 3 MW Pr MW Pr ΣPr 20 .7 -- 45 .504 45 .504 .504 15 .8 0 .3 -- 25 .216 35 .056 .560 10 .9 0 .2 20 .7 -- 30 .126 30 .126 .686 0 .3 -- 10 .054 sort 25 .216 .902 20 .7 -- 35 .056 20 .014 .916 0 .1 15 .8 0 .3 -- 15 .024 15 .024 .940 0 .2 20 .7 -- 20 .014 10 .054 .994 0 .3 -- 0 .006 0 .006 1.000 Use of a binary tree for the three generators to perform the convolution process

  18. .994 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 x (MW) .940 .916 .902 load not served .686 .560 Pr .504 generation .000 Graph of the 3 generator cumulative distribution for the probability that generation MW is > x)

  19. Unsuitability of the binary tree and Markov methods for large systems A system with 400 two-state generators has a total of: 400 120 210 states (combinations) This is greater than the number of atoms in the universe (~1080)!

  20. .994 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 x (MW) .940 .916 .902 load not served .686 .560 Pr .504 generation .000 The same cumulative distribution can be created one generator at a time. The function is updated as each generator is added.

  21. 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 x (MW) load not served Pr Starting with a blank distribution

  22. ∞ 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 x (MW) .900 load not served Pr .000 Cumulative distribution for generator 1generator 1 = {10 MW, FOR=.1}

  23. 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 x (MW) 1.00 x .8 =.80 .900 x .8 =.72 Pr .900 x .2 =.18 Adding generator 2 scales and shifts the initial distribution for Pr=.8 (up) and Pr=.2 (down)generator 2 = {15 MW, FOR=.2}

  24. .80+.18=.98 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 x (MW) .80 .72 load not served Pr generation .000 Summing the two curves gives the combined generators 1 and 2 cumulative distribution

  25. 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 x (MW) 1.00x.7=.700 .98x.7=.686 .80x.7=.56 Pr .72x.7=.504 .98x.3=.294 .80x.3=.24 .72x.3=.216 Adding generator 3 scales and shifts the distribution for Pr=.7 (up) and Pr=.3 (down)generator 3 = {20 MW, FOR=.3}

  26. .294+.700=.994 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 x (MW) load not served .24+.70=.940 .216+.7=.916 .686 .216+.686=.902 .560 Pr .504 generation .000 Summing the two curves gives the combined generators 1, 2, and 3 cumulative distribution(same curve as the one using a binary tree)

  27. .994 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 x (MW) .940 .916 .902 load not served .686 .560 Pr .504 generation .000 Binary tree graph of the 3 generator cumulative distribution for the probability that generation is available at x MW

  28. 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 x (MW) load not served Pr generation Cumulative distribution Pr[generation is up] represented in discrete 1 MW steps

  29. 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 45 40 35 30 25 20 15 10 5 0 x (MW) Pr .496 .440 .314 load not served .098 .084 .060 .006 .000 Flip the function over and backwards and the distribution represents Pr[generation out of svc]This is the COPT or Capacity Outage Probability Table

  30. 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 x (MW) Pr load not served Representation of Pr[gen out of service] as piecewise linear increments

  31. 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 x (MW) Pr for any 3 points, interpolate between the left two points load not served Representation of Pr[gen out of service] as piecewise quadratic increments

  32. PL x=30% PQ x=30% PQ x=20% Relative per unit error introduced by numerous interpolations of piecewise linear (PL) and piecewise quadratic (PQ) distributions

  33. EUE = MWH not served (for each hour in the study) LOLE for one day = 1 - Pr[gen up] = Pr[load loss] = 1-.56 = .44 d/y 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 MW x – MW (daily or hourly) .994 .940 .916 .902 .686 Pr .560 .504 generation LOLE – loss of load expectation EUE – expected unserved energy

  34. LOLP (for a week) = 1 - Pr[gen up] = Pr[load loss] = 1-.56 = .44 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 MW x – MW weekly peak demand .994 .940 .916 .902 .686 Pr .560 .504 generation annual LOLP = 1-i(Pr[gen up]i) for all i weeks LOLP – loss of load probability

  35. .994 1.0 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 5 10 15 20 25 30 35 40 45 50 MW x – MW (daily or hourly) .940 .916 .902 .686 .560 .504 Pr generation Representing load uncertainty(each hour is a Normal distribution of MW values)

  36. generator # insufficient reserve? 1 xxx 2 xxx 3 4 xxxxx 5 6 xxx 7 8 xxxx 9 xxx 10 xxxx 11 1 5 10 15 20 25 30 35 40 45 50 52 week # during the year summer Generation scheduled maintenance

  37. generator installed MW before maintenance 1 5 10 15 20 25 30 35 40 45 50 52 week # during the year with maint planning reserve MW demand insufficient reserve Generation scheduled maintenancereduces available generation – increases LOLE

  38. generator installed MW before maintenance 1 5 10 15 20 25 30 35 40 45 50 52 week # during the year with maint planning reserve MW demand Generation scheduled maintenanceto minimize overall LOLE

  39. Automatic scheduled maintenance methodology to minimize LOLE • Sort the MW unit sizes from largest to smallest. • Place the largest MW generator in a time slot with the greatest unused reserve margin. • Place the next largest generator in a time slot with the greatest unused reserve margin. • Repeat step 3 until all units are scheduled.

  40. 1999 load shape

  41. 1999 load shape

  42. actual more generators ~5000 MW ~9000 MW ~6% ~11% 12.5% 0 2000 4000 6000 8000 10000 12000 ERCOT generation (MW)forced out of service and derated

  43. DC tie considerations • probability of a DC tie failure is nearly 0 • probability of generation supply being available in the other region is expected to be nearly 1 • transmission constraints in the other region may reduce the probability of DC tie capability to less than 1 • DC tie capacity can be included or excluded from the LOLP calculations (affects the LOLE) • DC tie capacity may or may not be used to serve firm load in ERCOT (affects the reserve)

  44. DC tie in LOLP calculations Yes No LOLP: X MW DC CDR : X MW gen 12.5% reserve LOLP:100% DC MW CDR : X MW gen 11% x=0 to 12.5% for x=all firm LOLP: 0 MW DC CDR : 0 MW gen 12.5% reserve LOLP:100% DC MW CDR : 0 MW gen 11% reserve DC tie considerations DC tie with X MW firm generation DC tie with 0 MW firm generation

  45. Switchable generation considerations • Switchable generation capability must be available to ERCOT when called upon. • The same switchable MWs must be used in both the reserve calculation and the LOLP calculation.

  46. Self-serve generation considerations • Currently, both the self serve generation and self serve load (840 MW in the previous study) are omitted from the CDR and the LOLP calculations. • Alternately, the self serve generation and load could be included with the CDR and LOLP calculations with a negligible effect on LOLE. • Currently, self serve generation and load are included in the transmission load flow analysis as fixed MW values with 100% availability.

  47. Interruptible load considerations • The load can be modeled as two components, firm plus interruptible (i.e. two forecasts) • The LOLE for serving firm load can be calculated by using only the forecast for firm load in the computer simulation. • The LOLE for interruptible load can be calculated by using a forecast of firm load plus interruptible load in the computer simulation and then subtracting the LOLE results obtained for the firm load forecast.

  48. Data needed to perform single-area LOLP studies • hourly ERCOT loads for the (annual) study period (historical year hourly loads are scaled) • the annual peak demand forecast and the percentage of interruptible load • percentage of load forecast uncertainty • each generator’s seasonal MW (Pmax) capability, fuel type, type of unit, and maintenance periods (by beginning and ending week numbers or by total weeks needed) • FOR and DFOR of generator types such as gas, coal, nuclear, hydro, wind, etc. • identification of self-serve MW by generator

  49. Weeks 1-9 49-52 Weeks 38-48 Weeks 10-21 Weeks 22-37 Single Area Output Reports – Input Data SINGLE AREA GENERATOR DATA: SEASONAL CAPACITIES CDR FORCED PARTIAL-OUTAG SCHEDULED UNIT AREA WINT SPNG SUMM FALL CAP OUTAGE RATE DERATNG UNAVAILABLE NAME NAME MW MW MW MW % RATE % % % B1 D1 B2 D2 -------- -------- ---- ---- ---- ---- --- ------ ------ ------ -- -- -- -- STP1 ERCOT 1311 1311 1311 1311 100 6.90 2.30 5.50 5 4 0 0 STP2 ERCOT 1311 1311 1311 1311 100 6.90 2.30 5.50 4 4 0 0 CMPK 1 G ERCOT 1161 1161 1161 1161 100 6.90 2.30 5.50 11 4 0 0 CMPK 2 G ERCOT 1161 1161 1161 1161 100 6.90 2.30 5.50 3 12 0 0 DOW1 ERCOT 986 986 986 986 100 10.00 0.00 0.00 7 4 0 0 DOW2 ERCOT 917 917 917 917 100 10.00 0.00 0.00 3 12 0 0 DEC 1 G ERCOT 818 818 818 818 100 6.70 0.00 0.00 4 12 0 0 THSE 2 G ERCOT 818 818 818 818 100 6.70 0.00 0.00 48 4 0 0 LIM1 ERCOT 744 744 744 744 100 4.22 2.90 19.00 13 4 0 0 LIM2 ERCOT 744 744 744 744 100 4.22 2.90 19.00 3 12 0 0 MTNLK 1G ERCOT 727 727 727 727 100 4.22 2.90 19.00 43 4 0 0 MTNLK 2G ERCOT 727 727 727 727 100 4.22 2.90 19.00 48 4 0 0 MTNLK 3G ERCOT 727 727 727 727 100 4.22 2.90 19.00 49 4 1 8 MOSES3 G ERCOT 726 726 726 726 100 4.22 2.90 19.00 43 4 0 0 CB 3 ERCOT 703 703 703 703 100 6.70 0.00 0.00 13 4 0 0 DC-EAST ERCOT 700 700 700 700 0 0.01 0.00 0.00 3 12 0 0

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