Maximizing Gross Margin of a Pumped Storage Hydroelectric Facility Under Uncertainty in Price and Water Inflow. Thesis

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1 Maximizing Gross Margin of a Pumped Storage Hydroelectric Facility Under Uncertainty in Price and Water Inflow Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Akina Ikudo, B.S. Graduate Program in Industrial and Systems Engineering The Ohio State University 2009 Thesis Committee: Clark A. Mount-Campbell, Advisor Jerald Brevick ii

2 Copyright by Akina Ikudo 2009 iii

3 Abstract The operation of a pumped storage hydroelectric facility is subject to uncertainty. This is especially true in today s energy markets. Published models to achieve optimal operation incorporate mechanical details of generating and pumping units, but are deterministic. The purpose of this study is to establish a model that considers uncertainty in the market price of electricity and the water inflow rate from river streams to the reservoirs. Due to the nature of the stochastic problems, a dynamic programming approach is taken. The model balances the amount of water used in the current stage and the amount of water saved for the future stages in order to maximize the total gross margin. The meta-models are constructed to evaluate the amount of water used and pumped in the current stage, and the gross margin for different schedules. The uncertainty in the market price of electricity is handled by solving for different price scenarios and estimating the meta-model error. The amount of available water for the future stages is not perfectly predictable either, as the water inflow rate to the reservoirs is subject to uncertainty. For our model, the transition in water inflow rate is simulated by a Markov process. The resulting schedule turned out to be more conservative compared to the deterministic counterpart. ii

4 Acknowledgments My utmost gratitude goes to my thesis advisor, Dr. Clark Mount-Campbell for his guidance, expertise, and most of all, for his kindness. I have been pressured for time throughout this project, but his humor always made me smile. I am really glad that I could know my weaknesses and work on them with Dr. Mount-Campbell. I would like to thank my thesis committee members, including Dr. Jerald Brevick. He always welcomed me with his big smile, and helped me with planning the course work. I thank Dr. Dave Dippold for his patience and understanding. His appreciation of my work helped me go through this project. Lastly, I thank my fiancé, Alex, for the encouragement and the editing. Without his support, I could not finish this project. iii

5 Vita March Associate in Engineering, Tokuyama College of Technology B.S. Industrial and Systems Engineering, The Ohio State University Graduate Research Associate, Industrial and Systems Engineering, The Ohio State University Fields of Study Major Field: Industrial and Systems Engineering iv

6 Table of Contents Abstract... ii Acknowledgement... iii Vita... iv List of Tables... vi List of Figures... vii Chapter 1: Introduction...1 Chapter 2: Literature Review...4 Chapter 3: The Current Model...7 Chapter 4: The New Model...13 Chapter 5: Meta-Model for Water Usage and Gross Margin...21 Chapter 6: Inflow Rate Simulation...28 Chapter 7: Results and Future Research...32 References...39 v

7 List of Tables Table 1. Price Set...33 vi

8 List of Figures Figure 1. System Configuration...8 Figure 2. Reservoir Volume Limitations and Unit Discharge Rates...11 Figure 3. Dynamic Programming Model...16 Figure 4. Effect of Threshold Price and LMP on Unit On/Off Decisions...23 Figure 5. Residuals of Logistics Regression Using All Shapes...24 Figure 6. Threshold Price and Number of Hours of Generation...25 Figure 7. Residuals of Logistics Regression Using Shapes in the Same Group...26 Figure 8. Correlation between Inflows into Smith Mountain and Leesville...29 Figure 9. Simulated and Actual Inflow Rate...30 Figure 10. Gross Margin and Smith Mountain Reservoir Volume...34 Figure 11. Gross Margin and Leesville Reservoir Volume...34 Figure 12. Threshold Price with Small Leesville Volume and Increasing LMP...36 Figure 13. Threshold Price with Large Leesville Volume and Increasing LMP...36 Figure 14. Threshold Price with Small Leesville Volume and Decreasing LMP...37 Figure 15. Threshold Price with Large Leesville Volume and Decreasing LMP...37 vii

9 Chapter 1: Introduction A pumped storage hydroelectric facility is capable of storing energy in the form of water at high altitude. It is often used to flatten out load variation on the power grid by pumping when electrical demand is low and releasing water through a turbine when electrical demand is high. Although the pumping process is not as efficient as the generating process, the difference in the electricity price during peak and off-peak periods of electrical demand results in the positive net gross margin. Each pumped storage hydroelectric facility is unique in terms of configuration and objectives of the operation. The number and size of the reservoirs, the differences in elevations of upper and lower reservoirs, the degrees of correlation between reservoirs, the number and capacities of generating and pumping units, the magnitude of natural inflows from rivers to reservoirs, and other configurational factors all govern the scheduling strategy. The facilities are sometimes considered a part of a larger system that operates thermal electric plants or nuclear power plants as well. Some of the common objectives are to minimize the total operating cost while meeting the electrical demand, maximize the energy supply during the electrical demand peak, and so forth. In this paper, we construct a short-term model for a system with two reservoirs and several generating and pumping units. The operational purpose is to maximize the gross margin, which is the difference between the revenue from selling the generated electricity and the cost of 1

10 buying energy to operate the pumps. This is a simple and small system, and the formulation and acquisition of the optimal schedule is not very hard with proper assumptions. However, the operation of a hydroelectric facility is subject to uncertainty, and the realization of the random variables almost always differs from the predicted values. That is, the performance of the schedule obtained from a deterministic model can be significantly worse than that of the optimal schedule when the reality deviates from the forecast. Nevertheless, due to its complexity in mechanical details, published models for the pumped storage hydroelectric facility are deterministic. The purpose of this paper is to establish a model that incorporates the uncertainty in the electricity price and water inflow rate as well as some mechanical details. The number of variables increases exponentially as the time horizon lengthens when several possible realizations of random events are considered. To keep the problem solvable, reservoir volumes are discretized, and a dynamic programming scheme is applied. The problem in each stage is to balance the amount of water to be used in the current stage and the amount of water to be reserved and pumped for the future stages, in order to maximize the total gross margin. The gross margins from the future stages for different amounts of available water in the future are stored, as the problem is solved backwards. In each stage, once the market price of electricity becomes known, we should be able to evaluate the gross margin, or, equivalently, the revenue and cost, from the current stage. We will also be able to evaluate the amount of water used for generation and the amount of water pumped for different schedules. This can be directly calculated using the electricity price and the schedule. However, by constructing a meta-model for the evaluation of the gross margin and the amount of water used and pumped, these factors can be expressed as functions of electricity price 2

11 and schedule. The meta-model makes the evaluation of the value function easier and also reduces the computational time. Though the amount of water used and pumped for a stage can be calculated by a meta-model, the amount of available water for the future stages is not revealed until the amount of the water inflows from the rivers in the next stage is realized. For this reason, the water inflow rate is treated as a state variable in the model. The transition probability from one state to another can be estimated more precisely when the forecast and the observations from the past stages are taken into consideration, but it is assumed to follow the Markov process in our model. When the resulting schedule is compared against its deterministic counterpart, more conservative policy is observed, as is expected. The model can be enhanced by changing each part while keeping the overall structure. The rest of paper is organized in the following way. In Chapter 2, the problem we are focusing on is reintroduced, and compared against the relevant literature. In Chapter 3, the deterministic model for a two-reservoir system is explained. A new model that incorporates uncertainty in the electricity price and water inflow rate is discussed in Chapter 4. The following two chapters, Chapter 5 and Chapter 6, are dedicated to detailed explanations of the meta-model and inflow rate simulation, respectively. Finally, in Chapter 7, the schedules obtained from the model are compared to its deterministic counterparts, and possible future research is suggested. 3

12 Chapter 2: Literature Review In this chapter, the literature on optimization of hydro facility operation is studied. The models are not limited to the pumped storage hydroelectric facilities as the uncertainty in the water inflow rate is a common issue among the reservoir operations. However, the techniques found in the literature need to be examined carefully before applying, as they are established to solve problems particularly in the water management field, whose operational purposes are often different from that of a hydroelectric facility. There are only a few pieces of literature discussing pumped storage hydroelectric facilities, and none of them consider uncertainty in the market price of electricity. In the water management field, it was common practice to use forecasted values for the water inflow rates. When the importance of incorporating uncertainty in the model started being emphasized, the Markov process came to be widely used. The drawback of the Markov process is that one needs to use discretized states. Also, expressing correlation among reservoirs is difficult. As a hydro facility can be composed of a number of reservoirs, this had been a serious problem. Kelman et al. (1990) suggested the sampling stochastic dynamic programming approach. It employs a scenario based method, overcoming the shortcomings of the Markov decision process. It considers many of the stream flow scenarios simultaneously, instead of solving one at a time. When the model is applied to the Feather River hydroelectric system in California operated by 4

13 PG&E, the benefit function, which measures a dollar value for hydropower production that includes avoided thermal costs, is improved. Another approach is a Bayesian stochastic dynamic programming introduced by Karamouz and Vasiliadis (1992), which uses both the forecast and current inflow rates, reducing the effects of natural and forecast uncertainties in the model. BSDP includes the inflow, storage, and forecast as state variables and uses Bayesian decision theory to incorporate new information. When tested against an alternative stochastic dynamic programming model and a classical stochastic dynamic programming model, using real-time reservoir operation simulation models, the rules generated by the BSDP model outperformed. The loss incurred in a simulated reservoir operation by applying the optimal operating policies in a real-time operation is used for a measure. Various techniques for incorporating the uncertainty in correlated inflows for multiple reservoirs have been summarized in the review article by Labadie (2004). Though those methods have been firmly established in the water resource management field, the focus is often on meeting demand for irrigation. This is one of the reasons that a correlation between inflow rates is emphasized. However, when it comes to the pumped storage hydro facility, whose primary purpose is to maximize the gross margin, hardly any literature is found. The following is the literature somewhat related to our study. Nolde et al. (2007) suggests a predictive control model which minimizes cost for a given inflow rate and demand scenario. The system is comprised of water reservoirs with hydro-power plants and thermal-power plants. The objective is to minimize the total cost. They examine the effect of the shape of scenario trees and find that both the horizon length and the number of branch nodes affect the performance and the probability of infeasibility. 5

14 Borghetti et al. (2008) establish a mixed integer linear program which considers ramp transitions and the operating head. The objective is to maximize the gross margin. Their model accurately represents the pumped storage hydroelectric facility. It is computationally solvable for a time horizon of one week. However, this is a short term model, and a forecasted price is used. Other researchers who focus on the mechanical details of the facility are Zhao and Davison (2008). The model is constructed for a pumped storage hydroelectric facility. They suggest an algorithm in which both the energy supply and gross margin are maximized. Deterministic prices and water inflows are used, and they claim that the two objectives are not very different when prices are not variable. Though the literature above discusses how to structure a specific part of the model for the pumped storage hydroelectric facilities, none of them describe how to organize the whole problem. Uncertainty in the electricity price is not discussed either. Since our objective is to maximize the gross margin, as opposed to meeting water or energy demands, incorporating uncertainty in the electricity price is critical for robust scheduling. The purpose of this study is to establish a model that incorporates uncertainty in both the inflow rate and electricity price, considers operating head, and finds a near optimal schedule in a reasonable amount of time. This study is built upon a previous study conducted by the author. For details, refer to Ikudo (2009). 6

15 Chapter 3: The Current Model In this chapter, an actual configuration of the pumped storage hydroelectric facility is first described. The deterministic model that is currently applied to the facility is then displayed and explained. Illustrations are provided to help readers connect the parameters, variables and constraints in the model to the physical system. As the constraints of the system are the same for both the current model and the new model, which is described in the next chapter, the knowledge about the current model will provide the basis to understand the new model. The weakness of the current model is also discussed to present the problems that the new model is trying to solve. The system we use for a construction of the model is the Smith Mountain pumped storage hydroelectric facility operated by American Electric Power (AEP). The system configuration is shown in Figure 1. The system consists of two reservoirs; the Smith Mountain reservoir (upper reservoir) and the Leesville reservoir (lower reservoir). Two rivers, Roanoke and Blackwater, flow into the Smith Mountain reservoir, and one river, Pigg, flows into the Leesville reservoir. The water that cannot be stored in the reservoir and does not go through the turbines, leaves the reservoirs as spillage. There are five generating units below the Smith Mountain reservoir, totaling 590MW in generating capacity. Three of these units can pump water back into the Smith Mountain reservoir. There are two smaller generating units below the Leesville reservoir, which have a total capacity of 44MW. 7

16 Figure 1. System Configuration The model currently applied by AEP is a deterministic model that uses the forecasted LMP and inflow rate. The time horizon of two weeks is discretized into one hour intervals,. It maximizes the gross margin (GM), the difference between the revenue gained by selling generated electricity and the cost needed to operate pumps. The wattage of electricity generated at the Smith Mountain, generated at the Leesville, and used for pumping at the Smith Mountain, separately, for each hour are to be determined. The model is shown below. 8

17 (The Current Model) (1) Subject to: (2) (3) (4) (5) (6) (7) (8) (9) Where: = Locational Marginal Price at Smith Mountain reservoir, in [$/MW], during hour t. = Locational Marginal Price at Leesville reservoir, in [$/MW], during hour t. = Wattage of electricity generated at Smith Mountain by unit i, in [MW], during hour t. = Wattage of electricity generated at Leesville by unit j, in [MW], during hour t. = Wattage of electricity used for pumping by unit k, in [MW], during hour t. = Capacity of Smith Mountain generator i, in [MW/hr]. = Capacity of Leesville generator j, in [MW/hr]. = Capacity of Smith Mountain pump k, in [MW/hr]. = Volume of Smith Mountain reservoir, in [ft 3 ], at the end of hour t. = Volume of Leesville reservoir, in [ft 3 ], at the end of hour t. = Water inflow to Smith Mountain reservoir, in [ft 3 ], during hour t. = Water inflow to Leesville reservoir, in [ft 3 ], during hour t. = Spillage from Smith Mountain reservoir, in [ft 3 ], in hour t. 9

18 = Spillage from Leesville reservoir, in [ft 3 ], in hour t. = Efficiency of Smith Mountain generator i, in [ft 3 /MW]. = Efficiency of Leesville generator j, in [ft 3 /MW]. = Efficiency of Smith Mountain pump k, in [ft 3 /MW]. = Minimum and maximum limits of Smith Mountain reservoir, in [ft 3 ]. = Minimum and maximum limits of Leesville reservoir, in [ft 3 ]. The total gross margin in (1) is a sum of the gross margin in each hour. The revenue is calculated by multiplying the wattage of electricity generated and the LMP. Similarly, the cost is a product of the wattage of electricity used to pump and the LMP. Constraint (2)-(4) says that units can operate within their capacities. Another restriction on the Smith Mountain units is that they are not allowed to generate and pump at the same time. That is, if one unit is generating, the other unit cannot pump in the same hour. This is taken care of by the constraint (5). The volumes of water at the Smith Mountain and Leesville reservoirs are conserved from one hour to the next hour, and are calculated using the inflow rate forecast as shown in (6) and (7). The volumes of the water in the reservoirs have to be within the upper and lower limits every hour. When the lower or upper limits are violated, the units cannot operate. As is implicitly indicated by equation (6), upper limits will not be violated, as the extra water in the reservoir can leave it as spillage. Violation of the lower limits, however, results in violation of the physical constraints of operating units, and it may also have an impact on the environment. If the five units at Smith Mountain generate at their capacities, the volume of water in the reservoir drops from the upper limit to the lower limit in about two days. Similarly, if the three pumping units at the Smith Mountain operate at their capacities, the volume of the water in the upper reservoir rises from the lower limit to the upper limit in about five days. The storage 10

19 capacity of the Leesville reservoir is about one third of that of the Smith Mountain reservoir, and can be exploited in about three days if the two units generate at their capacities. It is reasonable to use a 336 hour, or two-week, time horizon, because the ending volumes of the reservoirs can be adjusted to any levels regardless of the initial volumes. The inflow rates and the symbols used for reservoir volume limits, unit capacities and discharge rates are shown in Figure 2. Symbols are used for confidential reasons. Figure 2. Reservoir Volume Limitations and Unit Discharge Rates 11

20 As the model uses LMP and the water inflow rate forecast, and ignores the uncertainty, the schedule given by the model may be totally different from the optimal schedule when the actual LMP and water inflow rate deviate from the forecast by a large amount. Though this is a shortterm model, it is not very easy to forecast the LMP or water inflow rate in two weeks, and there will always be forecast error. Since the operation of a hydroelectric facility requires continuous decision making, the strategy that uses probabilities should perform well in the long run. Another problem with the current model is that the efficiencies of the generator and pump are assumed to be constant. This is false, as it is known to be a nonlinear function of the operating head and discharge rate. Since the operating head is a function of the reservoir volumes, when the effect on the efficiency from the operating head is relatively large, it may have an impact on the overall scheduling strategy. That is, the model has to balance the gain from using water while LMP is high and the gain from maintaining a high elevation of the upper reservoir. On the other hand, the discharge rate is an hour by hour decision. Taking the effect of the discharge rate on the efficiency into account results in nonlinear constraints. If a piecewise linear approximation is used, a number of integer variables will be added to the model. As is explained in the sections above, the current model maximizes the gross margin within the restrictions of unit capacity, reservoir volume limits, and water balance equations. These properties must be considered in the new model. Since the LMP and water inflow rate forecasts are not perfect, they need to be replaced by other methods so that the resulting schedule is fairly insensitive to the unexpected events. Incorporating the effect of the operating head on the efficiencies of the generators and pumps is also an important subject as it is associated with the tradeoff between hours. 12

21 Chapter 4: The New Model In this chapter, the changes made to the current model to incorporate uncertainty in LMP and water inflow rate are discussed. The purpose of this chapter is to give an overall picture of the new model, and explain how the structure of the new model is decided. Though the efficiency is approximated by a piecewise function of the operating head, its independency of the discharge rate is still a valid assumption in the new model. Using this assumption the number of decision variables is significantly reduced. The formulation is given at the end of this chapter. If a set of possible realizations of the LMP and the water inflow rate is considered, the number of variables grows exponentially as the time horizon lengthens. To avoid this situation, the model was reformulated as a dynamic programming model, each stage representing one day. Since the model balances the amount of water to be used in the current stage and the amount of water to be reserved and pumped for the future stages, the reservoir volumes become the state variables. The volume of the Smith Mountain reservoir is discretized into nine states, and the elevation of Leesville reservoir is discretized into eight states. The volumes that correspond to the integer elevation are chosen so that the calculation of the operating head becomes easy. In order to split the available water between the current stage and the future stages in a way such that the total gross margin is maximized, the amount of water used for generation and pumping, along with the revenue and cost, needs to be estimable. For this purpose, meta-models are constructed. The amount of water released through turbines and pumped, and the gross 13

22 margin are both expressed as a function of schedule. This facilitates the calculation of value functions. These details are discussed in the following chapter. Uncertainty in the inflow rate is assumed to follow a Markov process. It can be simulated more accurately if the forecast and the information from the past days, instead of just the previous day, are carefully examined and utilized. However, the assumption that the transition of the water inflow rate follows a Markov process is not unreasonable for a system with small inflows, because the impact on the gross margin is also assumed to be small. Furthermore, unlike a huge multipurpose reservoir, the correlation between inflow rates is not very important. These details are described in Chapter 6. Another improvement is that the new model considers the effect of the operating head on the units efficiency. The independency of the efficiency of the discharge rate is restated as follows: the units are assumed to operate at their capacity. The merit of generating below a units capacity is that the wattage generated per unit volume of water is larger at a sub-maximum generating level. That is, one can use water more efficiently. However, when efficiency s independency of discharge rate is assumed, revenue is maximized by generating as much as possible when LMP is high. Now, with the assumptions above, the decision is reduced to whether to generate, pump, or do nothing for each unit for each hour. For the sake of simplicity, we require all the units to be in the same operational state. That is, units at Smith Mountain reservoir are considered as one big unit, and so are the units at Leesville reservoir. Of course, this will result in a sub-optimal schedule since some units are high in efficiency compared to others, and we are not taking this advantage. However, the transformation between the model that makes an on/off decision for each unit and the model that considers a group of units as one big unit is very simple. 14

23 It is intuitive that the optimal schedule is to generate when LMP is high, pump when LMP is low, and do nothing when LMP is moderate, within constraints on the amount of available water. This leads to another way to look at decisions. There exists a threshold price for generating when LMP is above it. It is profitable for units to generate at their capacities during those hours. Similarly, a threshold price for pumping can be found, when LMP is below, for which it is profitable for units to pump at their capacities. It is worth noting that the threshold price for generating is always higher than that for pumping for the same unit. If units generate and pump at the same time, hypothetically, at the same LMP, the gross margin is negative due to the difference in the generating and pumping efficiencies. Using the new scheme described in the sections above, the new model is constructed. The objective is to maximize the expected gross margin over possible realizations of LMP and inflow rates. The reservoir volume at the end of each stage is subject to uncertainty, and the water balance equations similar to (6) and (7) are applied to each of the possible realizations of LMP. Some of the resulting reservoir volume might violate constraint (8) or (9). Due to the nature of the problem, there is always a small possibility of violating constraints, and the probability of violating constraints is controlled by a penalty function. It is worth noting that the reservoir volume constraint is checked only at the end of each stage, or day, instead of at the end of each hour. One stage of the dynamic programming model is illustrated in Figure 3. 15

24 Figure 3. Dynamic Programming Model The state variables are the reservoir volumes and the inflow rate of the current stage. The relation between the beginning and ending volumes is given by the water balance equation, and the relation between the inflow rate in the current stage and the next stage is given by a transition matrix. The decision variables are the threshold prices for generating at Smith Mountain, generating at Leesville, and pumping at Smith Mountain, which are adjusted according to the LMP and the amount of available water. The uncertainty in LMP can be expressed as a combination of average LMP and shape, which is explained in the next chapter in detail. Different average LMPs create different relations between the threshold price, the gross margin, and the amount of water used or pumped. The variability in the shapes appears as errors in the meta-model. The new model is described by the following formulation. 16

25 (The New Model), +, +, 1,,, 1,, 1 (10) Subject to: (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) 17

26 Where: = Threshold price for generating at Smith Mountain units, in [$], in stage t. = Threshold price for generating at Leesville units, in [$], in stage t. = Threshold price for pumping at Smith Mountain units, in [$], in stage t. = Revenue from generating at Smith Mountain, in [$], in stage t for i th price shape group. = Revenue from generating at Leesville, in [$], in stage t for i th price shape group. = Cost of pumping at Smith Mountain, in [$], in stage t for i th price shape group. = Amount of water used for generating at Smith Mountain, in [ft 3 ], in stage t for i th price shape group. = Amount of water used for generating at Leesville, in [ft 3 ], in stage t for i th price shape group. = Amount of water pumped at Smith Mountain, in [ft 3 ], in stage t for i th price shape group. = Operating head at the end of stage t, in [ft]. = Sum of 24 LMPs, in [$/(MW/hr)], in stage t for i th price shape group. = Volume of water in Smith Mountain reservoir at the end of stage t, in [ft 3 ], for i th price shape group. = Volume of water in Leesville reservoir at the end of stage t, in [ft 3 ], for i th price shape group. = Volume of water flowed into Smith Mountain reservoir during stage t, in [ft 3 ]. = Penalty imposed for violating the Smith Mountain reservoir volume constraint at the end of stage t, in [$], for i th price shape group. = Penalty imposed for violating the Leesville reservoir volume constraint at the end of stage t, in [$], for i th price shape group. = Discharge rate of Smith Mountain generator, in [ft 3 /hr]. = Discharge rate of Leesville generator, in [ft 3 /hr]. = Discharge rate of Smith Mountain pump, in [ft 3 /hr]. 18

27 = Capacity of Smith Mountain generator, in [MW/hr]. = Capacity of Leesville generator, in [MW/hr]. = Capacity of Smith Mountain pump, in [MW/hr]. Ω = A set of average LMP scenarios. Subscript omitted. π = A set of price shape groups. R = A set of possible inflow rates for the next stage. k = A scalar to convert the inflow rate at Smith Mountain to the inflow rate at Leesville. P = A scalar to define the penalty function, in [$/ft 3 ]. The problem is solved over a set of average LMP scenarios, and the expected value is taken. At each stage, the value function (10), which is a sum of the gross margin from the current stage and the expected gross margin from the next stage onward, is maximized over decision variables. Depending on which price shape group is being considered, a different meta-model is used to evaluate the gross margin and the amount of water used and pumped. The value from the next stage onward is an expectation over a set of possible water inflow rates in the next stage. The revenue is evaluated using equation (11) and (12), and the cost is calculated according to equation (13). Similarly, the amount of water used and pumped is expressed as a function of the threshold price as shown in (14)-(16). The constraints (17) and (18) are identical to the water balance equations (6) and (7), except that the water inflow to Leesville reservoir is estimated by scaling the water inflow to Smith Mountain. Finally, when reservoir volume constraints are violated, the function is penalized as shown in (19) and (20). In this chapter, the structure of the new model is explained. The dynamic programming approach is taken, and the water inflow transition is assumed to follow a Markov process. Assuming that the units always operate at their capacity, the operating schedule can be fully described by three threshold prices. The problem is solved over a set of average LMP scenarios, 19

28 and price shape specific meta-models are constructed to incorporate uncertainty in the LMP. The recursive equation is expressed in a closed form. 20

29 Chapter 5: Meta-Model for Evaluation of Water Usage and Gross Margin In this chapter, how the meta-model is constructed is explained. The purpose of the meta-model is to simplify the evaluation of a value function by replacing the simulation with a regression. The relation between the LMP, threshold price, and the gross margin (the amount of water used or pumped) is obtained. The method of simulating LMPs is first discussed. This is a part of data preparation for the meta-model construction. The meta-model for the amount of water used for generating is explained in detail, and then how the method is modified to obtain the other meta-models is discussed. There are two parameters that define a set of LMPs for 24 hours, in terms of the relation between threshold price and LMP. One is the average LMP. For the same threshold price, it is assumed that the more hourly LMPs exceed the threshold price when the average LMP is high compared to when the average LMP is low. However, if the hourly LMPs are flat, it is possible that none of them exceeds the threshold price even when the average LMP is high. On the contrary, if hourly LMPs have peaks, some of them may exceed the threshold price. So the other component that defines daily LMPs is the shape. To construct a meta-model for the amount of water used, we need various combinations of average LMPs and shapes because they affect the number of hourly LMPs that exceed the threshold price. However, the amount of historical data is relatively small, and observations for only specific combinations of average LMPs and shapes are available. In order to create LMPs 21

30 that look like LMPs observed in the past, sets of 24 LMPs in the historical data are first standardized by subtracting the average LMP of the 24 LMPs, μ, and then dividing by standard deviation, σ. The subscript h denotes the hour. (21) To obtain a set of LMPs that have a specific average and shape, standardized LMPs are scaled up using the following formula: (22) Where μ and σ are the desired average and standard deviation of the new set of LMPs. The standard deviation is calculated as a linear function of the average in our case. In this manner, LMPs that mimic the historical data and have the desired average and shape are created to provide enough LMPs for regressions. When the average LMP and shape are known, the values of 24 LMPs can be calculated. For each threshold price, the number of LMPs that exceed the threshold price, and therefore the amount of water used during a day can be calculated. Figure 4 shows relation among threshold price for generating, the average LMP, and the fraction of time that the unit is generating. All the shapes in the historical data that appeared in May, Tuesday are aggregated to calculate the fraction of time since the shapes that fall in the same day in the week in the same month tend to be similar. It can be seen that, for the same average LMP (same line), as threshold price for generating increases, the fraction of time that units are generating decreases. When compared within a fixed threshold price, as the average LMP increases, the fraction of time that the unit is generating also increases. 22

31 Fraction of Time Unit is Generating Threshold Price, LMP, & % Generation Threshold Price [$] LMP=30 LMP=40 LMP=50 LMP=60 LMP=70 LMP=80 Figure 4. Effect of Threshold Price and LMP on Unit On/Off Decision The criteria for the regression of the amount of water used for generation against the threshold price is that the fraction of time that unit is generating is 0 when the threshold price is set to its maximum, and the fraction of time that unit is generating is 1 when the threshold price is set to its minimum. The S-shape of the graph shown in Figure 6 also needs to be maintained. Considering these requirements, logistics regression was chosen to be a regression model form: (23) The operating head, h t-1, is a difference in elevation of the Smith Mountain reservoir, which is a function of volume, and the elevation of the Leesville reservoir. Though the operating head changes throughout a day due to generating and pumping, the discharge rate is assumed to be constant throughout a day. A piecewise linear approximation is used. 23

32 Residual Originally, the average LMP is included in the meta-model as an independent variable together with the threshold price. When the resulting regression is examined, however, it turned out that for extreme average LMPs, the fraction of time that the unit is generating never reaches 0 or 1 even when the threshold price is set to its maximum and minimum. For this reason, the average LMP is eliminated from the meta-model. To accommodate uncertainty in the average LMP, a two-week long problem is solved for many average LMP scenarios. By maximizing the likelihood of input values, the coefficients in equation (23) are obtained. The Pearson goodness-of-fit test yields a p-value = 1.00, which confirms the correctness of the model form. However, as hourly LMPs from different days, or shapes, are aggregated to fit the regression model, extreme shapes tend to yield large residuals. Residuals for three shapes are shown in Figure 5. It seems that residuals follow a specific pattern. Threshold price vs. number of hours of generation in Figure 6 corresponds to this observation. Residuals vs Predicted Value Shape1 Shape2 Shape Predicted Fraction of Time Generating Figure 5. Residuals of Logistics Regression Using All Shapes 24

33 Hours of Generation [hr] Threshold Price vs Hours of Generation Threshold Price [$] Shape1 Shape2 Shape3 Figure 6. Threshold Price and Number of Hours of Generation To make the residuals small, shapes are clustered into several groups based on similarity. As we are interested in the relation between the threshold price and the number of hours of generation, the maximum difference in the number of hours of generation for a given threshold price is used to measure the similarity of shapes. When shapes whose maximum difference is less than four hours are grouped and regressed, the residual vs. predicted value plot in Figure 7 is obtained. Residuals seem to be randomly scattered, and almost all the residuals are within ±0.1. By reducing the threshold number of hours of difference, residuals are expected to become even smaller. 25

34 Residual Resisual vs Predicted Value Predicted Fraction of Time Generating Shape1 Shape2 Shape3 Shape4 Figure 7. Residuals of Logistics Regression Using Shapes in the Same Group The main idea is to replace the error term of the regression with other regressions that have very small errors. This is analogous to integrating over the error term. The motivation is to eliminate the need to consider a correlation between the error of water usage regression and the error of revenue regression. They are positively correlated, but not perfectly. When correlation is taken into consideration, the calculation of the value function becomes even harder. However, if shapes are clustered into groups of similar shapes, regressions and consequent water balance equations can be applied to each group. Since each shape group is associated with a probability of a shape falling into that group, expected water usage and revenue over shape groups can be calculated. The regression of revenue is acquired in a similar manner. Since the logistics regression returns a value between 0 and 1, revenue is expressed as a fraction of maximum revenue that is achievable for a given set of LMPs for each stage. This is simply a sum of the 24 LMPs, and it 26

35 corresponds to a schedule where unit generates 24 hours. The equation is given by (11) in the previous chapter. For the meta-models for the amount of water pumped, the regression for the amount of water used is applied. The fraction of time that units are to generate is equal to the fraction of time that the LMP is higher than the threshold price. Since each of the 24 LMPs is either higher than the threshold price or lower than the threshold price, the fraction of time that the LMP is lower than the threshold price is calculated by subtracting the fraction of time that the LMP is higher than the threshold price from 1. This gives the fraction of time that the units are to pump. A similar argument holds for the cost. In this chapter, it is explained that the defining parameters of a set of 24 LMPs is the average and the shape. The LMPs that are used to construct meta-models are generated by using these parameters. The variability in the average LMP appears as a set of scenarios in the model, and the variability in the shape corresponds to the error in the meta-model. Instead of being numerically integrated, the error is estimated by regressions, whose error is reduced by allowing groups of similar shapes to have its own regression. Logistics regression is fitted to estimate the fraction of time the LMP is above the threshold price, or the fraction of time the units are to generate for a given threshold price. Using the relation between threshold price and LMPs, the same regression is used for the amount of water used and pumped. 27

36 Chapter 6: Inflow Rate Simulation In this chapter, how the uncertainty in the water inflow rate is incorporated in the model is discussed. The inflow rate is assumed to be constant within a stage, but the inflow rate for the next stage is not known. The purpose of this chapter is to give a brief explanation on the setup in our model so that it can be used as a basis for a more complicated simulation. There are two rivers that go into Smith Mountain reservoir, Roanoke and Blackwater, and one river that goes into Leesville, Pigg. As it is mentioned in Chapter 3, the water inflow rate from the rivers is much smaller compared to the discharge rate of generators and pumps. However, in the event of storms, the inflow rate from the rivers becomes comparable to the pump rate. This motivated us to incorporate the uncertainty in the water inflow rate. It is ideal to combine the information from the forecast and the observation from the historical data. However, the water inflow rate is affected by numerous factors, and examining the relation among them is beyond our scope. We will leave this matter to the experts, and focus on the utilization of the information obtained from the observations in the past. When special events such as storms are not expected, this strategy is expected to give a reasonably good simulation. As there are only three rivers, and their contribution to the increase in the reservoir volumes is somewhat trivial, the correlation between inflow rates among rivers is not very important. Considering that the discretization of states is less likely to cause problems, a Markov process is assumed to yield reasonable variability in inflow rate. 28

37 Inflow Rate [cft/s] The water inflows from the two rivers that go into Smith Mountain reservoir, Roanoke and Blackwater, are combined because individual inflow rate is not one of our interests. When this combined inflow rate is plotted together with the inflow rate from Pigg river, which goes into Leesville reservoir, they seem to synchronize fairly well. The illustration is shown in Figure 8. Actually, the correlation coefficient between the combined inflow rate and inflow rate of Pigg river is For our purpose, it is reasonable to assume the synchronization between the two inflow rates. That is, the inflow rate is simulated for the combined inflow rate only, and will be multiplied by a constant to obtain an inflow rate for Pigg river. The constant is chosen to be the median of day by day ratios of the inflow rate for Pigg river to that of combined stream flow Inflow Rate: June Day Roanoke Blackwater Pigg R+B Figure 8. Correlation between Inflows into Smith Mountain and Leesville 29

38 Inflow Rate [cft/s] The historical inflow rates are divided into ten groups of equal sizes, and the median of each group is used as a discretized value. Then, the probability of transitioning from one state to another state is calculated based on the historical data. Using the resulting transition matrix, the simulated inflow rates shown in Figure 9 are obtained. It is worth noting that the value function is maximized over possible inflow rates using the probability of transitioning from one state to another. Hence, the simulated inflow rate that may look extreme is not a problem Markov Process Simulation Day Actual Simulated Simulated2 Figure 9. Simulated and Actual Inflow Rate Considering the fact that the water inflows from the rivers are relatively small compared to the discharge rate of generators and pumps, one of the simplest techniques to incorporate uncertainty, a Markov process, is applied. Though it gives reasonable variability to the water 30

39 inflow rate, it heavily relies on historical observations. It is strongly recommended to combine the Markov process with the forecast and recent observations. 31

40 Chapter 7: Results and Future Research In this chapter, computational results are reported. The computational time is given and possible changes to reduce the time are suggested. The model behavior is discussed and the threshold prices are compared against its deterministic counterpart. The schedule given by our model turned out to be more conservative. Finally some changes on the meta-model and water inflow simulation are suggested. The model is applied to the Smith Mountain reservoir. In each stage, the cyclic coordinate is used for maximization of the value function, and a Golden section is employed for the line search. There are nine states for the Smith Mountain reservoir volume, eight stages for the Leesville reservoir volume, and ten states for the inflow rate, totaling 720 distinct states. The time horizon is 14 days, and the value function is maximized over one average LMP scenario. The data is given on a spreadsheet, and the problem is solved by VBA. The computational time was 1 hours and 55 minutes, using AMD Turion TM 64 x2 Mobile Technology TL GHz, 894 MB RAM. To reduce the computational time, the cyclic coordinate can be replaced by another algorithm. Exploring and calibrating the stopping tolerance in both a multidimensional and linear search will get rid of the redundant iterations. Reducing the number of states and choosing states that give a good approximation to the values between them when interpolated is also a way to reduce the computational time. 32

41 To study the behavior of the model, the model is run from stage 1 through stage 14. The average price scenario used for this run is given in Table 1. The gross margin is calculated for each combination of three state variables. Some of the values are summarized in Figure 10 and Figure 11. As shown in Figure 10, for a fixed Leesville reservoir volume, the gross margin increases as the Smith Mountain reservoir volume increases. The high inflow rate also contributes to the large gross margin. Figure 11 indicates that the gross margin tends to increase as the Leesville reservoir volume increases. It is interesting that the gross margin is not monotonically increasing for the lower Leesville reservoir volumes. The cause may be the operating head effect or a violation of the reservoir volume constraint. It is also worth noting that the effect of the Smith Mountain reservoir volume on the gross margin is much larger compared to that of the Leesville reservoir. This is due to the higher unit capacity at the Smith Mountain reservoir. Table 1. Price Set Stage Average LMP [$] Stage Average LMP [$]

42 Gross Margin [M$] Gross Margin [M$] Leesville Volume Medium Level Smith Mountain Reservoir Volume Inflow Rate Low Inflow Rate High Figure 10. Gross Margin and Smith Mountain Reservoir Volume 1.8 Smith Mountain Volume Medium Level Leesville Reservoir Volume Inflow Rate Low Inflow Rate High Figure 11. Gross Margin and Leesville Reservoir Volume 34

43 Using the expected gross margin from stage 2 through 14, the threshold prices for the first stage are then compared to examine how uncertainty in the price shape affects the operational decisions. A moderate inflow rate is used, and threshold prices for generating at the Smith Mountain reservoir are compared at different levels of the Smith Mountain reservoir volume. The case with a small Leesville reservoir volume is summarized in Figure 12, and the case with a large Leesville reservoir volume is summarized in Figure 13. In both cases, at each level of the Smith Mountain reservoir volume, the threshold price that considers the price shape uncertainty is higher. That is, for the same Smith Mountain reservoir volume, when the price shape uncertainty is taken into account, less water is allocated for the current stage and more water is saved for the future stages. Another interesting observation is that the threshold prices are higher when there is more water in the Leesville reservoir. In other words, when the Leesville reservoir volume is small, more water is used for generating at the Smith Mountain reservoir and flows into the Leesville reservoir. The price set given in Table 1 is then reversed. That is, the old LMP for stage 14 is now the LMP for the first stage. The same analysis is employed for the new set of prices. The result is summarized in Figure 14 and Figure 15. As the LMP is decreasing, using more of the available water during the first stage is profitable, and the threshold prices are shifted upwards. Again, the threshold prices that are determined with a consideration to the uncertainty in the price shape are higher than the threshold prices that ignore the variability in the price shape. 35

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