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Adaptive mixed-integer programming unit commitment strategy for determining the value of forecasting
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APPLIED ENERGY Applied Energy 85 (2008) 171–181 www.elsevier.com/locate/apenergy
Adaptive mixed-integer programming unit commitment strategy for determining the value of forecasting Erik Delarue, William D’haeseleer
*
Division of Applied Mechanics and Energy Conversion, University of Leuven (K.U.Leuven), Celestijnenlaan 300 A, B-3001 Leuven, Belgium Accepted 13 July 2007 Available online 19 November 2007
Abstract This paper presents the development of a method to determine the value of forecasting (for load, wind power, etc.) in electricity-generation. An adaptive unit commitment (UC) strategy has been developed for this aim. An electricity generator faces demand with a given uncertainty. Forecasts are made to meet this load at the lowest cost. The adaptive UC strategy can be described as follows. Each hour, the generating company constructs a new forecast for a fixed number of hours. We assume that the first forecasted hour is in fact predicted correctly. For these forecasted hours, an optimal UC schedule is determined (given the on/off states of power plants for the current hour). The solution for the first hour (i.e., the one that was predicted correctly) is retained, and a new forecast is made. A 15,000 MW power system is used in a 168 hour (one-week) schedule. The UC problems presented in this work are solved through a Mixed-Integer Linear Programming (MILP) approach. In the first case, the effect of limited (correct) forecasting is investigated. Forecasts are made 100% correctly, but the UC scheme is built modularly and compared with the reference case, where the UC problem is solved for the one-week problem as a whole. Depending on the number of forecasted hours, solutions differ by up to 0.5% with the reference case. In a second case, when a certain error is imposed on the forecasts made (up to 5%), the deviations from the optimal solution become larger and amount in certain cases to almost 1%. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Electricity-generation modelling; Unit commitment (UC); Mixed-integer linear programming (MILP); Forecasting
1. Introduction Forecasting plays an important role in current electricity-generation systems. It is important to have accurate weather (wind, hydro, temperature) and accompanying load forecasts in order to determine an optimal unit commitment (UC). UC optimization determines which power plants should be activated and/or shut down over time, in order to be able to meet demand at the lowest cost. When forecasts (e.g., load, wind power) differ from the actual situation, presupposed UC solutions, obtained with wrong forecasts, will yield sub-optimal solutions. If the load is ‘over forecasted’, too many *
Corresponding author. Tel.: +32 16 32 25 11; fax: +32 16 32 29 85. E-mail addresses: [email protected] (E. Delarue), [email protected] (W. D’haeseleer).
0306-2619/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2007.07.007
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power plants will be brought online, which results in a more expensive operation. On the other hand, if the load were to be ‘under forecasted’, it is possible that the electricity-generating company will still be able to meet demand with the sub-optimal UC scheme (sub-optimal solution), or, if not, it will have to purchase additional expensive power on the spot market [1,2]. Even if the load for electricity could be perfectly forecasted, other uncertainties are often present. For instance, a generating company may own a significant share of intermittent unpredictable power-sources, such as a wind farm. The output of this farm is wind dependent and therefore not 100% predictable. If wind-generated electricity output is expected to differ from the real one in the upcoming hours, power-plant scheduling could differ from the optimal one, i.e., for which wind would be perfectly forecasted. In the rest of this work, we will, however, only talk about load forecasting.1 Studies have already been performed on load forecasting itself, i.e., on how to improve load forecasts. Corresponding to better forecasts, power generation will become less expensive and savings will occur. For an overview of load-forecasting strategies, we refer to Ref. [3]. In our work, however, the focus lies on how UC decisions should change with respect to newly-made forecasts, rather than how these forecasts were established. Hobbs et al. [1] determined the value of load forecasting in a 24 h forecast UC scheme. A UC schedule is made each day for the coming 24 h, based upon the predicted load. This load forecast has certain errors. Then, each hour, units are dispatched for the correct load. The total cost of meeting demand in this way is compared with the cost when the load would have been forecasted perfectly 24 h in advance (having then an optimal UC schedule). They found a decrease in generation cost of 0.1–0.3% when their ‘forecast mean absolute percentage error’ (MAPE) is reduced within 1%. Teisberg et al. [4] extended this investigation and determined the economic value of temperature forecasts this way. Park et al. [5] present a UC algorithm anticipating a given load-uncertainty. This method should yield better results than the deterministic classic UC approach. Miyagi et al. [6] presented an adaptive UC strategy, dependent on sudden load-changes, in order to obtain lower generation costs. This paper attempts to develop a method to compare the UC solution for which load is perfectly known in advance for a large number of hours with the UC outcome in a case where the upcoming load is only known correctly a few hours in advance. To make a reasonable comparison, the supplied power should in the end be the same in both cases, while solutions are being established in a different way. Dependent on the forecasts made, the final solution of the latter case will be sub-optimal, and significant profits will be lost. The adaptive UC strategy in our study differs from the above mentioned references in the fact that it is built gradually: each hour a new forecast is made, and the UC scheme is adapted. This paper will focus mainly on the effects and the results of having limited and inaccurate forecasts. How these forecasts are being established and what causes them (e.g. load uncertainties, and wind fluctuations) are not the subject of research in this work. The paper starts with a brief description of the mixed-integer linear programming (MILP) model used to solve the UC problems. A reference scenario (i.e., optimal solution) is also defined. Next, the implications of limited forecasting are investigated. In a final step, the effects of inaccurate limited forecasting are described, and some final conclusions are drawn.
2. Model description The UC problem is a well described in the literature. Several methods exist to tackle this issue: priority listing, dynamic programming, Lagrange relaxation, mixed-integer programming, genetic algorithms, etc. For an extended overview and classification of these methods, we refer to Ref. [7–9]. The UC model used in this study is a MILP model [10–12]. The UC problem is therefore written as an optimization problem with a single objective-function, i.e., a cost-minimization function. The unknown states (activation level of power plants, and production quantities) are declared as variables and a commercial solver is used to determine the optimal value of these variables, taking into account a whole set of technical con1
In the case of unpredictable wind, wind power is subtracted from the load to be delivered (by classic units), and therefore, this unpredictability of wind will be reflected in a load unpredictability.
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straints. Through the use of MILP, the variables can be limited to integer (e.g. binary) values [13,14]. The MILP model in fact solves both the UC and the economic dispatch problem. The objective function (i.e., the cost function to be minimized) can be written as the sum of fuel costs and start-up costs, over all power plants and all time periods.2 Here, for each power plant, a step-wise linear fuelcost function is used (approximating to a typical quadratic cost function [7]). For the implementation of these costs, and for the technical constraints on the power plants, we refer to Ref. [12]. The same model (computer code) is used for the reference UC optimization as for the adaptive UC strategy. Special attention should, however, be devoted to the inclusion of minimum up- and downtimes (see further). The model is implemented partly in Matlab [15] and partly in GAMS [16] (using the Matlab/GAMS link [17]) and is solved using the Cplex 10.0 solver [18]. In all simulations, the Cplex relative optimality criterion [19] was set at 0.001. 3. Reference case In a first reference case, the optimization is performed for the entire time span of one-week (168 hours) as a whole, i.e., a typical UC optimization. This can be seen as the case where demand is perfectly predicted. A realistic one-week demand pattern is used [20]. The composition of the power system used is presented in Appendix.3 The total installed power equals 15,000 MW. The power system can be equipped with a pumping-storage facility. The properties of this pumping-storage facility are also presented in Appendix. Simulations are performed for a case with and without the pumping unit. Fuel prices are taken from IEA [21]. As said before, the equations that monitor minimum up- and down-time in the model require special attention. With z(i, j) being the binary variable reflecting the commitment state of each power plant i during each time period j, these equations take the following form in the reference scenario4: 8i 2 I; 8j 2 J ; 8k 2 ½1; 2; . . . ; muti 1: zði; jÞ zði; j 1Þ þ zði; j þ k 1Þ zði; j þ kÞ 6 1
ð1Þ
8i 2 I; 8j 2 J ; 8k 2 ½1; 2; . . . ; mdti 1: zði; j 1Þ zði; jÞ þ zði; j þ kÞ zði; j þ k 1Þ 6 1
ð2Þ
with I, the set of power plants (index i); J, set of time periods, i.e., hours (index j); muti, minimum uptime of power plant i; mdti, minimum downtime of power plant i; z(i, j) the variable that defines the on/off state of power plant i in period j (binary variable: 1 if committed, 0 if not). These two equations will take a different form in the adaptive UC strategy. The MILP model is used to solve the UC problem as a whole for the power system with and without the storage facility, so two reference scenarios are considered. The total generation cost amounts to 30.33 M€ and 30.19 M€ respectively, for the one-week problem. 3.0.1. Power system without a pumping-storage facility As an example, Fig. 1 shows the activation of the power plants for all 168 h, for the power system without a pumping unit. In this figure, no. 1 ! 7 presents the nuclear plants, 8 ! 18 the gas-fired CC plants, 19 ! 34 the coal-fired plants, 35 ! 39 the gas-fired Rankine units, 40 ! 44 the gas-fired Brayton units, 45 ! 64 the oilfired Brayton units and 65 ! 84 the diesel units. As can be seen, the nuclear plants are always online. The coal-fired plants are also found in base load and are constantly activated. The highly efficient combined-cycle (CC) gas-fired plants are online during the daytime of working days, while the less efficient gas-fired Rankine and Brayton units and the diesel engines are activated only sporadically, mostly when demand peaks. 2
To solve the UC problem, the time dimension is discretized in time periods. In this study, one time period reflects one hour. This power system corresponds to some extent with the composition of the Belgian power system. 4 For the first and last time periods, corrections have to be made on these equations, e.g. to allow a solution with a certain power-plant online in the last time periods for a number of hours lower than its minimum uptime. 3
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UC status 1 8
plant no
19
35 40 45
65
84 24
48
72
96
120
144
168
hour [h] On
Off
Fig. 1. Unit commitment of power plants in the reference case (perfect forecast): power system without storage facility.
3.0.2. Power system with pumped storage The power system is now equipped with a pumping unit. To illustrate the use of this pumping unit, Fig. 2a presents the demand correction made by this storage facility. Fig. 2b presents the electricity-generation by fuel type. The optimal use and the technical constraints involving this storage unit are incorporated in the MILP optimization process.
(a) use of pumping unit
load [GW]
15
pumping up releasing 10
24
48
72
96
120
144
168
(b) electricity generation generation [GW]
15 oil 10
gas coal
5
nuclear
0 24
48
72
96
120
144
168
hour [h]
Fig. 2. (a) Use of pumping unit and (b) electricity-generation by fuel type in the reference case (perfect forecast): power system with pumped-storage facility.
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4. Adaptive UC strategy 4.1. Algorithm Now, in a second case, the same UC problem will be solved differently. The same load-profile is used, but we assume that only a limited number of hours can be predicted for each time step. So in this case, in each time step an optimization is executed for the coming predictable hours. From this optimization, with a given state in the first hour, only the solved state of the second hour is retained. This state serves as the initial state (i.e., hot start) of the next optimization. This way, solving a UC problem of 168 h requires 168 optimizations. This process is illustrated in Fig. 3. To obtain in the end the exact same load profile and actual generation as in the reference scenario, the first predicted hour should always be the exact value of the load during that hour. In a first case, we will investigate the effect of limited forecasting, so whole forecasts are made correctly. In a second case, forecasting with an error (‘inaccurate forecasting’) will be investigated. The first forecasted hour will, however, always be the correct actual load-value. In this algorithm, minimum up- and downtimes are incorporated. So, in each new forecast, the current upand downtimes of the solved states prior to the forecasted hours are passed on to the solver in order to respect minimum up- and downtimes in the new solution. The equations differ from Eqs. (1) and (2), and are as follows5: 8i 2 I : zði; 0Þ ¼ 1 ) 8j 2 ½1; 2; . . . ; muti cutðiÞ: zði; jÞ ¼ 1 8j 2 ½muti cutðiÞ þ 1; muti cutðiÞ þ 2; . . . ; jlast ; 8j 2 J ;
8k 2 ½1; 2; . . . ; muti 1:
zði; jÞ zði; j 1Þ þ zði; j þ k 1Þ zði; j þ kÞ 6 1 8k 2 ½1; 2; . . . ; mdti 1:
zði; j 1Þ zði; jÞ þ zði; j þ kÞ zði; j þ k 1Þ 6 1 zði; 0Þ ¼ 0 ) 8j 2 ½1; 2; . . . ; mdti cdtðiÞ:
ð3Þ
zði; jÞ ¼ 0 8j 2 ½mdti cdtðiÞ þ 1; mdti cdtðiÞ þ 2; . . . ; jlast ; 8k 2 ½1; 2; . . . ; mdti 1: zði; j 1Þ zði; jÞ þ zði; j þ kÞ zði; j þ k 1Þ 6 1 8j 2 J ;
8k 2 ½1; 2; . . . ; muti 1: zði; jÞ zði; j 1Þ þ zði; j þ k 1Þ zði; j þ kÞ 6 1
ð4Þ
with cut(i) the current uptime of power plant i; cdt(i), current downtime of power plant i; jlast the index of the last hour. As can be seen, a distinction has to be made now, as to whether or not the unit is committed in the current time period (period 0). If a certain unit is committed in the current time period, and the current number of operating hours of that unit is cut(i), then the unit should be activated for an additional muti–cut(i) number of hours. If the unit is not activated in the current time period, the same applies for the minimum downtime. 4.2. Consequences of limited forecasting This section presents results for the limited-forecast case. The UC scheme is built gradually out of correct forecasts of a limited number of hours. Again, the power system with and without the pumped-storage facility is considered.
5
Again, for the last time periods, corrections have to be made to these equations.
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1
2
3
4
5
6
7
… make forecast
2
3
4
5
6 solve problem for forecasted hours; first hour is given
2
3
4
5
6 retain solution of second solved hour (in this case hour 3)
1
2
3
4
5
6
7
8
… make new forecast
3
4
5
6
…
7
…
Fig. 3. Methodology of developed algorithm. The black numbers present solved states; the grey numbers are not solved yet. Squares with a circle represent a final solved state (i.e., final solution).
4.2.1. Power system without pumped-storage facility Table 1 presents absolute and relative differences in total cost for the several power systems in different forecasting scenarios (each with a different number of predicted hours for each optimization). From this table, it is clear that with shorter forecasts for each optimization, the total cost becomes higher. The activation of power plants in the 4-h forecast case is presented in Fig. 4. This scheme has to be compared with that for Fig. 1. As can be seen, clear differences occur. In the limited forecast case, more gas-fired Rankine units are activated during daytime, while peak units are less often used. Because of the short limited forecast, no optimal planning is achieved. 4.2.2. Power system with pumped-storage In this case, the power system is equipped with a pumped-storage unit. The results for this case are presented in Table 2. As can be seen from this table, differences with the reference scenario6 (i.e., the perfect forecast) are larger than in the case with no pumped-storage unit. The limited forecasting makes it very hard to predict how these units should be deployed optimally. The benefit of having a storage facility is largely lost. As an example, Fig. 5 presents results of the 4-h forecast case. The use of the pumping unit is shown, together with the electricity-generation. This figure should be compared with Fig. 2. In Fig. 5, the clearly worse use of the pumping unit is noticed. 4.3. Consequences of inaccurate forecasting In the next step, simulations can be performed, similarly as before, but now, with an error added to the forecasted hours. Thus, these hours are in fact forecasted incorrectly.7 This wrong forecast can have a serious impact on the use of the pumping unit (if present) and efficient planning of activation of power plants. Several simulations have been done, with different possible error magnitudes.
6
In this case, the reference scenario with the pumping unit is referred to. As stated before, the first hour of each forecast is always predicted correctly. Thus the overall electricity demand is the same as in the previous cases and results can be compared with the reference case. 7
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Table 1 Absolute and relative differences for the power system without pumped-storage facility in several forecast scenarios
2h 4h 8h 16 h
Abs diff (k€)
Rel diff (%)
42.9 33.5 17.2 9.8
0.141 0.110 0.057 0.032
UC status 1 8
plant no
19
35 40 45
65
84 24
48
72
96
120
144
168
hour [h] On
Off
Fig. 4. Unit commitment status of power plants in 4-h forecast case: power system without storage facility.
Table 2 Absolute and relative differences for the power system equipped with pumped-storage facility in several forecast scenarios
2h 4h 8h 16 h
Abs diff (k€)
Rel diff (%)
167.7 121.3 59.7 67.2
0.552 0.400 0.197 0.222
We have investigated the influences of two kinds of errors on forecasts. A first case assumes a linear increasing error in the forecast. In the last hour of the forecast, the error indicated is reached. The second case on the other hand assumes a constant-error on the whole forecast (except for the first hour). Again, simulations have been performed on the power system, with and without the pumped-storage facility. 4.3.1. Power system without the pumped-storage facility Fig. 6 presents the deviation from the optimal solution. Each node on this figure presents the outcome for the UC for the 168 h period. In such a simulation, each hour, an optimization is performed with an error on the forecast. This error is equal to the number indicated on the X-axis.8 Fig. 6a presents results for the linearlyincreasing error case, while Fig. 6b provides results for the constant-error case.
8
It should be noted that the solutions shown for 0% error (in the middle of the graphs) correspond to the ‘limited forecast case’ (i.e., no error).
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E. Delarue, W. D’haeseleer / Applied Energy 85 (2008) 171–181 (a) use of pumping unit
load [GW]
15
pumping up releasing 10
24
48
72
96
120
144
168
(b) electricity generation generation [GW]
15 oil 10
gas coal
5
nuclear
0 24
48
72
96
120
144
168
hour [h]
Fig. 5. (a) Use of pumping unit and (b) electricity-generation by fuel type in the 4-h forecast case: power system with pumped-storage facility.
deviation from optimal solution [%]
(a) linear increasing error 0.5 2 hour forecast 0.4
4 hour forecast
0.3
8 hour forecast 16 hour forecast
0.2 0.1 0 —5
—2.5
0
2.5
5
absolute error of last forecasted hour [%]
deviation from optimal solution [%]
(b) constant error 0.5 2 hour forecast 0.4
4 hour forecast
0.3
8 hour forecast 16 hour forecast
0.2 0.1 0 —5
—2.5
0
2.5
5
absolute error of all forecasted hours (except first) [%]
Fig. 6. Deviation from optimal solution in system with no pumped-storage. (a) Results in a case where the error in each forecast is linearly-increasing, while (b) presents results for the case where the error is the same for the whole forecast (except the first hour, which is forecasted correctly).
As can be seen, the differences stay below 0.5% in all cases. In the linearly-increasing error case, the 2 h forecast case yields the poorest results. When a negative error is made in the forecasts, this results in more expensive electricity-generation (i.e., less operating capacity scheduled, resulting in operating more expensive peak power).
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deviation from optimal solution [%]
(a) linear increasing error 1 2 hour forecast 0.8
4 hour forecast
0.6
8 hour forecast 16 hour forecast
0.4 0.2 0 —5
—2.5
0
2.5
5
absolute error of last forecasted hour [%]
deviation from optimal solution [%]
(b) constant error 1 2 hour forecast 0.8
4 hour forecast
0.6
8 hour forecast 16 hour forecast
0.4 0.2 0 —5
—2.5
0
2.5
5
absolute error of all forecasted hours (except first) [%]
Fig. 7. Deviation from optimal solution in system with pumped-storage: (a) results in a case where the error in each forecast is linearlyincreasing, while (b) presents results for the case where the error is the same for the whole forecast (except for the first hour, which is forecasted correctly).
In the constant-error case, all 4 forecasted scenarios yield similar results when imposing serious negative errors on the forecasts.9 When imposing a positive error, a distinction can now be seen between the several forecasted scenarios. In general, the more extended forecasts tend to move to solutions where too many power plants are activated, and therefore, lead to more expensive electricity-generation. 4.3.2. Power system with pumped-storage Fig. 7 presents results for the power system with the pumping-storage facility. As can be seen from Fig. 7, forecasting with a negative error (i.e., under forecasting) yields the worst solutions. Differences amount in several cases up to almost 1% (272 k€). This is mainly due to a serious sub-optimal use of the pumped-storage facility, together with the rather limited operating base-load capacity committed. When imposing a positive error (i.e., over forecasting), only the 8 and 16 h forecast simulations show an increasing deviation from the optimal solution. In the linearly-increasing error case, the use of the storage facility is slightly modified. In the constant-error case, the deviation is stronger. The use of the storage facility is now seriously worsened in most cases. 5. Conclusions The consequences of limited-load forecasting in electricity-generation are investigated. Especially the implication on UC schedules has been examined. A UC scheme has been constructed by solving a UC problem for each time step for a limited number of forecasted hours and retaining each time the solution for the first hour. A MILP model has been used for a 15,000 MW power-system in a one-week (168 h) problem. Initially, the influence of limited (but correct) forecasting has been investigated. The composed solutions of the adaptive UC strategy are compared with the normal reference case (i.e., one single optimization over all time periods). Several forecasted scenarios are examined. Differences of 0.14% (42.9 k€) and 0.55% (167.7 k€) 9
The 2 h forecast cases are in fact the same in the linearly-increasing error and the constant-error case.
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Table 3 Composition of power system. PWR: pressurized-water reactor; CC: combined cycle; R: rankine cycle; B: Brayton cycle; D: diesel engine Fuel
Type
# Units
Unit size (MW/unit)
Full load efficiency (%)
Nuclear Nuclear Gas Gas Coal Coal Coal Coal Coal Coal Gas Gas Gas Oil Oil
PWR PWR CC CC R R R R R R R R B B D
4 3 4 7 2 1 2 5 3 3 1 4 5 20 20
1000 600 300 400 370 345 340 208 250 125 250 125 20 13 8
35 35 52 50 40 38 38 37 36 36 38 36 35 35 35
Table 4 Properties of the pumping-storage facility
Pumping-storage unit
Power (MW)
Energy (MWh)
Efficiency (%)
1300
3200
80
occur from the optimal solution for the power system, with and without a storage facility respectively (for the one-week problem). In a second step, the consequences of forecasting with a certain error, are determined. In general, an increasing error yields a worse solution. When the load is under forecasted (negative error), not enough base load remains online to ensure the optimal electricity-generation. When the load is over forecasted, too many plants could be activated. When the power system is equipped with a pumped-storage facility, the use of this facility is also seriously worsened. Differences amount in some cases to 0.9% (272 k€). Overall, it has to be noted that the efficient use of the pumping-storage facility is lost to a large extent, when forecasts are limited (in both limited correct and incorrect forecast scenarios). Appendix Table 3 presents the composition of the power system. Table 4 presents the properties of the pumped-storage facility. References [1] Hobbs B, Jitprapaikulsarn S, Konda S, Chankong V, Loparo K, Maratukulam D. Analysis of the value of unit commitment for improved load forecasts. IEEE Trans Power Syst 1999;14:1342–8. [2] Saksornchai T, Lee W, Methaprayoon K, Liao J, Ross R. Improve the unit commitment scheduling by using the neural-networkbased short-term load forecasting. IEEE Trans Ind Appl 2005;41:169–79. [3] Alfares H, Nazeeruddin M. Electric load forecasting: literature survey and classification of methods. Int J Syst Sci 2002;33:23–34. [4] Teisberg T, Weiher R, Khotanzad A. The economic value of temperature forecasts in electricity generation. Bull Am Meteorol Soc 2005;86:1765–71. [5] Park J, Moon Y, Kook H. Stochastic analysis of the uncertain hourly load-demand applying to unit commitment problem. Power Eng Soc, Summer Meet, IEEE 2000;4:2266–71. [6] Miyagi T, Senjyu T, Saber A, Urasaki N, Funabashi T. In: Proceedings of the 13th international conference on intelligent systems application to power systems, IEEE, 2005, 6 pp. [7] Wood A, Wollenberg B. Power generation, operation and control. 2nd ed. New York: Wiley; 1996. [8] Padhy N. Unit commitment – a bibliographical survey. IEEE Trans Power Syst 2004;19:1196–205.
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[9] Sen S, Kothari D. Optimal thermal-unit commitment: a review. Electr Power Energy Syst 1998;20:443–51. [10] Carrion M, Arroyo J. A computationally efficient mixed-integer linear formulation for the thermal-unit commitment problem. IEEE Trans Power Syst 2006;21:1371–8. [11] Hobbs B, Rothkopf M, O’Neill R, Chao H. The next generation of electric-power unit commitment models. Boston: Kluwer Academic Publishers; 2001. [12] Delarue E, D’haeseleer W. Advanced priority listing versus mixed integer programming in solving the unit commitment problem. Energy, submitted for publication. [13] Winston W. Operations research, applications and algorithms. 3rd ed. Belmont: International Thomson Publishing; 1993. [14] Wolsey L. Integer programming. New York: Wiley Inc.; 1998. [15] The Mathworks, Matlab. Website available at: http://www.mathworks.com/products/matlab/. [16] General Algebraic Modelling System (GAMS). Website available at: http://www.gams.com/. [17] Ferris M. MATLAB and GAMS: interfacing optimization and visualization software. Available at: http://www.cs.wisc.edu/mathprog/matlab.html. [18] ILOG Cplex solver. Available at: http://www.ilog.com/products/cplex/. [19] GAMS. The solver manuals. Washington, DC: GAMS development Corporation; 2006. [20] Beroepsfederatie van de Elektriciteitssector (BFE). Statistisch jaarboek 2002, 2002. Available [in Dutch] at: http://www.synergrid.be/ Media/StatistiekenBFE2002.pdf. [21] IEA. World energy outlook 2005, middle east and north Africa insights. Paris: International Energy Agency/OECD; 2005.
APPLIED ENERGY Applied Energy 85 (2008) 171–181 www.elsevier.com/locate/apenergy
Adaptive mixed-integer programming unit commitment strategy for determining the value of forecasting Erik Delarue, William D’haeseleer
*
Division of Applied Mechanics and Energy Conversion, University of Leuven (K.U.Leuven), Celestijnenlaan 300 A, B-3001 Leuven, Belgium Accepted 13 July 2007 Available online 19 November 2007
Abstract This paper presents the development of a method to determine the value of forecasting (for load, wind power, etc.) in electricity-generation. An adaptive unit commitment (UC) strategy has been developed for this aim. An electricity generator faces demand with a given uncertainty. Forecasts are made to meet this load at the lowest cost. The adaptive UC strategy can be described as follows. Each hour, the generating company constructs a new forecast for a fixed number of hours. We assume that the first forecasted hour is in fact predicted correctly. For these forecasted hours, an optimal UC schedule is determined (given the on/off states of power plants for the current hour). The solution for the first hour (i.e., the one that was predicted correctly) is retained, and a new forecast is made. A 15,000 MW power system is used in a 168 hour (one-week) schedule. The UC problems presented in this work are solved through a Mixed-Integer Linear Programming (MILP) approach. In the first case, the effect of limited (correct) forecasting is investigated. Forecasts are made 100% correctly, but the UC scheme is built modularly and compared with the reference case, where the UC problem is solved for the one-week problem as a whole. Depending on the number of forecasted hours, solutions differ by up to 0.5% with the reference case. In a second case, when a certain error is imposed on the forecasts made (up to 5%), the deviations from the optimal solution become larger and amount in certain cases to almost 1%. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Electricity-generation modelling; Unit commitment (UC); Mixed-integer linear programming (MILP); Forecasting
1. Introduction Forecasting plays an important role in current electricity-generation systems. It is important to have accurate weather (wind, hydro, temperature) and accompanying load forecasts in order to determine an optimal unit commitment (UC). UC optimization determines which power plants should be activated and/or shut down over time, in order to be able to meet demand at the lowest cost. When forecasts (e.g., load, wind power) differ from the actual situation, presupposed UC solutions, obtained with wrong forecasts, will yield sub-optimal solutions. If the load is ‘over forecasted’, too many *
Corresponding author. Tel.: +32 16 32 25 11; fax: +32 16 32 29 85. E-mail addresses: [email protected] (E. Delarue), [email protected] (W. D’haeseleer).
0306-2619/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2007.07.007
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power plants will be brought online, which results in a more expensive operation. On the other hand, if the load were to be ‘under forecasted’, it is possible that the electricity-generating company will still be able to meet demand with the sub-optimal UC scheme (sub-optimal solution), or, if not, it will have to purchase additional expensive power on the spot market [1,2]. Even if the load for electricity could be perfectly forecasted, other uncertainties are often present. For instance, a generating company may own a significant share of intermittent unpredictable power-sources, such as a wind farm. The output of this farm is wind dependent and therefore not 100% predictable. If wind-generated electricity output is expected to differ from the real one in the upcoming hours, power-plant scheduling could differ from the optimal one, i.e., for which wind would be perfectly forecasted. In the rest of this work, we will, however, only talk about load forecasting.1 Studies have already been performed on load forecasting itself, i.e., on how to improve load forecasts. Corresponding to better forecasts, power generation will become less expensive and savings will occur. For an overview of load-forecasting strategies, we refer to Ref. [3]. In our work, however, the focus lies on how UC decisions should change with respect to newly-made forecasts, rather than how these forecasts were established. Hobbs et al. [1] determined the value of load forecasting in a 24 h forecast UC scheme. A UC schedule is made each day for the coming 24 h, based upon the predicted load. This load forecast has certain errors. Then, each hour, units are dispatched for the correct load. The total cost of meeting demand in this way is compared with the cost when the load would have been forecasted perfectly 24 h in advance (having then an optimal UC schedule). They found a decrease in generation cost of 0.1–0.3% when their ‘forecast mean absolute percentage error’ (MAPE) is reduced within 1%. Teisberg et al. [4] extended this investigation and determined the economic value of temperature forecasts this way. Park et al. [5] present a UC algorithm anticipating a given load-uncertainty. This method should yield better results than the deterministic classic UC approach. Miyagi et al. [6] presented an adaptive UC strategy, dependent on sudden load-changes, in order to obtain lower generation costs. This paper attempts to develop a method to compare the UC solution for which load is perfectly known in advance for a large number of hours with the UC outcome in a case where the upcoming load is only known correctly a few hours in advance. To make a reasonable comparison, the supplied power should in the end be the same in both cases, while solutions are being established in a different way. Dependent on the forecasts made, the final solution of the latter case will be sub-optimal, and significant profits will be lost. The adaptive UC strategy in our study differs from the above mentioned references in the fact that it is built gradually: each hour a new forecast is made, and the UC scheme is adapted. This paper will focus mainly on the effects and the results of having limited and inaccurate forecasts. How these forecasts are being established and what causes them (e.g. load uncertainties, and wind fluctuations) are not the subject of research in this work. The paper starts with a brief description of the mixed-integer linear programming (MILP) model used to solve the UC problems. A reference scenario (i.e., optimal solution) is also defined. Next, the implications of limited forecasting are investigated. In a final step, the effects of inaccurate limited forecasting are described, and some final conclusions are drawn.
2. Model description The UC problem is a well described in the literature. Several methods exist to tackle this issue: priority listing, dynamic programming, Lagrange relaxation, mixed-integer programming, genetic algorithms, etc. For an extended overview and classification of these methods, we refer to Ref. [7–9]. The UC model used in this study is a MILP model [10–12]. The UC problem is therefore written as an optimization problem with a single objective-function, i.e., a cost-minimization function. The unknown states (activation level of power plants, and production quantities) are declared as variables and a commercial solver is used to determine the optimal value of these variables, taking into account a whole set of technical con1
In the case of unpredictable wind, wind power is subtracted from the load to be delivered (by classic units), and therefore, this unpredictability of wind will be reflected in a load unpredictability.
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straints. Through the use of MILP, the variables can be limited to integer (e.g. binary) values [13,14]. The MILP model in fact solves both the UC and the economic dispatch problem. The objective function (i.e., the cost function to be minimized) can be written as the sum of fuel costs and start-up costs, over all power plants and all time periods.2 Here, for each power plant, a step-wise linear fuelcost function is used (approximating to a typical quadratic cost function [7]). For the implementation of these costs, and for the technical constraints on the power plants, we refer to Ref. [12]. The same model (computer code) is used for the reference UC optimization as for the adaptive UC strategy. Special attention should, however, be devoted to the inclusion of minimum up- and downtimes (see further). The model is implemented partly in Matlab [15] and partly in GAMS [16] (using the Matlab/GAMS link [17]) and is solved using the Cplex 10.0 solver [18]. In all simulations, the Cplex relative optimality criterion [19] was set at 0.001. 3. Reference case In a first reference case, the optimization is performed for the entire time span of one-week (168 hours) as a whole, i.e., a typical UC optimization. This can be seen as the case where demand is perfectly predicted. A realistic one-week demand pattern is used [20]. The composition of the power system used is presented in Appendix.3 The total installed power equals 15,000 MW. The power system can be equipped with a pumping-storage facility. The properties of this pumping-storage facility are also presented in Appendix. Simulations are performed for a case with and without the pumping unit. Fuel prices are taken from IEA [21]. As said before, the equations that monitor minimum up- and down-time in the model require special attention. With z(i, j) being the binary variable reflecting the commitment state of each power plant i during each time period j, these equations take the following form in the reference scenario4: 8i 2 I; 8j 2 J ; 8k 2 ½1; 2; . . . ; muti 1: zði; jÞ zði; j 1Þ þ zði; j þ k 1Þ zði; j þ kÞ 6 1
ð1Þ
8i 2 I; 8j 2 J ; 8k 2 ½1; 2; . . . ; mdti 1: zði; j 1Þ zði; jÞ þ zði; j þ kÞ zði; j þ k 1Þ 6 1
ð2Þ
with I, the set of power plants (index i); J, set of time periods, i.e., hours (index j); muti, minimum uptime of power plant i; mdti, minimum downtime of power plant i; z(i, j) the variable that defines the on/off state of power plant i in period j (binary variable: 1 if committed, 0 if not). These two equations will take a different form in the adaptive UC strategy. The MILP model is used to solve the UC problem as a whole for the power system with and without the storage facility, so two reference scenarios are considered. The total generation cost amounts to 30.33 M€ and 30.19 M€ respectively, for the one-week problem. 3.0.1. Power system without a pumping-storage facility As an example, Fig. 1 shows the activation of the power plants for all 168 h, for the power system without a pumping unit. In this figure, no. 1 ! 7 presents the nuclear plants, 8 ! 18 the gas-fired CC plants, 19 ! 34 the coal-fired plants, 35 ! 39 the gas-fired Rankine units, 40 ! 44 the gas-fired Brayton units, 45 ! 64 the oilfired Brayton units and 65 ! 84 the diesel units. As can be seen, the nuclear plants are always online. The coal-fired plants are also found in base load and are constantly activated. The highly efficient combined-cycle (CC) gas-fired plants are online during the daytime of working days, while the less efficient gas-fired Rankine and Brayton units and the diesel engines are activated only sporadically, mostly when demand peaks. 2
To solve the UC problem, the time dimension is discretized in time periods. In this study, one time period reflects one hour. This power system corresponds to some extent with the composition of the Belgian power system. 4 For the first and last time periods, corrections have to be made on these equations, e.g. to allow a solution with a certain power-plant online in the last time periods for a number of hours lower than its minimum uptime. 3
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UC status 1 8
plant no
19
35 40 45
65
84 24
48
72
96
120
144
168
hour [h] On
Off
Fig. 1. Unit commitment of power plants in the reference case (perfect forecast): power system without storage facility.
3.0.2. Power system with pumped storage The power system is now equipped with a pumping unit. To illustrate the use of this pumping unit, Fig. 2a presents the demand correction made by this storage facility. Fig. 2b presents the electricity-generation by fuel type. The optimal use and the technical constraints involving this storage unit are incorporated in the MILP optimization process.
(a) use of pumping unit
load [GW]
15
pumping up releasing 10
24
48
72
96
120
144
168
(b) electricity generation generation [GW]
15 oil 10
gas coal
5
nuclear
0 24
48
72
96
120
144
168
hour [h]
Fig. 2. (a) Use of pumping unit and (b) electricity-generation by fuel type in the reference case (perfect forecast): power system with pumped-storage facility.
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4. Adaptive UC strategy 4.1. Algorithm Now, in a second case, the same UC problem will be solved differently. The same load-profile is used, but we assume that only a limited number of hours can be predicted for each time step. So in this case, in each time step an optimization is executed for the coming predictable hours. From this optimization, with a given state in the first hour, only the solved state of the second hour is retained. This state serves as the initial state (i.e., hot start) of the next optimization. This way, solving a UC problem of 168 h requires 168 optimizations. This process is illustrated in Fig. 3. To obtain in the end the exact same load profile and actual generation as in the reference scenario, the first predicted hour should always be the exact value of the load during that hour. In a first case, we will investigate the effect of limited forecasting, so whole forecasts are made correctly. In a second case, forecasting with an error (‘inaccurate forecasting’) will be investigated. The first forecasted hour will, however, always be the correct actual load-value. In this algorithm, minimum up- and downtimes are incorporated. So, in each new forecast, the current upand downtimes of the solved states prior to the forecasted hours are passed on to the solver in order to respect minimum up- and downtimes in the new solution. The equations differ from Eqs. (1) and (2), and are as follows5: 8i 2 I : zði; 0Þ ¼ 1 ) 8j 2 ½1; 2; . . . ; muti cutðiÞ: zði; jÞ ¼ 1 8j 2 ½muti cutðiÞ þ 1; muti cutðiÞ þ 2; . . . ; jlast ; 8j 2 J ;
8k 2 ½1; 2; . . . ; muti 1:
zði; jÞ zði; j 1Þ þ zði; j þ k 1Þ zði; j þ kÞ 6 1 8k 2 ½1; 2; . . . ; mdti 1:
zði; j 1Þ zði; jÞ þ zði; j þ kÞ zði; j þ k 1Þ 6 1 zði; 0Þ ¼ 0 ) 8j 2 ½1; 2; . . . ; mdti cdtðiÞ:
ð3Þ
zði; jÞ ¼ 0 8j 2 ½mdti cdtðiÞ þ 1; mdti cdtðiÞ þ 2; . . . ; jlast ; 8k 2 ½1; 2; . . . ; mdti 1: zði; j 1Þ zði; jÞ þ zði; j þ kÞ zði; j þ k 1Þ 6 1 8j 2 J ;
8k 2 ½1; 2; . . . ; muti 1: zði; jÞ zði; j 1Þ þ zði; j þ k 1Þ zði; j þ kÞ 6 1
ð4Þ
with cut(i) the current uptime of power plant i; cdt(i), current downtime of power plant i; jlast the index of the last hour. As can be seen, a distinction has to be made now, as to whether or not the unit is committed in the current time period (period 0). If a certain unit is committed in the current time period, and the current number of operating hours of that unit is cut(i), then the unit should be activated for an additional muti–cut(i) number of hours. If the unit is not activated in the current time period, the same applies for the minimum downtime. 4.2. Consequences of limited forecasting This section presents results for the limited-forecast case. The UC scheme is built gradually out of correct forecasts of a limited number of hours. Again, the power system with and without the pumped-storage facility is considered.
5
Again, for the last time periods, corrections have to be made to these equations.
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1
2
3
4
5
6
7
… make forecast
2
3
4
5
6 solve problem for forecasted hours; first hour is given
2
3
4
5
6 retain solution of second solved hour (in this case hour 3)
1
2
3
4
5
6
7
8
… make new forecast
3
4
5
6
…
7
…
Fig. 3. Methodology of developed algorithm. The black numbers present solved states; the grey numbers are not solved yet. Squares with a circle represent a final solved state (i.e., final solution).
4.2.1. Power system without pumped-storage facility Table 1 presents absolute and relative differences in total cost for the several power systems in different forecasting scenarios (each with a different number of predicted hours for each optimization). From this table, it is clear that with shorter forecasts for each optimization, the total cost becomes higher. The activation of power plants in the 4-h forecast case is presented in Fig. 4. This scheme has to be compared with that for Fig. 1. As can be seen, clear differences occur. In the limited forecast case, more gas-fired Rankine units are activated during daytime, while peak units are less often used. Because of the short limited forecast, no optimal planning is achieved. 4.2.2. Power system with pumped-storage In this case, the power system is equipped with a pumped-storage unit. The results for this case are presented in Table 2. As can be seen from this table, differences with the reference scenario6 (i.e., the perfect forecast) are larger than in the case with no pumped-storage unit. The limited forecasting makes it very hard to predict how these units should be deployed optimally. The benefit of having a storage facility is largely lost. As an example, Fig. 5 presents results of the 4-h forecast case. The use of the pumping unit is shown, together with the electricity-generation. This figure should be compared with Fig. 2. In Fig. 5, the clearly worse use of the pumping unit is noticed. 4.3. Consequences of inaccurate forecasting In the next step, simulations can be performed, similarly as before, but now, with an error added to the forecasted hours. Thus, these hours are in fact forecasted incorrectly.7 This wrong forecast can have a serious impact on the use of the pumping unit (if present) and efficient planning of activation of power plants. Several simulations have been done, with different possible error magnitudes.
6
In this case, the reference scenario with the pumping unit is referred to. As stated before, the first hour of each forecast is always predicted correctly. Thus the overall electricity demand is the same as in the previous cases and results can be compared with the reference case. 7
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Table 1 Absolute and relative differences for the power system without pumped-storage facility in several forecast scenarios
2h 4h 8h 16 h
Abs diff (k€)
Rel diff (%)
42.9 33.5 17.2 9.8
0.141 0.110 0.057 0.032
UC status 1 8
plant no
19
35 40 45
65
84 24
48
72
96
120
144
168
hour [h] On
Off
Fig. 4. Unit commitment status of power plants in 4-h forecast case: power system without storage facility.
Table 2 Absolute and relative differences for the power system equipped with pumped-storage facility in several forecast scenarios
2h 4h 8h 16 h
Abs diff (k€)
Rel diff (%)
167.7 121.3 59.7 67.2
0.552 0.400 0.197 0.222
We have investigated the influences of two kinds of errors on forecasts. A first case assumes a linear increasing error in the forecast. In the last hour of the forecast, the error indicated is reached. The second case on the other hand assumes a constant-error on the whole forecast (except for the first hour). Again, simulations have been performed on the power system, with and without the pumped-storage facility. 4.3.1. Power system without the pumped-storage facility Fig. 6 presents the deviation from the optimal solution. Each node on this figure presents the outcome for the UC for the 168 h period. In such a simulation, each hour, an optimization is performed with an error on the forecast. This error is equal to the number indicated on the X-axis.8 Fig. 6a presents results for the linearlyincreasing error case, while Fig. 6b provides results for the constant-error case.
8
It should be noted that the solutions shown for 0% error (in the middle of the graphs) correspond to the ‘limited forecast case’ (i.e., no error).
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load [GW]
15
pumping up releasing 10
24
48
72
96
120
144
168
(b) electricity generation generation [GW]
15 oil 10
gas coal
5
nuclear
0 24
48
72
96
120
144
168
hour [h]
Fig. 5. (a) Use of pumping unit and (b) electricity-generation by fuel type in the 4-h forecast case: power system with pumped-storage facility.
deviation from optimal solution [%]
(a) linear increasing error 0.5 2 hour forecast 0.4
4 hour forecast
0.3
8 hour forecast 16 hour forecast
0.2 0.1 0 —5
—2.5
0
2.5
5
absolute error of last forecasted hour [%]
deviation from optimal solution [%]
(b) constant error 0.5 2 hour forecast 0.4
4 hour forecast
0.3
8 hour forecast 16 hour forecast
0.2 0.1 0 —5
—2.5
0
2.5
5
absolute error of all forecasted hours (except first) [%]
Fig. 6. Deviation from optimal solution in system with no pumped-storage. (a) Results in a case where the error in each forecast is linearly-increasing, while (b) presents results for the case where the error is the same for the whole forecast (except the first hour, which is forecasted correctly).
As can be seen, the differences stay below 0.5% in all cases. In the linearly-increasing error case, the 2 h forecast case yields the poorest results. When a negative error is made in the forecasts, this results in more expensive electricity-generation (i.e., less operating capacity scheduled, resulting in operating more expensive peak power).
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deviation from optimal solution [%]
(a) linear increasing error 1 2 hour forecast 0.8
4 hour forecast
0.6
8 hour forecast 16 hour forecast
0.4 0.2 0 —5
—2.5
0
2.5
5
absolute error of last forecasted hour [%]
deviation from optimal solution [%]
(b) constant error 1 2 hour forecast 0.8
4 hour forecast
0.6
8 hour forecast 16 hour forecast
0.4 0.2 0 —5
—2.5
0
2.5
5
absolute error of all forecasted hours (except first) [%]
Fig. 7. Deviation from optimal solution in system with pumped-storage: (a) results in a case where the error in each forecast is linearlyincreasing, while (b) presents results for the case where the error is the same for the whole forecast (except for the first hour, which is forecasted correctly).
In the constant-error case, all 4 forecasted scenarios yield similar results when imposing serious negative errors on the forecasts.9 When imposing a positive error, a distinction can now be seen between the several forecasted scenarios. In general, the more extended forecasts tend to move to solutions where too many power plants are activated, and therefore, lead to more expensive electricity-generation. 4.3.2. Power system with pumped-storage Fig. 7 presents results for the power system with the pumping-storage facility. As can be seen from Fig. 7, forecasting with a negative error (i.e., under forecasting) yields the worst solutions. Differences amount in several cases up to almost 1% (272 k€). This is mainly due to a serious sub-optimal use of the pumped-storage facility, together with the rather limited operating base-load capacity committed. When imposing a positive error (i.e., over forecasting), only the 8 and 16 h forecast simulations show an increasing deviation from the optimal solution. In the linearly-increasing error case, the use of the storage facility is slightly modified. In the constant-error case, the deviation is stronger. The use of the storage facility is now seriously worsened in most cases. 5. Conclusions The consequences of limited-load forecasting in electricity-generation are investigated. Especially the implication on UC schedules has been examined. A UC scheme has been constructed by solving a UC problem for each time step for a limited number of forecasted hours and retaining each time the solution for the first hour. A MILP model has been used for a 15,000 MW power-system in a one-week (168 h) problem. Initially, the influence of limited (but correct) forecasting has been investigated. The composed solutions of the adaptive UC strategy are compared with the normal reference case (i.e., one single optimization over all time periods). Several forecasted scenarios are examined. Differences of 0.14% (42.9 k€) and 0.55% (167.7 k€) 9
The 2 h forecast cases are in fact the same in the linearly-increasing error and the constant-error case.
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Table 3 Composition of power system. PWR: pressurized-water reactor; CC: combined cycle; R: rankine cycle; B: Brayton cycle; D: diesel engine Fuel
Type
# Units
Unit size (MW/unit)
Full load efficiency (%)
Nuclear Nuclear Gas Gas Coal Coal Coal Coal Coal Coal Gas Gas Gas Oil Oil
PWR PWR CC CC R R R R R R R R B B D
4 3 4 7 2 1 2 5 3 3 1 4 5 20 20
1000 600 300 400 370 345 340 208 250 125 250 125 20 13 8
35 35 52 50 40 38 38 37 36 36 38 36 35 35 35
Table 4 Properties of the pumping-storage facility
Pumping-storage unit
Power (MW)
Energy (MWh)
Efficiency (%)
1300
3200
80
occur from the optimal solution for the power system, with and without a storage facility respectively (for the one-week problem). In a second step, the consequences of forecasting with a certain error, are determined. In general, an increasing error yields a worse solution. When the load is under forecasted (negative error), not enough base load remains online to ensure the optimal electricity-generation. When the load is over forecasted, too many plants could be activated. When the power system is equipped with a pumped-storage facility, the use of this facility is also seriously worsened. Differences amount in some cases to 0.9% (272 k€). Overall, it has to be noted that the efficient use of the pumping-storage facility is lost to a large extent, when forecasts are limited (in both limited correct and incorrect forecast scenarios). Appendix Table 3 presents the composition of the power system. Table 4 presents the properties of the pumped-storage facility. References [1] Hobbs B, Jitprapaikulsarn S, Konda S, Chankong V, Loparo K, Maratukulam D. Analysis of the value of unit commitment for improved load forecasts. IEEE Trans Power Syst 1999;14:1342–8. [2] Saksornchai T, Lee W, Methaprayoon K, Liao J, Ross R. Improve the unit commitment scheduling by using the neural-networkbased short-term load forecasting. IEEE Trans Ind Appl 2005;41:169–79. [3] Alfares H, Nazeeruddin M. Electric load forecasting: literature survey and classification of methods. Int J Syst Sci 2002;33:23–34. [4] Teisberg T, Weiher R, Khotanzad A. The economic value of temperature forecasts in electricity generation. Bull Am Meteorol Soc 2005;86:1765–71. [5] Park J, Moon Y, Kook H. Stochastic analysis of the uncertain hourly load-demand applying to unit commitment problem. Power Eng Soc, Summer Meet, IEEE 2000;4:2266–71. [6] Miyagi T, Senjyu T, Saber A, Urasaki N, Funabashi T. In: Proceedings of the 13th international conference on intelligent systems application to power systems, IEEE, 2005, 6 pp. [7] Wood A, Wollenberg B. Power generation, operation and control. 2nd ed. New York: Wiley; 1996. [8] Padhy N. Unit commitment – a bibliographical survey. IEEE Trans Power Syst 2004;19:1196–205.
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[9] Sen S, Kothari D. Optimal thermal-unit commitment: a review. Electr Power Energy Syst 1998;20:443–51. [10] Carrion M, Arroyo J. A computationally efficient mixed-integer linear formulation for the thermal-unit commitment problem. IEEE Trans Power Syst 2006;21:1371–8. [11] Hobbs B, Rothkopf M, O’Neill R, Chao H. The next generation of electric-power unit commitment models. Boston: Kluwer Academic Publishers; 2001. [12] Delarue E, D’haeseleer W. Advanced priority listing versus mixed integer programming in solving the unit commitment problem. Energy, submitted for publication. [13] Winston W. Operations research, applications and algorithms. 3rd ed. Belmont: International Thomson Publishing; 1993. [14] Wolsey L. Integer programming. New York: Wiley Inc.; 1998. [15] The Mathworks, Matlab. Website available at: http://www.mathworks.com/products/matlab/. [16] General Algebraic Modelling System (GAMS). Website available at: http://www.gams.com/. [17] Ferris M. MATLAB and GAMS: interfacing optimization and visualization software. Available at: http://www.cs.wisc.edu/mathprog/matlab.html. [18] ILOG Cplex solver. Available at: http://www.ilog.com/products/cplex/. [19] GAMS. The solver manuals. Washington, DC: GAMS development Corporation; 2006. [20] Beroepsfederatie van de Elektriciteitssector (BFE). Statistisch jaarboek 2002, 2002. Available [in Dutch] at: http://www.synergrid.be/ Media/StatistiekenBFE2002.pdf. [21] IEA. World energy outlook 2005, middle east and north Africa insights. Paris: International Energy Agency/OECD; 2005.