1. Introduction
Hydropower is a major green and renewable energy resource that can be massively developed and utilized at reasonable costs, and is also favorable in countries that are rich in it. An urgent deep decarbonation to ensure the sustainable development of economy and society places hydroelectricity in a critical role in energy structure [
1,
2]. In particular, the generation efficiency and water utilization rate can be substantially improved by a reasonable hydropower scheduling of cascaded reservoirs, which, however, is typically a complicated, high-dimensional, and multi-objective dynamic optimization problem that involves a nonlinear dynamic head, stochastic inflow, and hydraulic coupling between upstream and downstream hydropower plants [
3,
4]. Thus, it is essential to find an effective and efficient method or procedure to address this hydropower scheduling problem, which aims to maximize comprehensive benefits under some boundary conditions and other constraints.
The hydropower scheduling problem has long been studied in previous works, which experimented with many optimization algorithms including mathematical programming. The linear programming (LP), for instance, was employed by Feng et al. [
5] who presented a weekly hydropower scheduling problem of cascaded hydropower plants that was formulated into a mixed integer linear programming (MILP) model. Formulating a nonlinear optimization into an LP problem, however, is a skillful task, requiring the nonlinear functions to be linearized, as illustrated by Zheng et al. [
6], who presented a new three-triangle based method to linearly concave the hydropower output function (HOF), or in many cases, many integer variables to be introduced, causing the “combinatorial explosion” that makes it unlikely to derive a satisfactory solution in reasonable time. Nonlinear programming (NLP) is also a popular option, as applied by Barros et al. [
7], who formulated hydropower generation into a nonlinear function of the water discharge and head, but it could not guarantee a global optimum. The dynamic programming (DP) is particularly applicable to a stage-by-stage decision making process, just like in the reservoir operation, but was only limited to small scale hydro systems due to the well-known “dimensional difficulty”, which needs to be alleviated by improving the solution efficiency, as shown by Ji et al. [
8], who skillfully fitted the functional structure into the DP formulation to reduce the computation time.
Heuristic programming (HP), also known as intelligent algorithms, is another category of optimization algorithm that has been extensively applied in the hydropower scheduling of cascaded reservoirs. Wang et al. [
9], for instance, applied particle swarm optimization (PSO) to a multi-reservoir system to optimize the distribution of water storage. The evolutionary algorithms (EA), which starts its search from a randomly generated initial population to attain the global optimal solution, were explored by Arunkumar and Jothiprakash [
10], who introduced the chaotic technique to initiate a population randomly and also proposed a search strategy to enhance the performance of the EA in optimizing a multi-reservoir system. Bozorg et al. [
11] investigated the efficacy of the gravity search algorithm (GSA) based on the gravity law and mass interactions, and applied it to some benchmark functional objectives and operation problems involving a single and four reservoirs. Shang et al. [
12] made efforts to make the genetic algorithm (GA) more efficient when applied to maximize the hydropower production by proposing a proportional reproduction selection-based operator and a steady-state reproduction selection-based operator. HP is adaptable to a wide range of optimization problems, but very often with a volatile performance in securing a satisfactory solution in reasonable time.
The reservoir scheduling involves a variety of objectives including the power generation, flood control benefit, and ecological benefits. Diego et al. [
13], for instance, performed a comparative assessment on revenues from the short-term optimizations driven by two econometric models to enhance the expected revenue in hydropower systems, especially contributed by those with large storage capacity. Daniel et al. [
14] formulated a deterministic and a stochastic mathematical model to maximize the profits of selling energy produced by cascaded hydropower plants in a deregulated market. Yang et al. [
15] presented a multi-reservoir flood control model that incorporated operating rules to alleviate the risk incurred by a traditional flood control model. Wang et al. [
16] established a multi-objective optimization model to maximize the power generation while considering an ecological base flow, solved with a multi group cooperation operation particle swarm optimization (MGCL-PSO) model based on population cooperation. Wei et al. [
17] proposed a multi-objective optimization method and a multi-attribute decision making strategy to balance the objectives involving flood control, power generation, water supply, and ecology in reservoir operation. An optimal flood control operation model of cascaded reservoirs was proposed by Zhou et al. [
18] to coordinate the upstream reservoirs to reduce the downstream inflows at some flood control locations.
Table 1 shows the summary of abbreviations.
As one of the feasible direction methods, the Zoutendijk method, which is applicable to nonlinear programming problems with linear constraints, starts from an initial feasible solution, around which a descending feasible direction will be found to explore a better solution by searching along this direction [
19]. The Zoutendijk method has been applied to numerous optimization problems, showing very good performance in securing the global optimum and ensuring a high speed of convergence [
20]. The method, however, has not been reported yet, and was applied to a mid/long-term optimal scheduling of cascaded hydropower plants [
21].
This work will demonstrate how the Zoutendijk method can be applied to a monthly hydropower scheduling problem of six cascaded reservoirs located on the Lancang River, which involves nonlinear functions including the hydropower output and the capacity to generate discharge, which will be handled differently since the capacity of generating discharge is much less sensitive to the variation in the water head.
The following work is organized as follows.
Section 2 presents the monthly hydropower scheduling formulation with the objective to maximize the total hydropower production while ensuring the highest firm power output,
Section 3 shows how to convert this scheduling model into one that Zoutendijk is ready to solve,
Section 4 validates the model and its methods with a case study, and
Section 5 concludes the work.
2. Problem Formulation
Traditionally, a monthly hydropower scheduling takes into account the nonlinear dynamic head, hydraulic coupling, generating capacity, etc., with the objective to maximize the firm power output first and then the energy production during a planning horizon. Since the two objectives are different in priority, they are weighted differently in magnitude to convert the two objectives into an objective, expressed mathematically as
where W
1 and W
2 are the weights with
to prioritize the firm power output (
F) in MW over the total power output of all the hydropower plants;
i and
t are the subscripts for the reservoir and time-step, respectively;
Pit in MW is the power output in MW in time-step
t.
Constraints include:
with
where
Vit = the storage in hm
3 at the beginning of time-step
t of reservoir
i;
Qit = the outflow in m
3/s in time-step
t from reservoir
i;
Iit = the local inflow in m
3/s in time-step
t into reservoir
i;
= the set of reservoirs immediately upstream of reservoir
i;
= the number of days in time-step
t;
Viini and
Viend = the initial and target storages in hm
3 at the beginning and end of the planning horizon, respectively;
split = the spillage in m
3/s in time-step
t from reservoir
i;
qit = the generating discharge in m
3/s in time-step
t from hydropower plant
i.
The reservoirs on a monthly reservoir scheduling will have some storage capacity allocated for flood control during flood seasons, while they will have their forebay water levels remaining as high as possible during dry seasons to gain a higher water head and allow the hydro plants to produce more energy. The initial storage can usually be observed or forecasted when monthly generations are scheduled shortly after or before the beginning of the scheduling horizon, while the final storage is targeted in an over-year perspective.
- 2
Upper and lower bounds on storage and release, respectively,
where
= the dead storage in hm
3 of reservoir
i;
= the upper bound on the storage at the beginning of
t of reservoir
i, equal to the flood control limited storage during flooding seasons and the normal storage during dry seasons;
Qimin and
Qimax = the lower and upper bounds on the release from reservoir
i in time-step
t.
where
N = the number of hydropower plants/reservoirs.
- 4
Hydropower output and the capacity of generating discharge,
with
where
Ai = the power generating efficiency in MW; s/m
4;
hit = the water head in time-step
t of hydropower plant
i in m;
Gimax (.) = the capacity of generating discharge of
i in m
3/s, a function of water head;
Ziu (.) and
Zid (.) = the forebay and tailwater elevations in meters, dependent of the water storage and release, respectively, of reservoir
i.
3. Solution Techniques
The problem is typically a nonlinear programming, with the nonlinearity coming from constraints (6)–(8), among which the firm hydropower output, along with the capacity of generating discharge that generally has a small variety over the water heads, will be fixed as linear constraints, but updated during the solution with
with the water head estimated by
where
= the water head in meters in time-step
t of hydropower plant
i, estimated at the beginning of the
nth cycle of updating water heads;
and
= the storage and release at the solution derived at the beginning of the
nth cycle of updating water heads.
Figure 1 illustrates how the problem can be solved in a high-level flowchart, where the original problem with nonlinear constraints will be reformulated into a nonlinear optimization that involves only linear constraints, to which the Zoutendijk method is applicable. A new objective will be used to derive a solution feasible to the nonlinear optimization problem without considering the generating capacity (9), which will then be updated at this solution, which will then be modified to also meet the constraint on the generating capacity. Starting with a feasible solution, the LDP will be solved with the Zoutendijk method to obtain solutions used to update the generating capacity until convergence in updating the water heads.
3.1. Reformulation into Lagrange Dual Problem
With the forebay and tailwater elevations analytically expressed by
the water head can be determined as
and the hydropower output will be
where
,
,
and
= the coefficients/parameters to be calibrated to fit the relationship curve of storage versus forebay elevation of reservoir
i;
,
,
and
= the coefficients/parameters to be calibrated to fit the relationship curve of the release versus tailwater elevation of reservoir
i.
Thus, the problem will have the objective (1), expressed as
subject to (2)–(4), (9), (10), and (15). Substituting (15) for “
Pi (…)” in objective (16) gives a nonlinear programming problem with linear constraints, which allows its solution with methods of feasible directions.
3.2. Initial Feasible Solutions
(1) Initiate a feasible solution without generating capacity
The constraint of generating capacity needs to be estimated based on an initial solution, denoted as
where
represents a vector that consists of variables (
xit) for all the possible combinations (
i,
t), and
,
,
, and
that determine the water head (
),
that will be determined by solving a problem that
Subject to: (2)–(4) and the generating yield (
Y),
aimed to approximately maximize the firm hydropower output and the generating efficiency by operating reservoirs at their maximum water levels. To have all decision variables initiated, let
(2) Modify the solution to meet generating capacity
The solution used to estimate the generating capacity, when included or updated, may not meet the constraint on the generating capacity with a violation,
which is used to modify the solution to be feasible with
3.3. Application of the Zoutendijk Method
(1) Procedure of the Zoutendijk Method
The Zoutendijk algorithm involves two major steps: the first is to find a search direction () and the second to determine the best step () moving along this direction.
A nonlinear programming problem with linear constraints can be expressed as:
which has a total of
inequality constraints and
equation constraints. The solution steps of the Zoutendijk method can be summarized as follows:
Step 1: Initiate a feasible solution [] and let ;
Step 2: Decompose the into submatrices: and , and accordingly into: and so that according to the constraints on ;
Step 3: Solve a linear programming:
to derive the optimum:
, with
enforced to avoid unbounded solutions and ensure an optimum to the linear programming.
Step 4: If precision is achieved with , then output the current as the optimum; otherwise, continue to Step 5.
Step 5: if , then seek the best by a one-dimensional search, and then let , then turn to step 2; otherwise, turn to Step 6.
Step 6: Let , calculate , then update the solution by solving , then let ; turn back to Step 2.
(2) Seek a feasible increasing direction
Start at the beginning of the
nth cycle of updating water heads, and initiate a solution for the Zoutendijk method to be applied,
where the superscript number in parentheses and square brackets represents the number of cycles of updating water heads and iterations in the Zoutendijk method, respectively. Start at the beginning of the
kth iteration,
where the Zoutendijk method seeks a feasible increasing direction (
d) by
where
subject to:
to derive the optimum, denoted as
d(k).
(3) Line search with the golden section method
The gold section method is a line search procedure that starts from a feasible solution and seeks the optimum on a feasible improving direction by
with the feasible range [
] determined to enforce:
5. Conclusions
This work demonstrates how the Zoutendijk method could be applied, for the first time, to a monthly hydropower scheduling problem of cascaded reservoirs, which was formulated into a nonlinear programming with linear constraints, where the nonlinear hydropower output was represented with an exponential function to ensure its partial derivability while the generating discharge capacity was enforced with an iterative strategy. The model and method were applied to the Lancang River, which involved six cascaded hydropower plants, and the case studies demonstrated the following:
- (1)
The Zoutendijk method is well applicable to the monthly hydropower scheduling of cascaded reservoirs, deriving results that are reasonable and reliable;
- (2)
The exponential functions used to fit the forebay and tailwater curves showed a very high fitting accuracy at more than 99.5%, showing a great prospect to make segmented curves derivable when formulating a hydropower scheduling problem.
- (3)
The Zoutendijk method can significantly increase the total hydropower production while ensuring the highest firm power output of the whole cascade.
- (4)
This solution procedure is very fast in securing the optimum to the problem.
This work, however, only conducted a preliminary study and verified the effectiveness and efficiency of the Zoutendijk method, which may need to be compared with other widely used heuristic algorithms to show its strength and to be tested in solving larger scale problems that have more complex constraints.