Constrained Cost Optimization Under Uncertainty for an Incompletely-Connected Electric Utility System

Transcription

1 Constrained Cost Optimization Under Uncertainty for an Incompletely-Connected Electric Utility System J. Andrew Howe & Chris Humphries TransAtlantic Petroleum & UK National Grid Presented at the Palisade Risk Conference 2013 Frankfurt, Germany Howe & Humphries Utility Cost Optimization June / 38

2 Presentation Outline Introduction Modeling Power Utilities Our Problem Constrained Optimization Model The LP Formulation The psuedo-lp Formulation Adding in Uncertainty Modeling with Palisade s Evolver, and VBA Optimization Efficiency Model Results What We Now Know Investment Decision Making Concluding Remarks Howe & Humphries Utility Cost Optimization June / 38

3 Introduction Modeling Power Utilities Like any other business, electric utilities have to operate under cost constraints - even public utilities that don t need to make a profit. In fact, they can be more constrained. If your local supermarket runs out of your favorite food, you re unhappy. If your power company runs out of power, it s an entirely different thing. When power prices rise dramatically (California, USA, 1990 s), all hell breaks loose. Furthermore, power utilities are constrained by more than just costs: If a region s winter is especially warm & dry, there will be little runoff for dams the ensuing spring. If the utility relies on hydrological power for cheap baseload, this will raise costs. If a region s populace turns against gas transport pipelines, gas-burning CC / CT peaking units (basically a jet engine mounted on the ground) may have to be left off. The utility may have to purchase more off-network electricity at much higher prices - if even possible. If enough people are swayed by the arguments against nuclear power, clean-burning nuclear generators must be shut down. The utility will then have to burn more coal or gas - at higher costs (and ultimately more pollution). Howe & Humphries Utility Cost Optimization June / 38

4 Introduction next: our problem In addition to usual operating costs, utilities often have further constraints on investments. If a utility s region expands outward dramatically, it can be much cheaper to extend high-voltage lines to the new outlying regions, instead of building new capacity. However, this requires extra wayleaves that landowners may fight. Furthermore, expansion by adding new power plants can face regulatory and environmental challenges, as well as the costs. This bring us to our specific modeling problem. Howe & Humphries Utility Cost Optimization June / 38

5 Introduction Our Problem In this case study, we demonstrate how Palisade and Evolver products can work together to perform cost optimization and inform investment decision modeling for a system of incompletely-connected power grids. The modeled utility system in the UK is composed of four notional power-generating regions. Each region has its own power demand and portfolio of types of power generating plants. Because of the variation in generating assets, the generating cost profile varies dramatically by region. Power transmission lines connect the four regions subject to the constraints that a) some regions are not connected at all, and b) the amount of electricity that is allowed to flow over each boundary is limited. Furthermore, power plant availability and total generating capacity vary stochastically, as a function of many factors. Howe & Humphries Utility Cost Optimization June / 38

6 Introduction next: the LP formulation We mathematically define an optimization problem that allows us to meet the demand of all regions under the transmission constraints while minimizing the total cost. We develop this optimization problem in three stages. It is first defined as a linear programming optimization, then extended to be psuedo-linear, then finally nonlinear. This is then implemented under the framework of uncertain generation capacity so we can make probabilistic statements about the costs and other relevant quantities. The results from the final model can be used to guide and inform investment decisions for demand reduction and capacity / transmission expansion. Howe & Humphries Utility Cost Optimization June / 38

7 Constrained Optimization Model The LP Formulation Power flow boundaries between notional regions North Ireland (N), Scotland (S), England & Wales (E), and Republic of Ireland (R) Howe & Humphries Utility Cost Optimization June / 38

8 Constrained Optimization Model The LP Formulation In formulating our problem, first we define the region-specific variables: Unit Cost per region in /MW: CN, C S, C E, C R ; Generating capacity per region (MW): MN, M S, M E, M R ; Peak power demand per region (MW): DN, D S, D E, D R. Next we define the maximum flow constraints in Megawatts between all pairs of regions (even those not physically connected get a variable): North Ireland Scotland: MNSN, North Ireland England & Wales: MNEN, North Ireland Republic of Ireland: MNRN, Scotland England & Wales: MSES, Scotland Republic of Ireland: MSRS, England & Wales Republic of Ireland: MERE. Howe & Humphries Utility Cost Optimization June / 38

9 Constrained Optimization Model The LP Formulation Finally, we define variables holding the total amount of power generated per region: GN = G NN +G NS +G NE +G NR, GS = G SN +G SS +G SE +G SR, GE = G EN +G ES +G EE +G ER, GR = G RN +G RS +G RE +G RR, The first subscript indicates the generation region and the second indicates the demand region. For example, G SE is the power generated by the Scotland region that flows into England & Wales; G ES is the exact opposite. Our LP optimization problem can then be formulated to minimize the total cost, subject to the inter-region flow constraints, by varying the G variables. This is shown on the next slide. Howe & Humphries Utility Cost Optimization June / 38

10 Constrained Optimization Model The LP Formulation minimize C N G N +C S G S +C E G E +C R G R subject to G 0 G NS,G SN M NSN G NE,G EN M NEN flow limits G NR,G RN M NRN G SE,G ES M SES G SR,G RS M SRS G ER,G RE M ERE { 0 G N M N 0 G S M S generation up to capacity 0 G E M E 0 G R M R G NN +G SN +G EN +G RN = D N G NS +G SS +G ES +G RS = D S demand is met G NE +G SE +G EE +G RE = D E G NR +G SR +G ER +G RR = D R Howe & Humphries Utility Cost Optimization June / 38

11 Constrained Optimization Model next: the psuedo-lp formulation In the first constraint, G indicates all power generation quantities previously mentioned. This constraint forces power generation to be positive The six constraints starting in the first block limit the amount of power that can flow over all possible boundaries. To prevent power flowing between two disjoint regions, the RHS here just needs to be set to 0 (such as North Ireland and England & Wales, M NEN = 0). The next block of four constraints force the total power generated in each region to not exceed the capacity. The last four constraints guarantee all region s power demands are met This works, and can be solved extremely easily with the Solver add-in to Microsoft Excel, but has one major shortcoming. Howe & Humphries Utility Cost Optimization June / 38

12 Constrained Optimization Model The psuedo-lp Formulation The major drawback of this optimization problem is that it will allow power to flow bidirectionally between two adjacent regions. For example, maybe Scotland will generate some power to meet the demand in England & Wales, while England & Wales sends power to Scotland. The resulting minimum cost is the same as if the two flows simply netted out, but the solution requires some manual adjustment to be realistic. To solve this, we instead formulate a psuedo-lp problem by adding six binary constraints Define G NSN = { 1 if G NS > 0 and G SN > 0 0 otherwise The quantities G NEN, G NRN, G SES, G SRS, and G ERE are similarly defined.. Howe & Humphries Utility Cost Optimization June / 38

13 Constrained Optimization Model next: adding in uncertainty We can then optimize the psuedo-lp problem shown here minimize C N G N +C S G S +C E G E +C R G R subject to G 0 G NS,G SN M NSN G NE,G EN M NEN flow limits G NR,G RN M NRN G SE,G ES M SES G SR,G RS M SRS G ER,G RE M ERE { 0 G N M N 0 G S M S generation up to capacity 0 G E M E 0 G R M R G NN +G SN +G EN +G RN = D N G NS +G SS +G ES +G RS = D S demand is met G NE +G SE +G EE +G RE = D E G NR +G SR +G ER +G RR = D R G NSN,G NEN,G NRN,G SES,G SRS,G ERE = 0 Howe & Humphries Utility Cost Optimization June / 38

14 Constrained Optimization Model Adding in Uncertainty But we aren t quite done yet - we must consider the uncertainty in generation capacity and cost. For a power plant, the Nameplate Capacity is the maximum amount of power it can generate safely under optimal conditions. A utility system s Total Nameplate Capacity would be the sum of the Nameplate Capacities for all the plants. The maximum power a plant can generate at any given time will probably be less than its Nameplate Capacity; the same can be said for the total utility system. A utility system s total generating capacity will generally vary from this theoretical maximum for reasons such as: planned outages (for maintenance) and / or unplanned outages (something breaks) if gas prices rise too high, gas-burning combustion turbine units may not be run if lower quality coal is used, a coal-burning plant may not be able to run at full power dam spillways maybe be partially closed during spawning season in spring to increase viability of fish roe Howe & Humphries Utility Cost Optimization June / 38

15 Constrained Optimization Model Adding in Uncertainty But we aren t quite done yet - we must consider the uncertainty in generation capacity and cost. For a power plant, the Nameplate Capacity is the maximum amount of power it can generate safely under optimal conditions. A utility system s Total Nameplate Capacity would be the sum of the Nameplate Capacities for all the plants. The maximum power a plant can generate at any given time will probably be less than its Nameplate Capacity; the same can be said for the total utility system. A utility system s total generating capacity will generally vary from this theoretical maximum for reasons such as: planned outages (for maintenance) and / or unplanned outages (something breaks) if gas prices rise too high, gas-burning combustion turbine units may not be run if lower quality coal is used, a coal-burning plant may not be able to run at full power dam spillways maybe be partially closed during spawning season in spring to increase viability of fish roe (damn fish and dam spillways!) Howe & Humphries Utility Cost Optimization June / 38

16 Constrained Optimization Model Adding in Uncertainty The cost to generate one Megawatt of power from a power plant is largely, but not solely, a function fuel prices. A very generic generating cost hierarchy for some major plant types in the UK would be Hydrological Nuclear < Gas < Coal < Oil < Diesel In other places, such as the US, coal and gas may trade places. If generating capacity from power plants that are lower in the hierarchy is curtailed or completely unavailable, the total cost will be increased. Hence, since generating capacity (M ) will vary from the Nameplate Capacity at any given period, so will unit cost (C ). In this case study, we have several types of power plants, with capacity modeled as shown on the next slide. 1 1 Plant running costs and availability data is for illustrative purposes only, and does not represent information for any particular generating plant. Howe & Humphries Utility Cost Optimization June / 38

17 Constrained Optimization Model Adding in Uncertainty Stochastic modeling of plant capacity; N = Nameplate (MW), U = # plants. Type Cost ( /MW) Capacity Offshore Wind 0.5 Discrete(Uniform) N Onshore Wind 0.3 Discrete(Uniform) N Wave & Tidal 0.7 Binom(n = U,p = 0.20) 30 Hydrological 0.1 Gauss(µ = 0.6, σ = 0.04) N Nuclear 0 Binom(n = U,p = 0.70) 600 France* 0 Binom(n = 4,p = 0.50) 500 Imera/BritNed* 0 Binom(n = 9, p = 0.50) 500 Base Gas 40 Binom(n = U,p = 0.80) 500 Base Coal 50 Binom(n = U,p = 0.80) 500 Biomass 55 Binom(n = U,p = 0.85) 100 Marginal Gas 70 Binom(n = U,p = 0.80) 500 Marginal Coal 70 Binom(n = U,p = 0.80) 500 Pumped Storage 120 Binom(n = U,p = 0.90) 300 Oil 200 Binom(n = U,p = 0.80) 500 OCGT 300 TGauss(µ = 0.95,σ = 0.03) 677 Howe & Humphries Utility Cost Optimization June / 38

18 Constrained Optimization Model Adding in Uncertainty Some plant types have slightly negative costs. These are designed to run either intermittently (Wind) or predictably (Marine / Hydro), and contractual obligations and fixed costs are structured accordingly. Though fuel costs are zero, there are commercial advantages for these plant to always run (hence the negative cost). The two starred sources indicate submerged power cables coming from the European continent into the England & Wales region. Note the cost difference between Base Gas / Coal and Marginal Gas / Coal. Perhaps the marginal units are older or thermally less efficient; hence they should be used less. OCGT is a group of auxillary peaking diesel generators used only when absolutely necessary. Note that the OCGT capacity is modeled using a Truncated Normal distribution. The availability rates vary seasonally; what we used was estimated by SMEs as availability rates for the winter. This table, with the simulated generating capacities, forms a Merit Order. Howe & Humphries Utility Cost Optimization June / 38

19 Constrained Optimization Model Adding in Uncertainty The Merit Order for each of the four generating regions is specific, with the number of each plants varying by region. For example, North Ireland has only 3 gas plants and no coal-burning generators; England & Wales has 66 and 28, respectively. Each time we simulate the complete Merit Order for a region, we can sum the capacities and costs, then compute the average unit cost. For example: Example England & Wales partial Merit Order Type Unit Cost Capacity Total Cost Nuclear 0 6, Base Gas 40 17, , 000 Base Coal 50 5, , 000 Biomass 55 1, , 500 Pumped Storage 120 2, , 000 Total M E = 32,200 1,287,500 Average C E = Howe & Humphries Utility Cost Optimization June / 38

20 Constrained Optimization Model Adding in Uncertainty Hence we come back to our psuedo-linear optimization problem, which we must now call nonlinear. We allow each region s Merit Order to be dynamically generated, in that the optimization routine can turn on / turn off individual plant types in each region independently. Thus, the generating capacity M and unit cost C in each region can be varied. To extend our example from the table on the previous slide, if the 2,100 MW of expensive Pumped Storage are turned off, the generating capacity drops to 30,100 MW and the unit cost to /MW. While this allows expensive power generators to be turned off to minimize the costs, this must be balanced against meeting all demand. Our final nonlinear optimization problem is shown on the next slide, where M d and C d indicate that generating capacities and unit costs are dynamic. Howe & Humphries Utility Cost Optimization June / 38

21 Constrained Optimization Model next: modeling with Palisade s tools minimize CN d G N +CS d G S +CE d G E +CR d G R subject to G 0 G NS,G SN M NSN G NE,G EN M NEN flow limits G NR,G RN M NRN G SE,G ES M SES G SR,G RS M SRS G ER,G RE M ERE { 0 G N MN d 0 G S MS d generation up to capacity 0 G E ME d 0 G R MR d G NN +G SN +G EN +G RN = D N G NS +G SS +G ES +G RS = D S demand is met G NE +G SE +G EE +G RE = D E G NR +G SR +G ER +G RR = D R G NSN,G NEN,G NRN,G SES,G SRS,G ERE = 0 Howe & Humphries Utility Cost Optimization June / 38

22 Modeling with Palisade s Evolver, and VBA We implement all this in a Microsoft Excel spreadsheet model, using Palisade s Decision Tools Suite is used for the stochastic modeling and analysis. Evolver is used for the constrained optimization. Let s see these worksheets. Howe & Humphries Utility Cost Optimization June / 38

23 Modeling with Palisade s Evolver, and VBA We implement all this in a Microsoft Excel spreadsheet model, using Palisade s Decision Tools Suite is used for the stochastic modeling and analysis. Evolver is used for the constrained optimization. Let s see these worksheets. Challenge: Evolver requires static inputs and RiskOptimizer requires static constraints Howe & Humphries Utility Cost Optimization June / 38

24 Modeling with Palisade s Evolver, and VBA We implement all this in a Microsoft Excel spreadsheet model, using Palisade s Decision Tools Suite is used for the stochastic modeling and analysis. Evolver is used for the constrained optimization. Let s see these worksheets. Challenge: Evolver requires static inputs and RiskOptimizer requires static constraints Solution: Create a hybrid solution with VBA to bridge and Evolver Howe & Humphries Utility Cost Optimization June / 38

25 Modeling with Palisade s Tools next: optimization efficiency The modeling algorithm we designed is: i) At the beginning of the simulation, initialize the Evolver model and set convergence criteria ii) Simulate generation capacities for the Merit Order on the generation worksheet iii) Copy Merit Order to model worksheet iv) Execute the Evolver model to optimize the total system cost v) Record results and go back to Step ii) model measures the empirical distributions of several optimized quantities: Regional Generating Capacity (M d ) Total System Generating Capacity ( Md ) Regional Unit Cost (C d ) Total System Cost ( Cd G ) Power Flow Across Boundaries (G ) It also records information about the optimization model: Convergence Failure Total Number Trials & Efficiency Time Required Let s see a short demonstration. Howe & Humphries Utility Cost Optimization June / 38

26 Modeling with Palisade s Tools Optimization Efficiency Evolver has two optimization engines to choose from: the Genetic Algorithm and the OptQuest engine. Manually selecting the GA led to very poor results, so we let Evolver pick OptQuest. Actually, we run Evolver twice, to let it better search the solution space: the first run uses a maximum of 400 trials, while the second goes for a maximum of 600: 1,000 in total. With every trial, Evolver compares the current best to the best from 200 trials previous. If the improvement in Total System Cost is less than 0.001, it stops. Running model for 1,000 simulations takes less than 3 hours. This could be decreased by further tuning of the Evolver parameters. The next two slides show the empirical distributions of optimization efficiency measures. Howe & Humphries Utility Cost Optimization June / 38

27 Modeling with Palisade s Tools Optimization Efficiency Howe & Humphries Utility Cost Optimization June / 38

28 Modeling with Palisade s Tools next: what we now know Howe & Humphries Utility Cost Optimization June / 38

29 Model Results What We Now Know Mean total system generating capacity is 65, 830MW 90% confidence interval (64, 141, 69, 180) Howe & Humphries Utility Cost Optimization June / 38

30 Model Results What We Now Know Mean total system cost is 2,290,786 90% confidence interval (1, 734, 205, 2, 771, 690) Howe & Humphries Utility Cost Optimization June / 38

31 Model Results What We Now Know Mean unit cost (per MW) = Scotland: 8.74, Republic of Ireland: 37.93, England & Wales: 38.89, North Ireland: The next slide has the the empirical distribution of power flows into North Ireland from Scotland. Howe & Humphries Utility Cost Optimization June / 38

32 Model Results next: investment decision making In nearly half the simulations, power flow from Scotland into North Ireland was limited by the constraint. A similar impact of the constraint between Republic of Ireland and North Ireland was not observed. Howe & Humphries Utility Cost Optimization June / 38

33 Model Results Investment Decision Making North Ireland s generating capacity is relatively low; it has an approximately 65% chance of being less than demand. Because of the high cost and this low capacity, North Ireland generates power for other regions infrequently and only in small amounts, so we can focus on the Scotland North Ireland link. We test the scenario where we increase transmission capacity from Scotland to North Ireland to 2, 000MW. Can we lower total system cost by investing in transmissiong capacity to allow more cheaper power to flow into North Ireland? Howe & Humphries Utility Cost Optimization June / 38

34 Model Results Investment Decision Making Howe & Humphries Utility Cost Optimization June / 38

35 Model Results Investment Decision Making In the base model, North Ireland supplied 1,523MW of its own demand. With the transmission constraints relaxed, this became 637MW. Howe & Humphries Utility Cost Optimization June / 38

36 Model Results Investment Decision Making Here we see the impact of relaxing the flow constraint; in nearly 90% of the simulations more power than the original constraint flowed across this boundary. Howe & Humphries Utility Cost Optimization June / 38

37 Model Results Investment Decision Making With transmission constraints relaxed, the mean cost dropped by 2% to 2,245,495. The 5 th percentile decreased by about 4%, and the 95 th by 2%. Howe & Humphries Utility Cost Optimization June / 38

38 Model Results Investment Decision Making UK National Grid has been proactive in contracting for and installing alternative power plants, including Wind, Tidal, and Nuclear. Offshore wind and tidal, specifically, still have much room for growth. What is the effect on total system cost of investing in an approximately 50% increase in offshore wind and tidal power? Increase of Offshore wind and tidal generating capacity (MW). Region Offshore Wind Wave & Tidal North Ireland MW 3 90MW Scotland 3, 185 4, MW MW England & Wales 13,375 20, MW 4 120MW Republic of Ireland MW 4 120MW Total 17,185 25, ,200 Howe & Humphries Utility Cost Optimization June / 38

39 Model Results Investment Decision Making The addition of almost 9,000MW of alternate power decreased the mean total system cost by 7.7% while the variability jumped up by 25%; the 5 th percentile dropped by 20%. Howe & Humphries Utility Cost Optimization June / 38

40 Model Results Investment Decision Making Many utilities have been recently investing in programs to decrease demand - especially peak demand. This includes rebates for energy efficient smart appliances, education programs, interruptibility/curtailage agreements, and the Smart Grid. We assume that if overall peak demand is reduced, all regions will be affected the same, so the relative proportions remain unchanged. What is the effect on total system cost of investing in consumer programs that lead to a modest decrease of 6.25% in peak demand to 60, 000MW? Howe & Humphries Utility Cost Optimization June / 38

41 Decreasing demand by 6.25% dropped the total system cost by 12% to 2,019,410. Variability dropped by 7%, and the 5 th and 95 th percentiles dropped by 14 and 11 percent. Howe & Humphries Utility Cost Optimization June / 38 Model Results next: concluding remarks

42 Concluding Remarks In this case study, we have demonstrates how Palisade and Evolver products can work together to perform cost optimization for a system of incompletely-connected power grids. We began by mathematically defining an optimization problem that allows us to meet the aggregate demand of all regions under the transmission constraints while minimizing the total cost. This was then implemented under the framework of uncertain generation capacity, so the optimization can take into consideration the fact that capacity is a stochastic quantity. The resulting Monte Carlo simulation model allowed us to make probabilistic statements about regional generating capacity and cost, total system generating capacity and cost, and inter-regional power flows. Finally, we demonstrated how this optimization model can be used to guide and inform investment decisions. The results about the impact on power generation costs could be used as inputs to an investment model which would consider the capital, contractual, regulatory, and other costs. Howe & Humphries Utility Cost Optimization June / 38