Nonlinear optimization method for short-term hydro scheduling
considering head-dependency
J. P. S. Catala˜o
1*
,y, S. J. P. S. Mariano
1, V. M. F. Mendes
2and L. A. F. M. Ferreira
31Department of Electromechanical Engineering, University of Beira Interior, 6201-001 Covilha, Portugal 2
Department of Electrical Engineering and Automation, Instituto Superior de Engenharia de Lisboa, 1950-062 Lisbon, Portugal
3Department of Electrical Engineering and Computers, Instituto Superior Te´cnico, 1049-001 Lisbon, Portugal
SUMMARY
This paper is on the problem of short-term hydro scheduling, particularly concerning head-dependent reservoirs under competitive environment. We propose a new nonlinear optimization method to consider hydroelectric power generation as a function of water discharge and also of the head. Head-dependency is considered on short-term hydro scheduling in order to obtain more realistic and feasible results. The proposed method has been applied successfully to solve a case study based on one of the main Portuguese cascaded hydro systems, providing a higher profit at a negligible additional computation time in comparison with a linear optimization method that ignores head-dependency. Copyright#2008 John
Wiley & Sons, Ltd.
key words: short-term hydro scheduling; nonlinear optimization; cascaded reservoirs; head-dependency
1. INTRODUCTION
In this paper, the short-term hydro scheduling problem of a head-dependent cascaded hydro system is considered. In hydro power plants with a large storage capacity available, for instance, as it is the case in the Brazilian system, head variation has negligible influence on operating efficiency [1]. In hydro power plants with a small storage capacity available, also known as run-of-the-river hydro power plants, operating efficiency is sensitive to the head [2]. For instance, in the Portuguese system there are several cascaded hydro systems formed by several but small reservoirs. Hence, it is necessary to consider head-dependency on short-term hydro scheduling in order to obtain more realistic and feasible results. In a cascaded hydraulic configuration, where plants can be connected in both series and parallel [3], the release of an upstream plant contributes to the inflow of the next downstream plants, implying spatial–temporal coupling among reservoirs. Head-dependency coupled with the cascaded hydraulic configuration augments the problem complexity and dimension.
Hydro power plants particularly run-of-the-river hydro power plants are considered to provide a clean and environmentally friendly energy option, while fossil-fuelled power plants are considered to provide an environmentally aggressive energy option, but nevertheless still in nowadays a necessary option [4]. The Portuguese fossil fuels energy dependence is among the highest in the European Union. Portugal does not have endogenous thermal resources, which has a negative influence on Portuguese economy. Moreover, the Portuguese greenhouse emissions are already out of Kyoto target and must be reduced in the near future. Hence, promoting efficiency improvements in the exploitation of the Portuguese hydro resources reduces the reliance on fossil fuels and decreases greenhouse emissions.
In a competitive environment, such as the Norwegian case [5], the most advantageous management of the water available in the reservoirs for power generation, without affecting future operations, represents a major advantage for the hydroelectric utilities to
Published online in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/etep.301
*Correspondence to: J. P. S. Catala˜o, Department of Electromechanical Engineering, University of Beira Interior, R. Fonte do Lameiro, 6201-001 Covilha, Portugal. yE-mail: [email protected]
face competition. Short-term hydro scheduling models provide decision support for the operational task of bidding in the energy and system services markets [6].
In the short-term hydro scheduling problem, a time horizon of one to seven days is considered, usually divided in hourly intervals. Hence, the short-term hydro scheduling problem is treated as a deterministic one. The corresponding forecasts are used where the problem includes stochastic quantities such as inflows to reservoirs or electricity prices. The goal is to maximize the value of total hydroelectric power generation throughout the time horizon considered, satisfying all physical and operational constraints, and consequently to maximize the profit of the hydroelectric utility from selling energy into the market [7].
Dynamic programming is among the earliest methods applied to the short-term hydro scheduling problem [8,9]. Although, dynamic programming can handle the non-convex, nonlinear characteristics present in the hydro model, it suffers from the well-known curse of dimensionality, more difficult to avoid in short-term than in long-term optimization without losing the accuracy needed in the model [10]. For hydro systems with cascaded reservoirs, CPU-time and memory requirements expand exponentially with problem size making dynamic programming unsuitable.
Artificial intelligence techniques have also been applied to the short-term hydro scheduling problem [11–14]. However, a significant computational effort is necessary to solve the problem for cascaded hydro systems and a time horizon of 168 intervals. Also, a fixed head is usually assumed in order to simplify the problem [12].
A natural approach to short-term hydro scheduling is to model the system as a network flow model, because of the underlying network structure subjacent in cascaded reservoirs [15,16]. This network flow model is often simplified to a linear or piecewise linear one. Linear programming is a widely used method for short-term hydro scheduling [17]. Also, mixed-integer linear programming has been applied successfully to solve large-size scheduling problem in power systems, and is becoming frequently used for short-term hydro scheduling [18–24], where binary variables allow modeling of up costs to avoid unnecessary start-ups, and of discrete hydro unit-commitment constraints. However, linear programming typically considers that hydroelectric power generation is linearly dependent on water discharge, thus ignoring head-dependency. The discretization of the nonlinear relationship between power generation, water discharge, and head, used in mixed-integer linear programming to model head variations, augment the computational burden required to solve the short-term hydro scheduling problem. Furthermore, methods based on successive linearization in an iterative scheme depend on the expertise of the operator to properly calibrate the parameters. For instance, the selection of the best under-relaxation factor in [21–23] is empiric and case-dependent, rendering some ambiguity to these methods.
A nonlinear model has advantages compared with a linear one. A nonlinear model expresses hydroelectric power generation characteristics more accurately and head-dependency on short-term hydro scheduling can be taken into account [2,25].
In this paper, we propose a new nonlinear optimization method to solve the short-term hydro scheduling problem considering head-dependency. Head-dependency is taken into account in our study, which represents one of the main difficulties associated with the short-term hydro scheduling problem, usually ignored when using linear models or neglected for hydro power plants with a large storage capacity available. Although, there were considerable computational difficulties in the past to directly use nonlinear optimization methods to this sort of problem, we show as a new contribution that this disadvantage is eliminated by applying the proposed method to a realistically sized hydro system.
This paper is structured as follows. In Section 2, we present a mathematical formulation for scheduling hydro power plants considering head-dependency, modeled as a dynamic, nonlinear constrained optimization problem. In Section 3, the proposed nonlinear optimization method to solve the short-term hydro scheduling problem is shown. In Section 4, we present a case study based on one of the main Portuguese hydro systems with seven cascaded reservoirs, considering a time horizon of 168 hours and both dry and wet scenarios. Finally, Section 5 outlines the main conclusions.
2. PROBLEM FORMULATION
The problem of short-term hydro scheduling can be stated as to find out the periodic water discharges,qik, the water storages,vik, and
the water spillages,sik, for each reservoir,i¼1;. . .;I, at all hours of the time horizon,k¼1;. . .;K, that optimize an objective
function subject to constraints. The water storages at the end of the time horizon,vik, must be decided according to future operations.
In the short-term hydro scheduling problem under consideration, the objective function is a measure of the profit attained by the conversion of potential energy into electric energy, without affecting future operations. Thus, the objective function to be
maximized can be expressed as: J¼X I i¼1 XK k¼1 pkpi kþ XI i¼1 CiðviKÞ (1)
The objective function in (1) is composed of two terms. The first term represents the profit with the hydro system during the short-term time horizon, wherepkis the forecasted electricity price in hourkandpikis the power generation of plantiin hourk. The
second term expresses the value of the water stored in the reservoirs for the future operations. This second term is only considered if the final water storage is not specified as a constraint. An appropriate representation when this second term is explicitly taken into account can be seen for instance in [26].
The optimal value of the objective function is determined subject to constraints of two kinds: equality constraints and inequality constraints or simple bounds on the variables.
The constraints are indicated as follows:
(a) Water balance equation for each reservoir, given by vi k ¼vi;k1þai kþ
X m2Mi
ðqm kþsm kÞ qiksik; i2I; k2K (2Þ
wherevikis the water storage of reservoiriat end of hourk,aikthe inflow to reservoiriin hourk,qikthe water discharge of
plantiin hourk,sikthe water spillage by reservoiriin hourk, andMiis the set of upstream reservoirs of planti. Time-delay is a
difficult issue, depending on the distance between the reservoirs and on the water discharge, deserving particular attention and research. Time-delay can be accounted for by considering a model structure different for different flow levels in an iterative procedure, which is outside the scope of this paper. Also, in our case study, the time required for water to travel from a reservoir to a reservoir directly downstream is considered less than the one hour period, independently of water discharge, due to the small distance between consecutive reservoirs.
(b) Power generation equation: Power generation is considered as a function of water discharge and efficiency. We consider efficiency given by the output/input ratio, depending on the head
pik¼qikhikðhikÞ; i2I; k2K (3Þ
wherepikis the power generation of plantiin hourk,hikthe efficiency of plantiin hourk, andhikis the head of plantiin hour
k.
(c) Head equation: The head is defined as the difference between the water levels in the upstream and downstream reservoirs of the hydro power plant, depending on the water storages in the respective reservoirs
hik¼lfðiÞkðvfðiÞkÞ ltðiÞkðvtðiÞkÞ; i2I; k2K (4Þ
Typically for a powerhouse with a reaction turbine, where the tail water elevation is not constant, the head is modeled as in (4), and for a powerhouse with an impulse turbine, where the tail water elevation remains constant, the head depends only on the upstream reservoir water level as in [23]. Hence, tailrace effects can be considered by including a correction in the data regarding reservoir water levels.
(d) Water storage constraints: Water storage has lower and upper bounds
vmini vikvmaxi ; i2I (5Þ
wherevmini is the minimum storage andvmaxi is the maximum storage. (e) Water discharge constraints: Water discharge has lower and upper bounds
qmini qikqmaxik ðhikÞ; i2I; k2K (6Þ
whereqmini is the minimum discharge andqmaxik is the maximum discharge, which may be considered a function of the head. (f) Water spillage constraints: A null lower bound is considered for water spillage
sik0; i2I; k2K (7Þ
Normally, water spillage by the reservoirs occurs when without it the water storage exceeds its upper bound, so spilling is necessary to avoid damage. The initial water storages,vi0, and the inflows to reservoirs are known input data. Also, discharge
ramping constraints [18] should be included, for instance, for a reservoir with a task of navigation to keep a less and stead head variation. Another important aspect is start/stop of units, which has been considered using mixed-integer linear programming in [18–24], where binary variables allow modeling of start-up costs.
3. NONLINEAR OPTIMIZATION METHOD
The short-time hydro scheduling problem is written as a mathematical programming problem of the type:(8)
maximize FðxÞ ð8Þ
subject to bminA xbmax ð9Þ
xminxxmax (10)
wherexis the vector of the flux variables corresponding to the arcs of the underlying network structure in hydro chains, consisting of the water storages, the water discharges, and the water spillages,F(.) a nonlinear function of the vector of the flux variables,Athe constraint matrix,bminthe lower bound vector on the constraints,bmaxthe upper bound vector on the constraints,xminthe lower bound vector on the variables, andxmaxis the upper bound vector on the variables. The water balance in (2) is rewritten as in (9), setting the lower bound equal to the upper bound. The bounds on water storage, water discharge and water spillage in (5), (6), and (7), respectively, are rewritten as in the inequality constraints in (10).
Our nonlinear objective function is achieved by means of two linearizations: the first of them, efficiency as a function of head, is acceptable; the second one, water level as a function of water storage, implies reservoirs with vertical walls, which however is a good approximation for the run-of-the-river reservoirs, due to its small storage capacity, as our data have shown for our case study.
In (3) the efficiency depends on the head. We consider it given by:
hik¼aihikþhi0; i2I; k2K (11)
where the parametersaiandh0i are given by:
ai¼ ðhmax i hmini Þ ðhmax i hmini Þ ; i2I (12) h0i ¼hmaxi aihmaxi ; i2I (13)
In (4) the water level depends on the water storage. We consider it given by:
lik¼bivikþli0; i2I; k2K (14)
where the parametersbiandl0i are given by:
bi¼ ðlmax i lmini Þ ðvmax i vmini Þ ; i2I (15) l0i ¼lmaxi bivmaxi ; i2I (16)
Substituting (11) into (3) we have:
pik¼qikðaihikþh0iÞ; i2I; k2K (17)
Therefore, by substituting (4) and (14) into (17), power generation becomes a new nonlinear function of water discharge and water storage, given by:
where the parameter xiis given by:
xi¼aiðl0fðiÞl0tðiÞÞ þh0i; i2I (19)
This new nonlinear relationship for power generation in a short-term hydro scheduling problem, considering head-dependency, leads to parameters given by the multiplication ofa’sbyb’s. These parameters are of crucial importance for the behavior of head-dependent reservoirs in a cascaded hydro system, setting optimal reservoirs storage trajectories according to their position in the cascade and according to the physical data defining the hydro system [2].
The maximum dischargeqmax
ik may be different for each hourk, according to:
q1i qmaxik qimax; i2I; k2K (20) correspondingq1i tohmaxi , andqmaxi tohmini .
Therefore, the maximum discharge may be considered a function of the head, given by:
qmaxik ¼ dihikþqi0; i2I; k2K (21)
where the parametersdiandq0i are given by:
di¼ ðqmax i q 1 iÞ ðhmax i hmini Þ ; i2I (22) q0i ¼qmaxi þdihmini ; i2I (23)
Substituting (4) and (14) into (21), we have:
qmaxik ¼ di bfðiÞvfðiÞkbtðiÞvtðiÞk
h i
þgi; i2I; k2K (24)
where the parameter giis given by:
gi¼ diðl0fðiÞltðiÞ0 Þ þq0i; i2I (25)
4. CASE STUDY
The proposed nonlinear optimization method, considering head-dependency, has been applied on a realistically sized hydro system with seven cascaded reservoirs. The spatial coupling among reservoirs is shown in Figure 1.
The hydro power plants numbered 1, 2, 4, 5, and 7, in Figure 1, are run-of-the-river hydro power plants. The other hydro power plants numbered 3 and 6, in Figure 1, are storage hydro power plants. Hence, for these two hydro power plants, head-dependency is not so important, due two the small head variation during the short-term time horizon.
Two scenarios are considered in our case study for the inflows: a dry scenario of low inflows and a wet scenario of high inflows. Inflow is considered only on reservoirs 1 to 6. The final water storage in the reservoirs is constrained to be equal to the initial water storage, given as 80% of maximum storage. Hence, the value of the water stored in reservoirs for future operations, in (1), is not considered. Also, the minimum storage is constrained to be equal to 40% of maximum storage.
The numerical testing was performed on a 600 MHz-based processor with 256 MB of RAM using the optimization solver package Xpress-MP under MATLAB. The time horizon was of 168 hours, started on Monday and finished on Sunday. The electricity price profile considered over the time horizon is shown in Figure 2 ($ is a symbolic economic quantity).
The competitive environment coming from the deregulation of the electricity markets brings electricity prices uncertainty, placing higher requirements on forecasting. A good forecasting tool reduces the risk of under/over estimating the profit of the utilities and provides better risk management. Several forecasting procedures are available for forecasting electricity prices [27,28], but for the short-term hydro scheduling problem the prices are considered as deterministic input data.
Figure 2. Electricity price profile considered. Figure 1. Hydro system with seven cascaded reservoirs.
The optimal storage trajectories for the run-of-the-river reservoirs, dry scenario, are shown in Figure 3. The dashed lines denote the results obtained by a linear optimization method while the solid lines denote the results obtained by the proposed nonlinear optimization method.
The optimal storage trajectories for the run-of-the-river reservoirs, wet scenario, are shown in Figure 4. Again, the dashed lines denote the results obtained by a linear optimization method while the solid lines denote the results obtained by the proposed nonlinear optimization method.
The comparison between linear and nonlinear optimization methods occurs while satisfying the same hydro constraints, in order to reveal the influence of considering head-dependency on the behavior of the reservoirs.
The upstream reservoirs should operate at the highest possible storage level in order to maximize the operating efficiency of its associated plants. Nevertheless, it was essentially a compromise between inflow and water discharge, thus the storage level changes, but still this change is less with the proposed nonlinear optimization method than with the linear optimization method. In both scenarios, the average storage level in the last downstream reservoir is lower with the proposed nonlinear optimization method than with the linear optimization method, thereby improving the head for the immediately upstream reservoirs, because a higher efficiency of this last downstream plant is not important for the overall profit in this hydro system.
The optimal discharge profiles for the run-of-the-river reservoirs, dry scenario, are shown in Figure 5. The dashed lines denote the results obtained by a linear optimization method while the solid lines denote the results obtained by the proposed nonlinear optimization method.
The optimal discharge profiles for the run-of-the-river reservoirs, wet scenario, are shown in Figure 6. Again, the dashed lines denote the results obtained by a linear optimization method while the solid lines denote the results obtained by the proposed nonlinear optimization method.
The comparison between linear and nonlinear optimization methods reveals that the water discharge changes more quickly from the lower value to the upper value with the linear optimization method than with the proposed nonlinear optimization method, thus ignoring the head-dependency. The water discharge and consequently the hydroelectric power generation tend to follow the shape of the electricity price profile shown in Figure 2.
Table I summarizes an overall comparison between the results obtained by both optimization methods.
Although, the average water discharge is, as expected, the same for both optimization methods, the average storage is superior with the proposed nonlinear optimization method. Thus, with the proposed method, we have a higher total profit for the hydroelectric utility, about 5%. Moreover, the extra CPU-time required is negligible. This demonstrates that the proposed method provides better results for cascaded hydro systems where head-dependency plays a major role on the behavior of the reservoirs.
5. CONCLUSIONS
A new nonlinear optimization method is proposed for the short-term hydro scheduling problem, considering head-dependency. This method allows an efficient consideration of the nonlinear relationship between power generation, water discharge, and head. Numerical results show as a new contribution that the proposed method is computationally adequate for realistically sized hydro systems with run-of-the-river hydro power plants, since head-dependency has to be considered in order to obtain more realistic and feasible results. The additional CPU-time required is negligible, converging rapidly to the optimal solution. Even in scenarios with low inflows the proposed method achieved a higher average storage, thereby yielding higher profits for the hydroelectric utility, in comparison with a linear optimization method that ignores head-dependency.
6. LIST OF SYMBOLS
A constraint matrix
aik inflow to reservoiri in hourk
bmin,bmax lower and upper bound vectors on constraints
F nonlinear function of variables hik head of plantiin hourk
hmin i ; h
max
i head limits of planti I,i set and index of reservoirs
K,k set and index of hours in the time horizon lik water level in reservoiriin hourk
lmin
i ; lmaxi water level limits of reservoiri Mi set of upstream reservoirs of planti
pik power generation of plantiin hourk
qik water discharge of plantiin hourk
qmin
i ; qmaxik water discharge limits of planti sik water spillage by reservoiriin hourk
vik water storage of reservoiriat end of hour k
vmini ; vmaxi water storage limits of reservoiri
x vector of the flux variables corresponding to the arcs of the network
xmin,xmax lower and upper bound vectors on variables hik efficiency of plantiin hourk
hmini ; hmaxi efficiency limits of planti
pk forecasted electricity price in hourk
Ci future value of the water stored in reservoiri
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Table I. Comparison of results.
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Dry Linear 23.73 70.32 4567.32 2.25
Nonlinear 23.73 72.44 4796.09 4.14
Wet Linear 47.47 71.76 8543.64 2.25
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