CVAR-Constrained Multi-Period Power Portfolio Optimization

We consider power portfolio optimization of real and contractual assets, including derivative instruments in a multi-period setting. A model is introduced that incorporates fixed transmission rights in a three-node unidirectional network in order to evaluate the significance of transmission constraints. We use data from the PJM, which is located in the eastern United States for model implementation. The simulation results show that transmission constraints and fixed transmission rights can have a significant effect on the choices a utility will make when dealing with power procurement. Our results imply that companies should not only hedge the risk of unknown power prices but also unknown transmission congestion. INTRODUCTION The pre-deregulation electricity markets were characterized by predictable power prices and utility-owned power projects. In the 1990s, regulators in certain states began to explore deregulating the markets. They theorized that the increase in competition would result in lower power prices for consumers. By the late-90s, some states had deregulated their markets and others followed in the early 2000s. The result for utilities was the 30 MBAA Proceedings 2010 Papers introduction of competition for customers and more price volatility in power procurement. Companies that were interested in entering the power market could now apply to solicit customers and/or sell electricity on the wholesale market. All players in the electric market were faced with the risk of uncertain fuel prices, weather conditions and, therefore, unpredictable power prices due to electricity’s non-storable nature. In order to combat the uncertainty, companies have developed hedging strategies. Historically, models have accounted for differing time horizons, types of generation and risk. Many have commented that transmission constraints should be considered, but none has attempted to incorporate them into the model. This paper builds on previouslydeveloped models for the electric industry dealing with a multi-period portfolio optimization incorporating conditional value at risk (CVaR). Our model incorporates fixed transmission rights in a three-node unidirectional network in order to evaluate the significance of transmission considerations in power portfolio optimization. The stochastic nonlinear mixed-integer model presented shows that transmission constraints and fixed transmission rights can have a significant effect on the choices a utility will make when dealing with power procurement. It is demonstrated that the inclusions drastically decrease the value of the objective function. LITERATURE There has been a vast amount of research done in recent years dealing with power portfolio optimization. The non-storable nature of electricity and the increasing complexity of financial instruments as a tool for hedging against risk make the area of research very useful in the real world. Work done previously in this area provides models to help energy companies optimize profits, but very few incorporate transmission constraints into their models. The contribution of this research is to help companies not only hedge the risk of unknown power prices but also unknown transmission congestion. Literature on power portfolio optimization has used different measurements of risk: value at risk (VaR), CVaR and variation of spot price. While VaR had historically been used as a risk measure for electricity markets, the advantages of CVaR include more robust mathematical properties for a more accurate measure of extreme risk situations contained in the tail of the distribution. Likewise, different articles have incorporated financial 31 MBAA Proceedings 2010 Papers portfolios of varying scope. Some focus primarily on day ahead, forward, and spot prices. Other papers also include power purchase agreements and/or options. Some authors use generation location or performance (such as ramp rates, heat rates, etc.) as constraints while others focus on different execution and reservation prices for options. Varying timeframes are used in the models, too. Kleindorfer and Wu (2003) gave an excellent review of financial instruments and how they can be used in power markets. It integrated contracting and market structure with operational decisions and explained in detail why a company would choose the forward/options market over the spot market and vice versa, using a graph with corresponding costs. Basically, the more “make to order” businesses would favor the contracts market due to variability in product. Kleindorfer and Li (2005) developed a model that decreases the allowable time period for using the VaR measure from one year to one month, which allows for a more realistic decision timeframe. A Monte Carlo simulation was run to arrive at different portfolio combinations. Xu et al (2006) focused on the issue facing a utility of how to best procure power for its customers. They used semi-variances of spot market transactions to measure risk and offer a model that can analyze the procurement situation with different types of power generation and financial tools. Oum et al (2006) also attempted to aid utilities in finding the optimal financial portfolios given a set amount of available resources for additional capacity taking into account fluctuating demand. It addressed the problem of developing an optimal hedging portfolio consisting of forward and options contracts for a risk-averse load-serving entity when price and volumetric risks are present and correlated. While the focus of Oum et al (2006) was optimizing available resources, the Murphy and Smeers (2005) studied capacity expansion. Utilities seek to optimize their generation portfolios in order to have a sufficient amount of baseload, peaking and cycling capacity while minimizing costs. Here fuel costs can still be passed along to the customers and an oligopolistic market (where each player can influence prices) is assumed. The paper was presented based on a two-stage model: in the first stage investment decisions are made and in the second stage operational decisions are made. Kwon et al (2006) focused more on the agent that sells power either through long-term purchase agreements or through other financial arrangements. The model developed then 32 MBAA Proceedings 2010 Papers aides the selling agent in developing the optimal mix of custom contracts. The authors used a two-stage stochastic programming model where the first stage’s result is the quantity of forward contracts to buy, and the second stage gives the electric capacity to make or buy in future time periods. In a recent study, Conejo et al (2008) showed how a power producer can optimize its profits by utilizing forward contracts when they can be signed up to one year in advance. Only two tools were considered for power purchases in their model development: forward contracts and the “pool market” – day ahead, rather than real time. The decision of when to participate in the forward market was a complex one, involving lots of uncertainty over an extensive period of time. A particularly interesting article was written by Olmos and Neuhoff (2006) and deals with finding a balancing point in a transmission network where companies cannot utilize market power by owning transmission rights. Although an actual logical point was not found in the European Union’s network, the model is a good start to researching equitable fixed transmission rights . MODEL AND DATA The objective of our model is to determine the profit-maximizing combination of different power purchasing portfolios given transmission constraints and risk tolerance. Rather than using a flowgate constraint as a representation of transmission congestion, Fixed Transmission Rights (FTRs) have been utilized. FTRs are financial contracts that entitle the holder to a stream of revenues (or charges) based on the hourly energy price differences across the path. The model incorporates the following prices in power procurement optimization: power purchase agreements, forwards, options, and day-ahead prices. Hull (2006) provides a comprehensive review of derivative instruments. Power purchase agreements are considered the least risky way to procure power. Selling and buying agents agree on a $/MWh price for power purchased over a period of time. Some contracts last as long as twenty years and have provisions that increase the price of the contract to combat inflation. The forward market offers a contract for power whereby the seller and buyer agree on a price for an assessment period in the future. In general, a company will want to maximize its profits by determining the optimal way to purchase power at the beginning of the cycle. In January, decisions are made about 33 MBAA Proceedings 2010 Papers what instruments to use for power procurement for the summer peak season. In June, July and August come and the utility must use the instruments that were chosen to procure power. At the end of the year the company evaluates its decisions to determine if it has met its goals in terms of profit and risk. This problem seeks to maximize the expected profit. Revenue consists of each instrument multiplied by its respective price. The FTR auction price is then compared against the difference between the two nodes to arrive at whether the company won or lost by purchasing the FTR. The data that are analyzed with the model were obtained from Platts’ database. All data, with the exception of the Platts Megawatt Daily forward prices, are publicly available on the PJM or New York Mercantile Exchange (Nymex) websites. PJM is used for the analysis because it is a mature independent system operator with fixed transmission rights auctions and nodal pricing data. The data include PJM hourly day-ahead prices for summer 2007, PJM Financial Transmission Rights (FTR) auction prices for the assessment period of June, July and August of 2007, PJM daily forward prices for the summer months of 2007, New York Mercantile Exchange (Nymex) future prices for summer 2007, and PJM hourly load data from the summer of 2007. In order to determine which PJM nodes could be used in the model, first an analysis is performed on possible nodes with FTR data for the three summer months of 2007 that could be used as a simplistic three-node unidirectional network, which enables transmission constraint analysis. The summer of 2007 is used for the empirical implementation of the model because it was the most recent set of summer peaking season data available when this study was begun. Valid sources and sinks for the PJM FTR auction are limited to: hubs, zones, aggregates, interface buses, load buses and generator buses. PJM hubs are reference nodes at which standard energy goods are traded. Hubs serve as a common point, or reference price, for commercial trading. The hubs are fixed weighted averages of the LMP at a set of typical buses for the chosen area. Hub prices are demonstrative of the PJM market, are fairly steady under many system conditions and are not interfered with by local transmission confines or system topology variations. Zones are a collection of load-weighted LMPs and correspond to transmission zones. Each participating electric distribution company has its own transmission zone through which it supplies its customers. An aggregate node 34 MBAA Proceedings 2010 Papers represents a portion of the nodes that exist in the zones. The node is created at the request of the distribution utility and can be either generationor load-weighted. Interface buses are those which connect two adjacent transmission areas. Generator buses are located adjacent to the major generating units within PJM. The network is comprised of the nodes: Greenbri138 KV T1, Hinton 138 KV T1, and Roncever138 KV T1T3T5. These nodes are located in the American Electric Power (AEP) zone in West Virginia and belong to one of its subsidiary utilities, Appalachian Power. Greenbri138 KV T1, Hinton 138 KV T1 and Roncever138 KV T1T3T5 all represent load nodes. Each FTR is classified as an obligation FTR, which means that the selling entity is allocated the FTR based on its load. Since the focus of the problem is from a utility’s perspective, the sign on the peak prices is changed to represent what the FTR is worth to the selling agent. Table 1 shows the original data. Table 1: FTR Obligation Prices – 2007 Source Node Sink Node Month Peak Prices GREENBRI138 KV T1 HINTON 138 KV T1 June -512.85 July -290 August -211.41 HINTON 138 KV T1 RONCEVER138 KV T1T3T5 June 501.85 July 282 August 200 RONCEVER138 KV T1T3T5 HINTON 138 KV T1 June 11 July 8 August 11.41 35 MBAA Proceedings 2010 Papers The Platts Megawatt Daily forward prices are collected from a random anonymous selection of market participants. They indicate the trade date, the assessment period, whether the agreement is for peak/off peak power and the price in dollars per megawatthour. The nodes are managed by Appalachian Power, which is a subsidiary of AEP (American Electric Power). In order to obtain nodal level data for the forward prices, a spread between the day-ahead price for the AEP Hub and the price for each node is calculated. That spread is then multiplied with the AEP forward price in order to obtain a unique forward price for each node. The New York Mercantile Exchange (Nymex) provides monthly futures contracts to customers based on the daily floating price for each peak day of the month at the AEPDayton Hub. Additional hedging opportunities are offered through options on the contracts. Unfortunately, load data at the nodal level is not available for the PJM market. PJM only releases data at the load zone level, which only encompasses 18 entities. For this study’s purpose, the hourly load data for the AEP zone are chosen, since the given nodes all reside in that zone. Per PJM data, each of these three nodes represents around 0.08 percent of the zonal data. The fact that each node represents the same fraction does not lend itself to having unique demand data for each node. Using nodal pricing data to estimate demand at each node was considered, but that the estimation would not be correct because nodal price is set not only by demand but also by other factors such as system topology, weather, demand at other surrounding nodes, etc. Therefore, the assumption is made that demand at each node could vary by ten percent in either the positive or negative direction and a random number generator was applied so that demand at each node would have the possibility of being unique. The data represent the daily peak load for days when the market was open. In order to estimate different portfolio combinations, the load duration curve for AEP for the summer of 2007 is utilized. A load duration curve demonstrates a company’s load in megawatts from largest to smallest load for a given time. As a general rule, a utility may serve around 85 percent of its peak by more secure contracts like power purchase agreements; the remaining 15 percent would be obtained through forwards, options and the spot market (based on conversation with industry 36 MBAA Proceedings 2010 Papers experts). When the company makes procurement decisions, it does not know what the actual load will be. Therefore, it may not need to use the spot market because the load may not be high enough to warrant additional power purchases. It follows that two of the scenarios do not contain any spot purchases. Forward data are available for power bought up to 12 months before power delivery, but only the forwards that are purchased up to five months were used; it is assumed that a utility begins to think about summer peak season in January of the same year. In total eight scenarios are considered. The probability of each scenario being chosen is generated by Monte Carlo simulation (using an add-in tool called YASAI Simulation Version 2.0 in Microsoft Excel) for each of the three nodes, which gives the average profit for each scenario. The profit differential is then used to estimate the probability that each scenario will be used by the utility to purchase power. In order to calculate VaR and CVaR values, the Monte Carlo simulation method is used. Combinations of the eight scenarios for each node are evaluated for a total of 512 scenarios. The same random number seed is used for each scenario and each run has a sample size of 1,000. The simulation with an output of 512 scenarios is run a total of ten times because there is very little variation in output in each of the ten runs. The slight numerical difference among the ten simulation runs is probably due to the fact that historical data are used for the analysis. The variation in pricing and demand data that is normally observed is eliminated by using actual data from 2007. The estimated nodal demand and actual nodal prices are used to determine the profit for each of the scenarios. The probability for the combination of scenarios is calculated from the combined probabilities of each of the component scenarios. Figure six displays the expected profit distribution. The confidence level for the problem is set at 95 percent based on the level most-used by previous studies in the area of power portfolio optimization. The scenarios are then sorted in ascending order based on expected profit. In order to find the VaR, the probabilities of the scenarios (starting with the most negative profit) are added up until the five percent VaR is reached. The value is found to lie between the 88th and 89th records and is equal to a profit of around $1,216. The CVaR is then calculated basically as the weighted average of the tail from the most negative profit to the VaR profit value. This method is 37 MBAA Proceedings 2010 Papers used in order to take into account possible extreme behavior in the tail as shown in Sarykalin et al (2008). The value for the CVaR is calculated at a loss of $82. The fact that the CVaR is negative signifies that the VaR does not take into account extreme losses in the tail. Given a negative CVaR, the utility may want to decrease the confidence level to 90 percent. Table 2 is a snapshot of the Monte Carlo Simulation analysis. Table 2: Monte Carlo Simulation for VaR/CVaR Calculation Record Scenario Probability Observations Mean Profit 1 512 0.0156% 1000 $ (5,800.47) 2 448 0.0156% 1000 $ (5,111.05) 3 511 0.0313% 1000 $ (5,025.84) 4 504 0.0156% 1000 $ (5,025.11) 5 447 0.0313% 1000 $ (4,336.41) ... ... ... ... ... 84 310 0.0469% 1000 $ 1,187.08 85 380 0.1250% 1000 $ 1,188.34 86 303 0.0313% 1000 $ 1,199.12 87 407 0.0313% 1000 $ 1,199.75 88 352 0.0156% 1000 $ 1,214.14 89 442 0.3125% 1000 $ 1,260.75 38 MBAA Proceedings 2010 Papers RESULTS AND SENSITIVITY ANALYSIS The nonlinear stochastic mixed-integer model is run using the Solver tool in Microsoft Excel. Table 3 shows the vital statistics from the results. Table 3: Results from Optimization GREENBRI138 node (Scenario = Conservative (plan early)) %PPA %FOR1 %FOR2 %FOR3 %FOR4 %FOR5 %OPTION %SPOT 88% 2% 0% 0% 1% 4% 5% 0% Qppa Qfor1 Qfor2 Qfor3 Qfor4 Qfor5 Qcall Qspot 1,019.87 23.18 0.00 0.00 11.59 46.36 57.95 0.00 Profit Added Qspot for transmission constraint $ 8,983.75 0 RONCEVER138 node (Scenario = Conservative (plan early)) %PPA %FOR1 %FOR2 %FOR3 %FOR4 %FOR5 %OPTION %SPOT 88% 2% 0% 0% 1% 4% 5% 0% Qppa Qfor1 Qfor2 Qfor3 Qfor4 Qfor5 Qcall Qspot 1,021.24 23.21 0.00 0.00 11.61 46.42 58.03 0.00 Profit Added Qspot for transmission constraint $ 10,407.71 0 HINTON node (Scenario = Conservative (plan early)) %PPA %FOR1 %FOR2 %FOR3 %FOR4 %FOR5 %OPTION %SPOT 88% 2% 0% 0% 1% 4% 5% 0% Qppa Qfor1 Qfor2 Qfor3 Qfor4 Qfor5 Qcall Qspot 1,016.19 23.10 0.00 0.00 11.55 46.19 57.74 0.00 Profit Added Qspot for transmission constraint $ (5,465.44) 5.75