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Hydro Unit Commitment via Mixed Integer Linear Programming: A Case Study of the Three Gorges Project, China Xiang Li, Student Member, IEEE, Tiejian Li, Jiahua Wei, Guangqian Wang, and William W.-G. Yeh
Abstract—This paper develops a methodology for optimizing the hydro unit commitment (HUC) for the Three Gorges Project (TGP) in China. The TGP is the world’s largest and most complex hydropower system in operation. The objective is to minimize the total operational cost. The decision variables are the startup or shutdown of each of the available units in the system and the power releases from the online units. The mathematical formulation must take into account the head variation over the operation periods as the net head changes from hour to hour and affects power generation. Additionally, the formulation must consider the operation of 32 heterogeneous generating units and the nonlinear power generation function of each unit. A three-dimensional interpolation technique is used to accurately represent the nonlinear power generation function of each individual unit, taking into account the time-varying head as well as the non-smooth limitations for power output and power release. With the aid of integer variables that represent the on/off and operation partition statuses of a unit, the developed HUC model for the TGP conforms to a standard mixed integer linear programming (MILP) formulation. We demonstrate the performance and utility of the model by analyzing the results from several scenarios for the TGP.
Constants: Time period [h]. Water required to startup unit
[
Water required to shutdown unit
]. [
].
Maximum reservoir fore-bay water level [m]. Minimum reservoir fore-bay water level [m]. Maximum water volume in reservoir [
].
Minimum water volume in reservoir [
].
Maximum release from reservoir [
].
Minimum release from reservoir [
].
Maximum spillage from reservoir [ Maximum power output from unit
[MW].
Maximum power release from unit
[
Inflow in period [
Index Terms—hydro unit commitment, hydropower, mixed-integer linear programming, Three Gorges Project (China).
]. ].
].
Load demand in period [MW]. Reserve capacity in period [MW]. Maximum number of startups of a unit over the time horizon.
NOMENCLATURE The notation used in the modeling of the hydro unit commitment problem includes:
Variables: Total power release in period [
Sets
].
Set of indices of the time periods.
Total cost in terms of release from startup and shutdown of units in period [ ].
Set of indices of the units.
Total spillage in period [ Total release in period [
Manuscript received April 26, 2013; revised May 03, 2013, August 06, 2013, and September 24, 2013; accepted October 24, 2013. The work was supported by the National Key Technologies R&D Program #2013BAB05B03 in China. The first author is supported by a fellowship from the Chinese government for his visit to the University of California, Los Angeles. Partial support also is provided from an AECOM endowment. Paper no. TPWRS-00512-2013. X. Li is with the State Key Laboratory of Hydroscience & Engineering, Tsinghua University, Beijing 100084, China, and also with the Department of Civil and Environmental Engineering, University of California, Los Angeles, CA 90095 USA (e-mail:
[email protected]). T. Li, J. Wei, and G. Wang are with the State Key Laboratory of Hydroscience & Engineering, Tsinghua University, Beijing 100084, China (e-mail:
[email protected];
[email protected]; dhhwgq@ tsinghua.edu.cn). W. W.-G. Yeh (corresponding author) is with the Department of Civil and Environmental Engineering, University of California, Los Angeles, CA 90095 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TPWRS.2013.2288933 0885-8950 © 2013 IEEE
]. ].
Water volume at the end of period [ Average water volume in period [
]. ].
Reservoir fore-bay water level in period [m]. Reservoir tail-race water level in period [m]. Power release from unit
in period [
Power output from unit
in period [MW].
Penstock head loss of unit Net head on unit
].
in period [m].
in period [m].
Binary variable that is equal to 1 if unit is online in period and 0 otherwise.
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Binary variable that is equal to 1 if unit up and 0 otherwise.
is started
Binary variable that is equal to 1 if unit down and 0 otherwise.
is shut
Note that all constants are represented by upper cases and all variables by lower cases throughout the paper. I. INTRODUCTION
I
N general, the unit commitment (UC) problem involves determining when to startup or shutdown units and how to schedule the online units over a specific short-term time horizon (typically one day or one week) so as to minimize the operational cost or maximize the profit of a generating company. The UC problem is highly complex. As pointed out in [1], the problem generally can be formulated as a large-scale, mixed-integer, combinatorial, and nonlinear programming problem that may be difficult to solve. Over the past several decades, research has focused on building more accurate and computationally efficient models to represent real-world problems. Conventional methods developed for solving the UC problem include priority listing (PL) [2], dynamic programming (DP) [3] and Lagrangian relaxation (LR) [4]. Heuristic algorithms including simulated annealing (SA) [5], artificial neural networks (ANN) [6], genetic algorithm (GA) [7] and evolutionary programming (EP) [8] also have been used to solve the UC problem. Because of the advantages and disadvantages of different methods, there is a tendency to combine methods and take advantage of each one to develop a better solution technique for the particular problem under consideration [9]–[13]. For extensive literature reviews on the UC, refer to [13] and [14]. The HUC problem for hydropower production is an area of active research because of the potential cost savings or benefits. Major difficulties are the nonlinearity and non-concavity of the unit performance curves as well as the head effect on power production. In principle, the unit performance curves are functions of the net head, power release and power output. Mixed-integer linear programming (MILP) has gained increasing popularity because of the availability of more efficient and user-friendly software. MILP has been used to solve the short term hydro scheduling (STHS) problem in which the nonlinear unit performance curve is approximated by concave piecewise linear approximation under the assumption of constant head [15]. In [16], the unit performance curves of a unit are discretized into a small number of curves related to the water stored in the reservoir in order to model the head effect on power generation. Each curve is then modeled by a more generally non-concave piecewise linear approximation. Binary variables are used to determine the appropriate approximated curve in accordance with the water storage. A major contribution of [17] is its ability to further enhance the solution (power output) accuracy of the approach proposed in [16] through two-dimensional considerations of both water storages and power releases. A novel nonlinear programming (NLP) model has been proposed to solve the STHS problem [18], in which the complicated constraints caused by a head-sensitive cascaded hydro system
are addressed by assumptions and simplifications, including: 1) the power efficiency of a hydropower plant is a linear function of the head; 2) the head is a linear function of the water stored in the upstream and downstream reservoirs; and 3) the maximum total release is a linear function of the head. Consequently, power generation is expressed as a nonlinear function of power release and the storages of the upstream and downstream reservoirs. The maximum total release becomes a linear function of storages of the upstream and downstream reservoirs. The case study compares the results of NLP and linear programming (LP), which ignores the variable head effect. The results show that the NLP approach can produce more benefit with negligible increase in computation time. The NLP formulation is possible because the model does not consider the startup and shutdown statuses of units. This is a reasonable assumption for STHS, but not a valid assumption for the HUC problem. More recently, a mixed integer nonlinear programming (MINLP) model has been applied to optimal scheduling of a price-taker cascaded reservoir system [19]. The unit technical efficiency is estimated as a quadratic function of head and power release. By incorporating the technical efficiency, the power generation of a unit can be formulated as a continuous, nonlinear and non-concave function. The MINLP model outperforms the MILP model proposed in [16] in terms of solution accuracy. In principle, the MINLP model is the most accurate among all models in terms of capturing all of the nonlinear characteristics of the HUC problem. This paper develops an MILP model for solving the HUC problem of a large-scale, real-world, and highly complex multiunit hydropower system—the Three Gorges Project (TGP) in China. With MILP, constraints easily can be added, while nonlinearities can be represented accurately by means of binary variables and piecewise linear approximations [15]. The Itaipu in Brazil is a large-scale, multi-unit hydropower plant. As reported in [20], there were 18 identical 700 MW generating units in Itapu, with an installed capacity of 12 600 MW (Note that the Itaipu now has 20 identical units). The major goal of the study was to determine the trade-off between the objectives of minimizing the startup or shutdown of hydro generating units and minimizing power generation losses. However, the fore-bay water level was considered to be constant. Consequently, the continuity equation and relationship between fore-bay water level and storage were ignored. Additionally, the equal power release was assumed to dispatch the generating units because of the units’ identical performance. In this paper, we develop a practical, real-time optimization model for HUC for the world’s largest hydropower system with the intention of removing the assumptions and simplifications employed in the past. In brief, the model possesses three key features: 1) Although the TGP is a hydropower system with a large storage capacity reservoir behind the dam, the maximum allowable head change could reach as much as 5.0 m/day. As a result, the assumption of constant water head from hour to hour is inappropriate; 2) The relationships between head and storage as well as maximum power release and head are nonlinear; 3) The TGP has a large number of heterogeneous generating units and the operation status of each unit varies among them, so it is not possible to use just one power efficiency to represent the
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Fig. 1. Units in the hydropower plants of the TGP.
characteristics of all units, nor assume that the power efficiency is a linear function of the head. This paper concentrates on the development of an accurate and computationally efficient model addressing the HUC problem of the TGP. Its contributions are summarized in the following: 1) A three-dimensional interpolation method is used to simulate the nonlinear power generation function of a unit. We use nine data points to define the safe operation zone of a unit while taking into account the time-varying head effect, the non-smooth maximum and minimum power output limitations, as well as the corresponding power release limitations and the on/off and operation partition statuses of a unit; 2) The model takes individual heterogeneous generating units into account. The acceptable optimum can be obtained in an acceptable computation time. To our knowledge, this is the first attempt to accommodate a hydropower system of such size and complexity in the literature. Our proposed model considers each unit individually in a hydropower system with a large number of units of various types, the continuity equation as well as the time-varying head effect. This paper proceeds as follows. Section II introduces in detail the Three Gorges Project in China. Section III describes the mathematical formulation of the HUC problem. In particular, the two major nonlinearities, the net head on a unit and the unit performance curve, are discussed using the actual data from the TGP. Section IV analyzes the results obtained from several scenarios. Section V presents conclusions.
II. THREE GORGES PROJECT The Three Gorges Project (TGP) is located on the Yangtze River, upstream from the city of Yichang, Hubei Province, in central China. The TGP, operated by the Three Gorges Corporation, is the world’s largest hydropower system with an installed capacity of 22 500 MW. The TGP is a multi-purpose reservoir; its three main purposes are flood protection, power generation and navigation. A priority of the TGP’s operation is the regulation of fore-bay water level and release. During the non-flood season, the fore-bay water level of the TGP should be operated between 155.0 m–175.0 m; during the flood season, the fore-bay water level should be lowered and controlled at around 145.0 m, unless a large flood occurs. Thus, a large change in fore-bay water level will occur during the release and storage seasons. The minimum release from the TGP varies with different seasons and must be guaranteed for ecological and navigational purposes; the maximum release cannot exceed the safety discharge (56 700 ) at Zhicheng (a town downstream from the TGP) and the safety water level (43.0 m) at Shashi (a city
TABLE I MAIN CHARACTERISTICS OF THE TGP
downstream from Zhicheng) [21]. The main characteristics of the TGP are shown in Table I. The three hydropower plants in the TGP are of the dam-behind type, with a large storage capacity behind the dam. Since July 2012, all 32 main generating units installed have been in operation. Among the 32 main units, 14 are in the left-bank hydropower plant, 12 are in the right-bank hydropower plant, and six are in the underground hydropower plant. Although each main unit has the same capacity, not all unit performance curves are the same, since not all units are made by the same manufacturing company. Fig. 1 shows the units schematically in the three hydropower plants of the TGP. There are seven types of units installed in the three hydropower plants: #1–#3 and #7–#9 are the VGS-type, #4–#6 and #10–#14 are the ALSTOM I-type, #15–#18 are the ORIENTAL I-type, #19–#22 are the ALSTOM II-type, #23–#26 and #31–#32 are the HAERBINtype, #27–#28 are the ORIENTAL II-type, and #29–#30 are the ALSTOM III-type. At present, power generation of the TGP is under the jurisdiction of the power industry based on the day-ahead schedule. The power generated from the TGP supplies the electricity demand for nine provinces and two cities in China, signifying the strategic importance of the TGP to China. The power is transmitted to the State Grid Corporation of China and the China Southern Power Grid Co., Ltd with the currently constant energy price of $0.25 CNY/KWh. With an installed capacity of 22 500 MW, a small improvement in operation can translate into large benefit. Accordingly, there is an urgent need to develop a real-time operation model that can be used to optimize the startup or shutdown of the 32 generating units and the power generated from online units. The change in net head from hour to hour, the commitment of a large number of heterogeneous units, and the nonlinear power generation function of a unit are the major challenges that must be addressed. III. MATHEMATICAL FORMULATION Section III-A formulates the general mathematical model of the HUC problem. Sections III-B and III-C highlight the net head on a unit and simulation of unit performance curves, using the actual TGP data.
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A. Problem Formulation The goal of the model is to minimize the total operational cost, which is related to the volume of water released from storage. A principle of operation frequently used in China is minimizing the total amount of water released while meeting the specified load demands over the operation periods. The objective function is represented as (1) where (2)
(12) impose the upper and lower bounds, respectively, on power release and power output from a unit. Constraints (13) and (14) define the startup and shutdown statuses of a unit [15]. The frequent switching of hydro units in daily operation causes mechanical stress [10]. In general, the startup and shutdown costs of hydro generating units are expressed in the form of monetary units [10], [15]–[17], [19], [20]. Since the amount of water released is used as the objective, we introduce constraints (15) and (16) to indicate the minimum online time ( ) and minimum offline time ( ) of a unit [15] and use constraint (17) to limit the number of startups. Moreover, two nonlinearities exist in the modeling of the HUC problem. One is the net head on a unit, which can be expressed as
and
(18) (3)
where (19)
(4) (5) In (1), the total amount of water released ( ) is the sum of the power release ( ), the cost ( ) in terms of release from startup and shutdown of all units, and spillage ( ). Equations (3)–(5) define the power release, cost and spillage. The linear constraints of the model can be classified into reservoir constraints (6)–(8) and unit constraints (9)–(17). These constraints can be formulated as
The net head is a function of the reservoir fore-bay water level, which depends on the average storage of the reservoir; the reservoir tail-race water level, which varies in accordance with the total release from the reservoir; and the penstock head loss, which is a function of the power release from the unit. The three terms on the right-hand side of (18) are nonlinear functions of their corresponding variables. The other nonlinearity is the power generation curves of a unit, expressed as a nonlinear function of power release and net head:
(6)
(20)
(7) (8) (9) (10) (11) (12) (13) (14) (15) (16)
In (20), describes the hydropower generation characteristic of a unit, which involves the nonlinear relationship among power output, power release and net head. B. Net Head an a Unit First, we evaluate each term on the right-hand side of (18). Most of the published work to date involves the following assumptions: 1) The net head is a linear function of the storages in the upstream and downstream reservoirs [18]; 2) The net head is a linear function of its storage [19]; and 3) the net head is replaced with storage to simulate the unit performance curve [17]. Fig. 2 depicts the relationship of TGP storage and TGP fore-bay water level. As can be seen, it appears approximately linear. Thus, we formulate this relationship as (21)
(17) Constraint (6) is the continuity equation of a reservoir between two consecutive time periods. Constraints (7) and (8) establish the upper and lower bounds, respectively, on storage and total release for a reservoir in each time period. Constraint (9) requires that the power output from all units satisfies the specified load demand in each time period. Constraint (10) guarantees the reserve capacity for the power system. Constraints (11) and
where (22) It should be noted that though the fore-bay water level of the TGP varies from 145.0 m–175.0 m, more refined data should be used for a more accurate analysis. For instance, given that the initial TGP fore-bay water level is 160.0 m, the allowable
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Fig. 2. Relationship of TGP storage and TGP fore-bay water level (provided by the Three Gorges Corporation).
Fig. 3. Long-term relationship of TGP tail-race water level, TGP total release and GZB fore-bay water level (provided by the Three Gorges Corporation).
variation range in TGP fore-bay water level is during a single day, and thus the lower ( ) and upper ( ) bounds of TGP fore-bay water level can be specified as 160.0 m and 165.0 m or as 155.0 m and 160.0 m, or other acceptable bounds containing 160.0 m. The tail-race water level and the penstock head loss are important factors that directly influence the effective water head and, further, cause power loss in hydropower generation [22]. However, these factors are difficult to model and are either ignored or simplified [16]–[19]. For the TGP, there is a complication in modeling the tailrace water level, as this level also is affected by the fore-bay water level of the Gezhouba (GZB), a reservoir immediately downstream from the TGP. This relationship is highly unstable and extremely difficult to simulate for short time periods but is stable for longer time periods, as shown in Fig. 3. Fig. 4 depicts the relationship of penstock head loss and power release for 32 units in the TGP. The power releases from the underground hydropower plant units (i.e., #27–#32) incur more head losses than the other units (i.e., #1–#26). Directly fitting analytic expressions for the two relationships and integrating them into the model would introduce additional nonlinearities. This should be avoided, since in most cases the MINLP model is too complex to solve. Another way is to linearize these nonlinear relationships and then integrate the linearized formulations into the model, but this may produce thousands of additional variables, especially with the relationship between penstock head loss and power release. We use an iterative method to obtain the TGP tail-race water level by following this procedure:
5
Fig. 4. Relationship of penstock head loss and power release for 32 units in the TGP (provided by the Three Gorges Corporation).
Fig. 5. Hill chart of a typical unit in the TGP (provided by the Three Gorges Corporation).
1) Initialize the TGP tail-race water level at each period of the operation horizon; 2) Execute the MILP model to obtain the total release from the reservoir at each period; 3) Adjust the TGP tail-race water level at each period by plugging the optimized total release from the MILP model in Step 2 into a quartic polynomial function (23) [22], considering a two-dimensional interpolation based on the GZB fore-bay water level (known input data): (23) 4) Repeat 2)–3) until convergence occurs. Moreover, each of the 32 units in the TGP has a safe zone and the unit is constrained to operate in this zone (see Section III-C). We assume a constant penstock head loss, which is the average value within the safe zone. As shown in Fig. 4, this may introduce a maximum 0.5 m error on the penstock head loss of a unit. However, this assumption greatly reduces the computational burden. Additionally, the errors may cancel-out because a large number of units are operating in the system during a given time period. C. Unit Performance Curve The unit performance curves refer to a family of nonlinear curves, also known as the hill chart. Fig. 5 shows the hill chart (with 5 m as a net head interval) of a typical unit in the TGP, derived from the actual unit test results. Accurate modeling of unit power generation is difficult but crucial for the HUC problem.
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Fig. 6. Piecewise linear approximation of the hill chart.
The difficulties stem from the nonlinearity and non-concavity of the unit performance curve [16], [19]. Nevertheless, the curve can be modeled easily by an MILP formulation [15]. In the past, to avoid nonlinearity head variation was ignored by assuming a constant value for the net head [15]. A single unit performance curve with a concave piecewise linear approximation [15], or several best local efficiency points [16], were used to overcome the non-concavity. However, these simplifications may lead to inaccuracies [16]. To achieve a more accurate model, recent studies have taken into account the head effect on power generation [16], [17], as illustrated in Section I. In this study, we assume that the hill chart is symmetrical with various net heads. As Fig. 5 shows, this assumption is reasonable. In Section IV, the assumption will be validated by a numerical test. We only consider three unit performance curves for each unit, as shown in Fig. 6. The three curves (solid lines), representing the net head variations, are labeled as 1, 2 and 3. To avoid mechanical vibration, cavitation and low efficiency [23], the unit is constrained to operate in the safe zone. The safe zone is bounded by solid lines 1 and 3 as well as the minimum and maximum output limitations (dotted lines). Note that the minimum and maximum output or power release limitations vary with the net head on a unit. In the TGP case study, the time period used for scheduling is one hour, much longer than the startup or shutdown time required for a unit (i.e., one to five minutes) in practice. Consequently, it is possible to impose a hard constraint to avoid the forbidden and limited zones entirely and treat the safe zone as the feasible region for optimization. The curves 1, 2 and 3 of the safe zone are approximated by three two-piece linear segments so that a total of nine data points, denoted as , are used to represent the hill chart of a unit. Then the two-dimensional space of the hill chart is partitioned into eight triangles, as shown in Fig. 6. The advantages of this method are that a point within a triangle can be uniquely determined by weights on the three corner points and only three binary variables are required. The interpolation technique makes use of the following constants and variables:
set of indices of the curves; set of indices of the breakpoints; net head on unit
at data point
[m];
power release from unit at data point [ ]; power output from unit at data point [MW]; binary variable that is equal to 1 if the chosen cell is above the middle horizontal line for unit in period and 0 otherwise; binary variable that is equal to 1 if the chosen cell is right of the middle vertical line for unit in period and 0 otherwise; binary variable that is equal to 1 if the chosen cell is in the center diamond for unit in period and 0 otherwise; weight of data point . Accordingly, the unit performance curve in (20) can be transformed into several constraints: (24)
(25) (26) (27) (28) (29) (30) (31)
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(32) (33) (34) Constraint (24) sets the weighting sum of all nine data points; if a unit is online the weighting sum is equal to 1 and 0 otherwise. Constraints (25) and (26) compute the interpolation value for net head. If the unit is online, the interpolated value is equal to the sum of the weighted net head of each data point. Otherwise, constraint (25) is deactivated by the inclusion of the . Constraints (27) maximum reservoir fore-bay water level and (28) compute the interpolation values for power release and power output, respectively. Constraints (29)–(29)–(34) establish the principle of triangular tessellation in a two-dimensional space. The advantages of using this interpolation technique are summarized as follows: 1) Each unit in the TGP can be considered individually, with only six binary variables to simulate the power generation and ) of a unit. Among them, three variables ( , are used to simulate the on/off status of a unit, and three and ) are used to simulate the operation par( , tition of the unit. 2) The interpolation technique directly uses the values (i.e., net head, power release and power output) of nine data points from the three unit performance curves to define the safe operation zone of a unit, without calculating the slope [16] or power release and power output [17] of each block of a piecewise approximation. 3) The head effect on power generation and the on/off and operation partition statuses of a unit can be taken into account. The non-smooth maximum and minimum output limitations that vary with the net head easily can be considered. Accordingly, the non-smooth power release limitations can be included as well. As a result, constraints (11) and (12) can be removed because the equivalent considerations already have been established in the interpolation technique. For the TGP, the maximum power output (power release) curve is non-smooth with a breakpoint separation [see Fig. 6, data point (2, 3)]. The maximum power release increases as the net head increases, but decreases after the maximum power output limitation is reached. Similarly, note that the more refined data of unit performance curves should be selected for a more accurate analysis. For instance, given that the initial TGP fore-bay water level is 160.0 m and the TGP tail-race water level is 65.0 m, we can deduce that the allowable net head variation ranges approximately from 90.0 m to 100.0 m, so the three selected linear segments for applying the three-dimension interpolation can be specified as 90.0 m, 95.0 m and 100.0 m. At this point, the complete MILP model has been formulated. IV. RESULTS AND DISCUSSION The formulated MILP model presented in the previous section is solved with LINGO [24], a commercial optimization software package. The LINGO 14.0 64-bit used at the time of this study is the latest release. The branch-and-bound algorithm was called to solve the MILP model. Several tests were performed
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TABLE II COMPUTATIONAL DIMENSION AND REQUIREMENTS
TABLE III OPTIMIZATION RESULTS OF THE THREE GORGES PROJECT
on a ThinkPad W510 workstation containing an Intel (R) Core (TM) i7 CPU with 1.60 GHz and 4 GB of RAM. We consider hourly operation and a total of 24 hourly time periods as the time horizon. Although there is only one reservoir in the TGP that feeds water to all 32 units, this problem is difficult to solve because the multiple units are considered individually, introducing high-dimensionality. As Table II shows, the model has 14 816 variables, including 4608 binary variables and 13 329 constraints. We consider three days in August, September and October. Among the three scenarios, the days in August and September fall during storage season. We intend to test the model in a situation where large variation in the fore-bay water level of the TGP happens during a single day. In October, the fore-bay water level of the TGP is above 170.0 m and does not change much during a single day. Model input data includes forecasted inflows into the reservoir, forecasted load demands and initial reservoir fore-bay water levels. Since there is no spillage except during flood season for the TGP, we set in the optimization to accelerate convergence. The minimum online and offline times are assumed to be five hours and the number of startups is assumed to be two times over the time horizon. We set a time limit of 10 minutes as the stopping criterion because the operators should make a quick decision in practice. The optimization results obtained from various scenarios are shown in Table III, where the final gap is expressed as
The optimized objective function values are 13.27 , 23.84 and 9.01 , respectively. The narrowest final gap is 2.30% in Scenario October, while the widest final gap is 4.11% in Scenario August. We believe the results are quite acceptable because of the large number of variables in this problem. Furthermore, it is worth noting in particular that the final gap means the objective solution is at least within the percentage of the global optimal solution; the longer run may only tighten the bounds but not improve the objective solution. To validate the symmetry assumption proposed in Section III-C, we test the results of three scenarios numerically. First, we simulate the power generation performance of a unit for each net head with a quadratic polynomial function: (35)
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Fig. 7. Reservoir fore-bay water level, inflow and total release.
where , and are the fitting coefficients, as a function of unit type and head. Note that the fitted quadratic polynomial function matches the unit performance curve very well. Second, we input the optimization results of net head and power release for each unit at each period in (35) to compute the total power output during each period. Third, we compare the total power output for each period obtained by the MILP model and nonlinear simulation (35) with the coefficient of determination ( ) and mean relative error (MRE) (see the Appendix) to examine the process of total power output. The for the three scenarios are 0.997, 0.988 and 0.995, respectively; the MRE are , 0.0003 and 0.0034, respectively. All are close to 1.0 and all MRE are close to 0 for the three scenarios. This validates the symmetry assumption for the unit performance curve. We select Scenario August to illustrate the optimization results. Fig. 7 plots the reservoir fore-bay water level, inflow and total release for that scenario. It is evident that variation in the reservoir fore-bay water level is approximately 1.0 m over 24 hours. This impacts power generation significantly and cannot be overlooked. The mathematical formulation presented considers head variation over the operation periods. The on/off statuses of units in each period for Scenario August are shown in Fig. 8. Since constraints (15)–(17) are imposed, no unreasonable “flip-flop” occurs in this schedule. Fig. 9 shows the 24-hour hydro power generation schedule of the left-bank, right-bank and underground hydropower plant for Scenario August. In order to test the model’s applicability to a case where the units have discrete operating zones, we collected all relevant data from the Geheyan (GHY) hydropower plant on the Qingjiang River, China. The plant has yearly regulation capacity. The four identical generating units have one forbidden operating zone. Fig. 10 illustrates the piecewise linear approximation of the hill chart for a unit with discrete operating zones. We include additional constraints, adapted from [17], to avoid the forbidden zone: (36)
Fig. 8. Statuses of units in each period.
Fig. 9. Power outputs from the three hydropower plants.
Fig. 10. Piecewise linear approximation of the hill chart for a unit with discrete operating zones. (Note: the original curve is lengthened to power output equal to 0.)
(37) where is the forbidden power output of a unit in the plant, and is a binary variable that is equal to 1 if or 0 if . The model formulation for this plant is the same as the TGP, except that net head is assumed to be a linear function of storage
for simplification. Input data is chosen for three days in three different months (i.e., July, September and November). The minimum online and offline times are assumed as six hours. The time limit is 10 minutes. Table IV summarizes the optimization results, where both models with and without (36)–(37) are
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TABLE IV OPTIMIZATION RESULTS FOR UNITS WITH DISCRETE OPERATING ZONES
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ACKNOWLEDGMENT The authors would like to thank the editor and five anonymous reviewers for their in-depth reviews and constructive comments, which helped improve the paper greatly. REFERENCES
considered. The model considering the forbidden zone requires more water than the one that does not. For a small-sized UC problem such as the GHY, the global optimum is almost obtained in 10 minutes. This additional case study demonstrates the utility of our proposed MILP model. V. CONCLUSIONS In this paper, we developed a MILP model for solving the HUC problem of the TGP, the largest and most complex hydropower system in the world. The model considered the performance of individual units in a hydropower system containing 32 units of seven different types. We established a mathematical formulation to minimize the total operational cost subject to a set of reservoir and unit constraints. The formulation considers two main nonlinearities, the net head on a unit and the unit performance curve. In particular, the three-dimensional interpolation of the hill chart renders the simulation of power generation of an individual unit much more accurate. The interpolation scheme takes into account many aspects of power generation, such as the head effect, the non-smooth maximum and minimum power output limitations as well as the corresponding non-smooth power release limitations, and the on/off and operation partition statuses of a unit. The modern commercial optimization software LINGO solved the MILP model and obtained the acceptable optimum in an acceptable computation time. The successful applications of the developed model to several scenarios demonstrate its utility and efficiency for solving the HUC problem for a complex multi-unit hydropower system such as the TGP. APPENDIX The equations used to compute the coefficient of determination ( ) and mean relative error ( ) are given as follows:
(A1)
(A2) where is the simulated power output in period ; and and are the average values of simulated power output and load demand over the operation horizon, respectively.
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Xiang Li (S’13) is pursuing the Ph.D. degree at Tsinghua University, Beijing, China and was formerly a visiting graduate student at the University of California, Los Angeles, CA, USA. His research interests are in the fields of water resources planning and management, particularly in reservoir operation, and hydro unit commitment.
Guangqian Wang received the Ph.D. degree from Tsinghua University, Beijing, China, in 1989. He is a Professor at Tsinghua University and a member of the Chinese Academy of Sciences. His main research fields are fluvial dynamics and hydroinformatics.
Tiejian Li received the Ph.D. degree from Tsinghua University, Beijing, China, in 2008. He is currently an associate research fellow at Tsinghua University. His research interests are in the fields of river basin modeling and parallel computing in hydraulic engineering.
William W.-G. Yeh received the Ph.D. degree from Stanford University, Stanford, CA, USA, in 1967. He is currently a Distinguished Professor at the University of California, Los Angeles, CA, USA. He is a member of the US National Academy of Engineering. His research interests include groundwater modeling, conjunctive use planning of surface water and groundwater, and the development of methodologies and models for optimizing large-scale water resources systems.
Jiahua Wei received the Ph.D. degree from the University of Geosciences, Beijing, China, in 2001. He is an Associate Professor at Tsinghua University, Beijing, China. His research interests are in the fields of hydroinformatics and water resources planning and management.
