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Discussion on: “Profit Maximization of a Power Plant”
European Journal of Control (2012)1:55–57 © 2012 EUCA
Discussion on: “Profit Maximization of a Power Plant”g Trine Krogh Boomsma1,∗ , Stein-Erik Fleten2,∗∗ 1 2
Risø National Laboratory of Sustainable Energy, Technical University of Denmark, Roskilde, Denmark Norwegian University of Science and Technology, 7491 Trondheim, Norway
Kragelund et al. provides an interesting contribution to operations scheduling in liberalized electricity markets. They address the problem of profit maximization for a power plant participating in the electricity market. In particular, given that the plant has already been dispatched in a day-ahead market, the aim is to schedule production throughout an operation day while complying with the day-ahead commitments, referred to in the paper as tracking a predefined production reference. The authors refer to Fig. 2 for an example of a production reference plan for a power plant of DONG Energy. A minor objection applies to this figure. It should be noted that DONG Energy operates at Nord Pool, where such plans are piece-wise constant hour by hour. However, given both smaller and larger power ripples lasting for only minutes, the figure most likely shows the actual production of the plant. This indicates that the research may have been done at some distance to the reality of power operations. As opposed to formulating the tracking as a hard constraint, the authors penalize volume deviations between reference and actual production (tracking error) in the objective function. This is not entirely consistent with the balancing mechanisms in most electricity markets, since no (or only very small) deviations are acceptable. Still, one could argue that it is possible to estimate penalty costs from balancing prices. In many markets, such costs vary
∗ E-mail: [email protected] ∗∗ E-mail: stein-erik.fl[email protected]
linearly with volume, which would simplify the problem solved by Kragelund et al.1 The profit maximization problem is formulated as an optimal control problem. According to the authors, this economic perspective is rare in the optimal control literature. It is worth noting, however, that it is the basis of real options analysis, which commonly involves optimal control theory. For recent examples of profit maximization in real options analysis, see [5], [6], [7], [8], [9]. Several of these consider power plants. The problem is one of fuel optimization, since the power production is controlled through the consumption of fuel. The controls represent the scheduled flows of fuel, and with input-output time lags, the flow available for electricity generation are the state variables. Since the plant can switch between three types of fuel differing in efficiency and controllability, the problem is known as a so-called switching problem in optimal control. Such problems are also well known in the real options literature, where they are typically solved as coupled differential equations [2], [4], or by Monte Carlo simulation [1], [3]. These methods easily incorporate switching costs, which is listed as a topic for further research by Kragelund et al. However, it is a subject of ongoing discussion how such methods should handle state variables that are affected by the controls. The paper presents an appealing alternative to overcome this problem by using the Hamiltonian approach. Although applied to a conventional power plant, we hypothesize that the methodology could be used for many other technologies, for instance energy storage. Here, we sketch its application to hydropower reservoir operation.
1
One the other hand, balancing costs are unknown at the time of planning, and hence, could be modeled as stochastic.
56
Discussion on: “Profit Maximization of a Power Plant”
For simplicity, we confine ourselves to day-ahead market scheduling, and disregard any reference tracking. We consider a hydropower plant that can operate in only three different modes u = (u1 , u2 , u3 ), with corresponding water flows y = (y1 , y2 , y3 ) and efficiencies e = (e1 , e2 , e3 ), for example producing at minimum (mode 1), at the best efficiency point (mode 2), or at maximum (mode 3). We denote the reservoir storage level by z. The problem is to determine an operating policy the maximizes the total profits throughout the operation day, that is, T max f (z, t; u)dt, u∈U
0
where the profit f may consist of electricity sales revenues, production costs (although usually small), an estimated value of the water in storage, and possibly a value of controllability. Here, we assume that f (z, t; u) = f−u (z, t) + pR1 (t)yT diag(e)u(t), where f−u consists of the profit function terms that do not involve u. For hydro-power, the input space is U = {u ∈ {0, 1}3 | ymin (z, t) ≤ yT u ≤ ymax (z, t), γ T u = 1}. Profit maximization is further subject to the reservoir storage dynamics z˙ (t) = i(t) − yT u(t), where i denotes external reservoir inflow. Maximization of the Hamiltonian now gives us max H(z, λ, t; u) =
u∈U f−u (z, t) + λ(t)i(t)+ max(pR1 (t)yT diag(e) − λ(t)yT )u(t). u∈U
Hence, the switching function is σ(t)T = pR1 (t)yT diag(e)− λ(t)yT . By noting that max σ(t)T u(t) = u∈U
max
ymin (z,t)≤yT u≤ymax (z,t)
where
⎡
1 C2 = ⎣ 0 −1
{σ(t)T C2 u(t)},
⎤ 0 0 1 0 ⎦, −1 0
we can continue along the lines of the paper. It should be noted, though, that the short-term hydropower scheduling problem is often subject to complications such as generation being a non-linear function of flow rate and reservoir head, non-convexity due to the hydro generation characteristics below its best efficiency point, discrete features caused by the on/off status of the turbines and start-up costs, and water transportation delay between cascaded reservoirs.
The suggested algorithm proceeds as follows. An initial estimate of the state variables is obtained by solving a discrete-time version of the switching problem. Using this estimate in the adjoint equation, the switching function is determined. With a known switching function, the optimal policy is derived, which is again used to update the estimate of the state variables, before a new iteration is made. Unfortunately, this algorithm does not converge, and deviations between reference and actual production still occur after many iterations. To overcome the problem, it is necessary to introduce heuristics. However, despite the acclaimed convergence, the results continue to display tracking errors of a magnitude that would not be acceptable in practice, see Fig. 10. It is worth discussing the continuous-time formulation versus a discrete-time. In spite of a significant increase in profits when the authors compare the continuous-time model with the discrete-time version, it would be straightforward to obtain a more realistic discrete-time model by including non-constant fuel efficiencies, fuel consumption for start-up, ramping constraints that directly reflect the controllability considerations, etc. In this way, although the continuous-time provides some interesting insights, we also believe that the discrete-time formulation maintains its strengths.
Acknowledgments Trine Krogh Boomsma and Stein-Erik Fleten acknowledge support from the ENSYMORA project funded by the Danish Council for Strategic Research.
References 1. Boogert A, Jong C. Gas storage valuation using a Monte Carlo method. J Derivatives 2008; 15(3): 81–98. 2. Brennan MJ, Schwartz ES. Evaluating natural resource investments. J Bus 1985; 58(2): 135–157. 3. Carmona R, Ludkovski M. Gas storage and supply guarantees: A optimal switching approach. Available at www.orfe.princeton.edu/∼rcarmona. 4. Cortazar G, Casessus J. Optimal timing of a mine expansion: Implementing a real options model. Q Rev Econ Financ 1998; 38: 755–769. 5. Cortazar G, Schwartz ES, Salinas S. Evaluating environmental investments: A real options approach. Manage Sci 1998; 44(8): 1059–1070. 6. Rothwell G. A real options approach to evaluating new nuclear power plants. Energ J 2006; 27(1): 37–53. 7. Siddiqui A, Fleten S-E. How to proceed with competing alternative energy technologies: A real options analysis. Energ Econ 2010; 32(4): 817–830. 8. Thompson M, Davidson M, Rasmussen H. Valuation and optimal operation of electric power plants in competitive markets. Oper Res 2004; 52(4): 546–562. 9. Tseng C, Barz G. Short-term generation asset valuation: A real options approach. Oper Res 2002; 50(2): 297–310.
Discussion on: “Profit Maximization of a Power Plant”g Trine Krogh Boomsma1,∗ , Stein-Erik Fleten2,∗∗ 1 2
Risø National Laboratory of Sustainable Energy, Technical University of Denmark, Roskilde, Denmark Norwegian University of Science and Technology, 7491 Trondheim, Norway
Kragelund et al. provides an interesting contribution to operations scheduling in liberalized electricity markets. They address the problem of profit maximization for a power plant participating in the electricity market. In particular, given that the plant has already been dispatched in a day-ahead market, the aim is to schedule production throughout an operation day while complying with the day-ahead commitments, referred to in the paper as tracking a predefined production reference. The authors refer to Fig. 2 for an example of a production reference plan for a power plant of DONG Energy. A minor objection applies to this figure. It should be noted that DONG Energy operates at Nord Pool, where such plans are piece-wise constant hour by hour. However, given both smaller and larger power ripples lasting for only minutes, the figure most likely shows the actual production of the plant. This indicates that the research may have been done at some distance to the reality of power operations. As opposed to formulating the tracking as a hard constraint, the authors penalize volume deviations between reference and actual production (tracking error) in the objective function. This is not entirely consistent with the balancing mechanisms in most electricity markets, since no (or only very small) deviations are acceptable. Still, one could argue that it is possible to estimate penalty costs from balancing prices. In many markets, such costs vary
∗ E-mail: [email protected] ∗∗ E-mail: stein-erik.fl[email protected]
linearly with volume, which would simplify the problem solved by Kragelund et al.1 The profit maximization problem is formulated as an optimal control problem. According to the authors, this economic perspective is rare in the optimal control literature. It is worth noting, however, that it is the basis of real options analysis, which commonly involves optimal control theory. For recent examples of profit maximization in real options analysis, see [5], [6], [7], [8], [9]. Several of these consider power plants. The problem is one of fuel optimization, since the power production is controlled through the consumption of fuel. The controls represent the scheduled flows of fuel, and with input-output time lags, the flow available for electricity generation are the state variables. Since the plant can switch between three types of fuel differing in efficiency and controllability, the problem is known as a so-called switching problem in optimal control. Such problems are also well known in the real options literature, where they are typically solved as coupled differential equations [2], [4], or by Monte Carlo simulation [1], [3]. These methods easily incorporate switching costs, which is listed as a topic for further research by Kragelund et al. However, it is a subject of ongoing discussion how such methods should handle state variables that are affected by the controls. The paper presents an appealing alternative to overcome this problem by using the Hamiltonian approach. Although applied to a conventional power plant, we hypothesize that the methodology could be used for many other technologies, for instance energy storage. Here, we sketch its application to hydropower reservoir operation.
1
One the other hand, balancing costs are unknown at the time of planning, and hence, could be modeled as stochastic.
56
Discussion on: “Profit Maximization of a Power Plant”
For simplicity, we confine ourselves to day-ahead market scheduling, and disregard any reference tracking. We consider a hydropower plant that can operate in only three different modes u = (u1 , u2 , u3 ), with corresponding water flows y = (y1 , y2 , y3 ) and efficiencies e = (e1 , e2 , e3 ), for example producing at minimum (mode 1), at the best efficiency point (mode 2), or at maximum (mode 3). We denote the reservoir storage level by z. The problem is to determine an operating policy the maximizes the total profits throughout the operation day, that is, T max f (z, t; u)dt, u∈U
0
where the profit f may consist of electricity sales revenues, production costs (although usually small), an estimated value of the water in storage, and possibly a value of controllability. Here, we assume that f (z, t; u) = f−u (z, t) + pR1 (t)yT diag(e)u(t), where f−u consists of the profit function terms that do not involve u. For hydro-power, the input space is U = {u ∈ {0, 1}3 | ymin (z, t) ≤ yT u ≤ ymax (z, t), γ T u = 1}. Profit maximization is further subject to the reservoir storage dynamics z˙ (t) = i(t) − yT u(t), where i denotes external reservoir inflow. Maximization of the Hamiltonian now gives us max H(z, λ, t; u) =
u∈U f−u (z, t) + λ(t)i(t)+ max(pR1 (t)yT diag(e) − λ(t)yT )u(t). u∈U
Hence, the switching function is σ(t)T = pR1 (t)yT diag(e)− λ(t)yT . By noting that max σ(t)T u(t) = u∈U
max
ymin (z,t)≤yT u≤ymax (z,t)
where
⎡
1 C2 = ⎣ 0 −1
{σ(t)T C2 u(t)},
⎤ 0 0 1 0 ⎦, −1 0
we can continue along the lines of the paper. It should be noted, though, that the short-term hydropower scheduling problem is often subject to complications such as generation being a non-linear function of flow rate and reservoir head, non-convexity due to the hydro generation characteristics below its best efficiency point, discrete features caused by the on/off status of the turbines and start-up costs, and water transportation delay between cascaded reservoirs.
The suggested algorithm proceeds as follows. An initial estimate of the state variables is obtained by solving a discrete-time version of the switching problem. Using this estimate in the adjoint equation, the switching function is determined. With a known switching function, the optimal policy is derived, which is again used to update the estimate of the state variables, before a new iteration is made. Unfortunately, this algorithm does not converge, and deviations between reference and actual production still occur after many iterations. To overcome the problem, it is necessary to introduce heuristics. However, despite the acclaimed convergence, the results continue to display tracking errors of a magnitude that would not be acceptable in practice, see Fig. 10. It is worth discussing the continuous-time formulation versus a discrete-time. In spite of a significant increase in profits when the authors compare the continuous-time model with the discrete-time version, it would be straightforward to obtain a more realistic discrete-time model by including non-constant fuel efficiencies, fuel consumption for start-up, ramping constraints that directly reflect the controllability considerations, etc. In this way, although the continuous-time provides some interesting insights, we also believe that the discrete-time formulation maintains its strengths.
Acknowledgments Trine Krogh Boomsma and Stein-Erik Fleten acknowledge support from the ENSYMORA project funded by the Danish Council for Strategic Research.
References 1. Boogert A, Jong C. Gas storage valuation using a Monte Carlo method. J Derivatives 2008; 15(3): 81–98. 2. Brennan MJ, Schwartz ES. Evaluating natural resource investments. J Bus 1985; 58(2): 135–157. 3. Carmona R, Ludkovski M. Gas storage and supply guarantees: A optimal switching approach. Available at www.orfe.princeton.edu/∼rcarmona. 4. Cortazar G, Casessus J. Optimal timing of a mine expansion: Implementing a real options model. Q Rev Econ Financ 1998; 38: 755–769. 5. Cortazar G, Schwartz ES, Salinas S. Evaluating environmental investments: A real options approach. Manage Sci 1998; 44(8): 1059–1070. 6. Rothwell G. A real options approach to evaluating new nuclear power plants. Energ J 2006; 27(1): 37–53. 7. Siddiqui A, Fleten S-E. How to proceed with competing alternative energy technologies: A real options analysis. Energ Econ 2010; 32(4): 817–830. 8. Thompson M, Davidson M, Rasmussen H. Valuation and optimal operation of electric power plants in competitive markets. Oper Res 2004; 52(4): 546–562. 9. Tseng C, Barz G. Short-term generation asset valuation: A real options approach. Oper Res 2002; 50(2): 297–310.