INTEGRATING REAL-TIME PRICING INTO UNIT COMMITMENT PROGRAMMING

Transcription

1 INTEGRATING REAL-TIME PRICING INTO UNIT COMMITMENT PROGRAMMING Cedric De Jonghe ELECTA branch Erik Delarue TME branch William D haeseleer TME branch Ronnie Belmans ELECTA branch Abstract The European Commission has set a renewables target enforcing wind power investments. Simultaneously, a smart meter roll-out, in the context of smart grid developments, is said to facilitate the integration of this variable source of power generation. This paper analyzes to which extent a price responsive demand-side contributes to an efficient short-term system operation with a significant renewables share. First, a basic three nodal mixed-integer unit commitment model is described. Then, a methodology is suggested replacing fixed demand levels by elastic demand curves considering the ability to adjust demand levels in response to real-time prices. The improved model accounts for both operating costs and welfare received from consuming electricity. Model results emphasize the impact of demand response on the optimal commitment with an increase of base load power generation output and a decrease of peak load power generation. Finally, environmental benefits are described with more wind power injected into the system and reduced CO 2 emission levels. Keywords: Demand response real-time pricing unit commitment mixed integer linear programming 1 INTRODUCTION Fluctuations in the amount of wind power fed into the electrical power system require operational flexibility. Flexibility can be offered by generation units, energy storage facilities and transmission interconnections [1]. These options are offered at the supply-side of the system to instantaneously balance generation and demand. The expected roll-out of smart metering appliances is an opportunity to increase the flexibility at the demandside of the system. Consumers will be able to respond to real-time electricity prices reflecting the marginal cost of generating electricity. This responsiveness is expected to improve power generation efficiency and contribute to the sustainability of the power generation industry. This paper quantifies the impact of responsive consumers, emphasizing the benefits with respect to the integration of wind power generation, characterized by its variable power output. A Mixed Integer Linear Programming (MILP) Unit Commitment (UC) model is developed in section 2, minimizing short-term generation costs. Fixed demand levels in this model are replaced by price elastic demand levels representing the aggregation of residential, commercial and industrial customer classes. Including own-price elasticities into the aggregated demand curve allows quantifying the impact of consumers instantaneously adjusting demand to price levels. In high price moments electricity consumption is reduced and in low price moments, the initial level of electricity consumption is increased. Model results in section 3 quantify benefits of real-time demand. Conclusions are drawn in the final section. 2 MODEL DESCRIPTION This section first describes the basic unit commitment model and presents data and assumptions in subsection 2.2. Elastic demand curves are constructed in subsection 2.3. Finally, subsection 2.4 presents the methodology which is applied to include the elastic demand curve into the model. 2.1 Unit commitment model Unit commitment models optimize the operation of power generating units in an electric power system, typically in a time frame of a few hours up to one week [2]. Different unit commitment models are discussed in [3] and [4]. An overview of methods to solve UC models is given in [5]. The available generation units satisfy system load, referred to as the system energy balance, while considering operational constraints. Ramp rates, minimum on and off times, emissions, minimum run levels and capacity limits are considered in this model. The model represents one control area, composed of three nodes, indicated by index h. Those nodes are interconnected by three lines, indicated by index k (Figure 1). Different conventional generation units (G) and wind power generation are situated in each bus, as well as consumers represented by aggregated demand (D). Figure 1: Three nodal network Corresponding author: Cedric De Jonghe - Kasteelpark Arenberg 1 (PB 2445) 31 Heverlee Phone , Cedric.DeJonghe@esat.kuleuven.be

2 The unit commitment optimization is performed dayahead. The model outcome yields the optimal commitment of conventional units (i) involving a technology specific start-up cost (SC i ) whenever a unit is turned on. The on or off status of generation units is indicated for every hour (u) by binary variable z u,i, which equals 1 when the unit is on and when the unit is off. Additionally, variable generation costs have to be paid depending on the output level. Both generation technology specific marginal costs (MC i ) and an emissions cost (EC) constitute the variable generation costs. Finally, excessive wind power injections in a node could result in nodal imbalances or overloaded transmission interconnections. Reducing hourly wind power injections in order to prevent these situations can be realized by curtailing wind power in a certain node. For each MWh of wind power curtailment (curt u,h ), a curtailment cost (CC) is incurred, representing the opportunity cost of missing out on a subsidy such as green certificate. Considering the above mentioned cost aspects, the objective function of the UC model can be written as:, _,,, (1) This objective function is minimized subject to constraints, limiting the generation output levels. Generation output levels are restricted by minimum (PMIN i ) and maximum (PMAX i ) output levels, where the latter equals the nominal capacity of the unit. Additionally, the generation output fluctuations are restricted by ramp rates (RAMP i ) by including Eq. (3) and (4). These ramp rates indicate the maximum hourly output change as a percentage of the capacity of the corresponding generation unit.,,, (2),,, (3),,, (4) Once committed, generation units are required to be on for a minimum number of hours. Correspondingly, once turned down, generation units are required to remain off for a minimum number of hours. These requirements are technology specific and typically more stringent for base load and mid load generation technologies than for peaking units. The requirement of minimum on (MO i ) and minimum down (MD i ) times is enforced by Eq. (5) and (6), based on [6].,,,, 1,, 1, 2,, 1 (5),,,, 1,, 1, 2,, 1 (6) In order to ensure that start-up costs are only incurred when a unit is turned on, a variable (s_costs u,i ) is introduced in combination with Eq. (7). By enforcing that this is a positive variable, no negative costs will be incurred when turning down a unit. _,,,, (7) The network as shown above (Figure 1) allows balancing surpluses and deficits of injected power or demand between different nodes. Matrix GEN_NODE h,i ascribes generators i to nodes h whereas matrix LINE_NODE k,h defines the starting and ending point of transmission interconnections (incidence matrix). The nodal energy balance requirement is enforced by Eq. (8) such that demand in each node is satisfied by net imports, conventional generation and wind power injections after curtailment. This equation corresponds to Kirchhoff s current law [7]. Kirchhoff s voltage laws are satisfied by enforcing Eq. (9), assuming an equal reactance for all three lines (k). Additionally, the flow (flow u,k ) over each line is restricted by the interconnection capacity (CAP k ) as formulated in Eq. (1)., _, _,,,,,, (8),,, (9),, (1) 2.2 Data and assumptions Energy demand and wind power data is based on historical demand and wind power profiles 1 in order to represent a realistic variability. The total amount of wind power generation and the initial real-time demand levels are assumed to be fixed and perfectly known as shown in Figure 2 for a 48-hour time period. 2 The total amount of electricity demand and wind power injection in the area is equally divided over the three nodes. [MW] Initial demand Wind power injection Hours Figure 2: Input wind and demand data Technology-specific parameters are summarized in Table 1. Five different technologies are selected, inspired by the 24-bus IEEE Reliability Test System [8] and a modified IEEE 118-bus Test System [9]. Technology-specific emissions are based on [6]. Each node is assumed to be characterized by a different dominant generation technology. The first node is considered to be the most carbon intensive with two coal-fired power plants and three Oil-fired Combustion Turbines (OCT). 1 Historical wind power generation and energy demand data is easily accessible on the website of the Danish grid operator Energinet and has been modified. ( 2 Perfect knowledge of wind power injections is assumed, given a focus on demand-side flexibility and the resulting demand adjustments in this paper. Uncertainty about wind power injections, using different scenarios will be dealt with in further research.

3 The second node is dominated by nuclear power generation. Two nuclear power plants, two Gas-fired Combustion Turbines (GCT) and one OCT are situated in the second node. Node three is characterized by gas-fired power generation composed of three Combined Cycle Gas Turbines (CCGT) and three GCT plants. Each of the nodes is connected to the other nodes with a transmission interconnection capacity of each 15 MW. The cost of emissions is considered to be 1 /ton CO 2 and the cost of wind power curtailment is assumed equal to 3 /MWh. Nuclear Coal CCGT GCT OCT PMAX i [MW] PMIN i [MW] EMIS i [ton/mwh] MC i [ /MWh] SC i [ ] RAMP i [%/h] MO i [h] MD i [h] Number of plants Table 1: Technology-specific parameters 2.3 Elastic demand function This subsection illustrates how an elastic demand function is constructed in order to allow consumers to adjust consumption in response to deviations in electricity price levels. This function represents the aggregated demand of residential, commercial and industrial electricity consumption for each hour (u) in each node (h). 3 For simplicity and clarity, the methodology is only explained for one hour. First, the UC model is optimized for the 48-hour time period with fixed demand levels and the parameters as listed above. The model output defines the optimal commitment of the available generation units subject to the operational constraints. Based on the dual variable or shadow price of the nodal energy balance requirement in Eq. (8), the marginal price of electricity is found. 4 As a result of including operational constraints and a transmission network, the marginal price of electricity can differ from the marginal fuel cost [1]. This difference is referred to as the energy balance quality of supply premium. The hourly electricity prices relating to fixed demand levels are then used to calculate a weighted average energy price. This weighted average price is assumed to be the fixed or single tariff (P ) faced by consumers who are not under a real-time pricing structure. A similar methodology could be applied in order to calculate weighted average prices for a double tariff structure, distinguishing between specific blocks such as peak and off-peak as in [11]. 3 Different consumer classes may be characterized by a different ability to adjust demand levels in response to changing electricity prices. Distinguishing between different consumer classes will be dealt with in further research. 4 As the shadow price is the change in the objective value of the optimization problem obtained by relaxing the nodal energy balance by one unit, it corresponds to the electricity price. Next, the single tariff (P ) combined with the fixed hourly demand level (DEM u ) is assumed to constitute the market equilibrium (P, DEM u ) in hour u, defining the perfectly inelastic linear demand function (vertical curve in Figure 3). Afterwards, this equilibrium is used as an anchor point through which the linear elastic demand function can be drawn by changing the slope of the initial demand function. Changing the slope of the curve corresponds to increasing the demand own-price elasticity or the responsiveness of consumers with respect to price changes. The addition of exogenous ownprice elasticities (ε u ) results in a new short-term demand response function (Eq. (11)). This function is reformulated as D u (Eq. (12)), shown in Figure 3, with simplifying parameters a u and b u. The corresponding inverse demand function P u (Eq. (13)) is characterized by parameters e u and f u hour. Price P,u P D u DEM u Demand Figure 3: Construction of a short-term elastic aggregated demand function (11) Du: (12) Pu: (13) The construction of the elastic linear demand function is performed for each hour and each node. When responding to electricity price changes by adjusting demand conform this function, consumers have the ability to increase their benefits 5. A real-time price higher than the single tariff (p u >P ) results in decreased levels of electricity consumption. Correspondingly, realtime prices lower than the single tariff result in increased levels of electricity consumption. 2.4 Inclusion of elastic demand function UC models with fixed demand profiles typically pursue the reduction of system costs. When short-term demand response is integrated, fixed demand levels are replaced by hourly elastic demand functions as constructed in subsection 2.3. The model must define a new optimal solution characterized by an equilibrium price and demand for each hour and node. This solution maximizes consumer s welfare, being the integral of the demand function, at a minimum cost for producers. It 5 In this methodology, consumers face real-time wholesale prices, referred to as spot prices. This model abstracts from additional transmission and distribution grid charges or taxes.

4 can be found by reformulating the MILP model as a mixed integer quadratic problem or a mixed integer complementarity problem [12]. As these models are typically hard to solve, an alternative computational procedure is suggested given supply characteristics and a own-price elastic demand function. 6 The methodology, referred to as the stepwise integration, is based on [13] in the context of Project Independence. 7 In order to find a solution to this problem, perturbation y u is introduced, defined as y u = dem u -DEM u. This continuous, positive variable y u allows building a partition of the interval around the anchor point with the initial demand level DEM u. Given set N (n= 1,,m- 1,m), y + u,n constructs m partitions at the right-hand side of initial demand level DEM u and y - u,n constructs m partitions at the left-hand side of DEM u (Figure 4). Variables y + u,n and y - u, n are constrained by U + u,n and U + u,n respectively, being the partition size.,,,, (14),, (15) For each partition step around the initial demand levels, the inverse demand function P u gives the resulting price level P + u,n and P - u,n respectively.,, (16),, (17),, With:,,,, (18),, (19) Eq. (18) indicates that increasing the demand level, when y + u,n is greater than zero, increases consumer welfare. Correspondingly, decreasing demand levels, when y - u,n is greater than zero, results in a decreasing consumer welfare. This is illustrated by the gray rectangles in Figure 4. Equation (18) is added to the original objective function (Eq. (1)) in order to maximize total welfare equal to consumer s welfare minus generation costs (Eq. (2)). The original nodal balance requirement (Eq. (8)) is changed as well, accounting for demand adjustments indicated by y + u,n and y - u, n (Eq. (21)). The performed perturbation does not influence the formulation of operational constraints. Given the negative first derivatives of the linear demand function, convexity of Eq. (2) is guaranteed.,,,,,, _,,, (2),,,,,,,,,, j (21) As a solution to this model with optimal values for decision variables y + u,n and y - u, n, an integrated demand level is found, being an approximate solution to the welfare maximization. Initial demand levels are ex post recalculated by Eq. (22),, (22) Figure 4: Partitioning for stepwise integration Then, the integral calculating consumer welfare is approximated by a stepwise summarization of onedimensional integrals. 6 Non-zero cross-price elasticities can be added, assuming dominance of own-price elasticities. Dominance means that the aggregation of cross-price elasticities is lower than the absolute value of own-price elasticities. 7 This project was the initiative of U.S. president Nixon in 1973 in response to the OPEC oil embargo. Convergence of this algorithm is proven in [15]. Convexity ensures that it can not be welfare maximizing to simultaneously take a positive partition step to the left and to the right. Consequently, positive perturbation variables y + u,n and y - u,n are not simultaneously different from zero, reflected by Eq. (23).,,, (23) Furthermore, if y + u,n is positive, it can be equal to or less then partition size U + u,n. If the perturbation variable equals the partition size, the welfare maximizing equilibrium solution has not yet been found. If the perturbation variable is less than the partition size, the optimal number of partition steps n* has been found. The dual variable of the system balance constraint π u is an estimate of the (marginal) electricity price. In order to optimize solution time of this model, an iterative procedure can be suggested. After a first stepwise integration approximation, the partition size can be

5 reduced to increase accuracy. This is especially relevant when dealing with -1 binary variables, typically created to represent UC related operational constraints, such as start-up costs, minimum run levels and minimum up- and downtimes. The flowchart of this procedure is schematically shown in Figure 5 and described below. 3 RESULTS This section illustrates the impact of integrating short-term demand response into a UC. Input data are described in section 2.2. The model is written in Matlab, calling data from Excel and uses the Matlab-Gams interfacing optimization [14]. In order to solve the MILP model, Gams utilizes CPLEX Initial demand levels are equal in each of the nodes, being one third of total demand shown in Figure 2. The aggregated optimal UC is presented in Figure 6, assuming fixed demand levels. 8 This optimal commitment must be compared with Figure 7, assuming price responsive consumers with a -.2 own-price elasticity. Whereas the dashed line corresponds to the initial load profile, the full line indicates the aggregated electricity demand after consumers have adjusted their demand in response to real-time electricity prices. In the first part of day one, electricity consumption is increased by about 4 MW in order to deal with excess wind power generation. Initial demand levels are adjusted in such a way that total wind power generation, can be injected into the system. As a result, no more wind power needs to be curtailed. From hour 7 up to hour 15, a price elastic demandside positively impacts the operation of conventional generation units. On the one hand, generation output levels of nuclear power plants are increased due to lower electricity prices in the nuclear node. As a result, nuclear plants with lower marginal costs are operated at nominal capacity. Figure 5: Flowchart iterative procedure 1. Assume initial demand levels DEM u : These demand levels are commonly projected for the model without inclusion of demand elasticity. 2. Calculate specific price levels P + u,n and P - u,n for each step: Given the most recent estimated demand levels, price levels are calculated for different steps around that demand using the inverse demand function as in Eq. (16) and (17), for use in the supply LP model. Own-price elasticities are included to the inverse demand function. 3. Solve the LP to define the optimal values y + u,n, y - u,-n and marginal price estimate λ u : Based on the integration approximation described above, optimal values for the perturbation decision values are defined. If desired, the step size could be reduced in order to increase accuracy and prevent from finding a solution in between two binary optimal solutions. Therefore, a new iteration can be performed until the objective value is no longer impacted. 4. Define the equilibrium solution: By using Eq. (22) the optimal demand levels can be calculated given y + u,n, y - u,n. Figure 6: Optimal commitment without demand response Figure 7: Optimal commitment with -.2 own elasticity 8 Unused generation capacity is shown above the demand curve in Figure 6 and committed generators are represented as rectangles

6 On the other hand, the power output of CCGT and coal-fired power plants is reduced due to higher electricity prices in the respective nodes. As a result, demand response prevents from starting up a second CCGT and coal-fired power plant. In the first part of day two, valley filling occurs with increasing electricity demand between hour 24 and 32. Again, generation output levels of the nuclear power plant can be increased. Alternatively, peak reduction effects occur during the peak demand period from hour 32 until 44. This peak reduction prevents from starting up the GCT in hour 33 and the CT in hour 34. Also, in the following hours, the third CCGT plant does not need to be turned on, which again avoids start-up costs. Corresponding hourly electricity prices in each of the nodes are given in Figure 8. The upper graph shows prices, given fixed demand levels in each of the nodes. The optimal UC model outcome with fixed demand levels yields those prices as the dual variable of the system energy balance. The weighted average of those prices is assumed to be the node specific single tariff (P in section 2.3) faced by consumers who are not under a real-time pricing structure. Single flat tariffs of about 3, 42 and 47 /MWh are found in the nuclear, coal and gas node respectively. Alternatively, hourly nodal electricity prices with price responsive consumers are shown in the lower graph of Figure 8. When the nodal hourly electricity price is below the flat tariff, consumers increase their demand level and the opposite occurs when prices are above the flat tariff, both corresponding to the assumed -.2 own-price elasticity. Consequently, electricity demand adjustment in Figure 7 must be linked with deviations of hourly electricity prices from the initial flat tariff based on Figure 8. This new pricedemand pair composes the new hourly market equilibrium. Without demand response, negative electricity prices can be noticed in the first 6 hours, relating to the cost of curtailing wind power during excess power supply. Furthermore, due to increasing demand levels, electricity prices increase up to the marginal cost of GCT power generation (hour 33 and 35) and even OCT power generation in hour 34. Price [ /MWh] Price [ /MWh] coal node nuclear node ccgt node Hour coal node nuclear node ccgt node Hour Figure 8: Hourly nodal electricity prices without (upper graph) and with (lower graph) -.2 own-price elasticity Assuming price responsive consumers, no more wind power is curtailed and prices remain above 1 /MWh. Not only downward price spikes are reduced, also upward price spikes remain below 6 /MWh. Assuming only 15 MW interconnection capacity, only a limited number of hours with price convergence over the three nodes is found. Comparing Figure 6 with Figure 7 suggests that the nuclear power plant has on average a higher output level after consumers have adjusted their demand levels. Similarly, peak reduction effects of short-term demand response allows to reduce the average power output level of peak load generation plants. This conclusion is also suggested in Figure 9, showing capacity factors 9 of nuclear, coal-fired and CCGT plants with different levels of assumed own-price elasticity. Increasing the responsiveness of the aggregated demand yields an increase of nuclear plant capacity factor from 8% up to 9% in this 48-hour period. The capacity factor of the coal-fired power plant, operated as a mid load generation unit, is hardly changed. Finally, the capacity factor of the CCGT plant, operated as a peak load generation unit, can be reduced from above 45% to below 35%. A higher capacity factor of the less expensive nuclear power plant and a reduced capacity factor of the more expensive CCGT peak load plant yields a significant generation cost reduction, shown in Table 2. Furthermore, fewer plants have to be started up, given reduced demand levels during peak load moments. This yields an additional cost reduction, summarized in Table 2. Capacity factor [%] Nuclear Coal CCGT Figure 9: Capacity factor of nuclear, coal-fired and CCGT power plants given different levels of own-price elasticity Total cost Generation Start-up Emissions Curtailment [k ] cost [k ] cost [k ] [kton] [MW] 1, , , ,9.3 1, , , , , , , Table 2: Total cost and environmental impact of a priceresponsive demand-side with different levels of own-price elasticity 9 Capacity factors are calculated as the average power output of the generation units over the 48-hour period

7 Powered by TCPDF ( The inclusion of demand response into a UC model also has environmental benefits. In this illustrative example, price responsive consumers increase their demand levels during moments of excess wind power injections. As a result, the renewables share can be increased. Those wind power injections as well as lower consumption during peak moments reduces the power output of CO 2 emitting generation technologies. Consequently, price responsive consumers realize CO 2 emission reductions up to 1% given high levels of ownprice elasticity. 4 CONCLUSIONS Unit commitment models optimize short-term operation of available generation units, accounting for technical constraints. Typically, demand levels are assumed to be fixed and system flexibility fully originates from the generation side. With the ongoing wind power deployment, system flexibility requirements increase. However, the smart meter roll-out creates flexibility at the demand-side of the power system. Even though this is often argued to the main benefit of smart metering, little efforts have been done to quantify these benefits. This paper models the integration of short-term demand response in a unit commitment by replacing fixed demand levels by elastic demand curves. A stepwise integration approach is applied in order to internalize welfare effects of consuming electricity in the objective function. A methodological example is described in order to illustrate the impact of real-time demand response on the aggregated demand profile. With respect to the integration of fluctuating wind power generation, benefits are measured in terms of a larger base load generation power output and reduced peak load power generation. This cost reduction effect is supplemented with avoided startup costs during peak demand periods. Furthermore, wind power injections can be increased which has a carbon mitigation effect combined with lower conventional power generation output levels. Stated differently, wind power generation benefits from responsive electricity consumers. ACKNOWLEDGEMENT The corresponding author would like to thank the Electricity Policy Research Group (EPRG) at the University of Cambridge for great discussions, help and comments. The authors are grateful for suggestions on earlier versions of this work by Benjamin F. Hobbs (The Johns Hopkins University) and Adriaan Hendrik van der Weijde (VU Amsterdam). REFERENCES [1] C. De Jonghe, E. Delarue, R. Belmans, and W. D haeseleer, Determining optimal electricity technology mix with high level of wind power penetration, Applied Energy, vol. 88, 211, pp [2] B.F. Hobbs, Optimization methods for electric utility resource planning, European Journal of Operational Research, vol. 83, May. 1995, pp [3] B.F. Hobbs, M.H. Rothkopf, R.P. O Neill, and H.-po Chao, The next generation of electric power unit commitment models, KLUWER ACADEMIC PUBLISHERS, 21. [4] G.B. Sheble and G.N. Fahd, Unit commitment literature synopsis, IEEE Transactions on Power Systems, vol. 9, 1994, pp [5] N.P. Padhy, Unit commitment A bibliographical survey, IEEE Transactions on Power Systems, vol. 19, May. 24, pp [6] E. Delarue, Modeling electricity generation systems: Development and application of electricity generation optimization and simulation models, with particular focus on CO2 emissions, PhD Thesis, Katholieke Universiteit Leuven, 29, p [7] B.F. Hobbs and U. Helman, Complementaritybased equilibrium modeling for electric power markets, Modeling Prices in Competitive Electricity Markets, D.W. Bunn, ed., J. Wiley, Ch. 3, 24. [8] Q.B. Dam, a P.S. Meliopoulos, G.T. Heydt, and A. Bose, A breaker-oriented, three-phase IEEE 24-substation test system, IEEE Transactions on Power Systems, vol. 25, Feb. 21, pp [9] T. Li and M. Shahidehpour, Price-Based Unit Commitment : A Case of Lagrangian Relaxation Versus Mixed Integer Programming, IEEE Transactions on Power Systems, vol. 2, 25, pp [1] M.C. Caramanis, Optimal spot pricing: Practice and theory, IEEE Transactions on Power Apparatus and Systems, vol. PAS-11, 1982, pp [11] A. Paul, D. Burtraw, and K. Palmer, Haiku documentation: RFF s electricity market model, Resources for the Future, 29, p. 52. [12] T. Takayama, A note on spatial and temporal price and allocation modeling Quadratic programming or linear complementary programming?, Regional Science and Urban Economics, vol. 13, Nov. 1983, pp [13] W.W. Hogan, Energy policy models for Project Independence, Computers & Operations Research, vol. 2, Dec. 1975, pp [14] M.C. Ferris, MATLAB and GAMS : Interfacing Optimization and Visualization Software, 25. [15] B.-H. Ahn and W.W. Hogan, On convergence of the PIES algorithm for computing equilibria, Operations Research, vol. 3, Mar. 1982, pp