An efficient stochastic algorithm for midterm scheduling of cascaded hydro systems
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Abstract
Due to the stochastic and correlated attributes of natural inflows, the midterm generation scheduling problem for cascaded hydro systems is a very challenging issue. This paper proposes a novel stochastic optimization algorithm using Latin hypercube sampling and Cholesky decomposition combined with scenario bundling and sensitivity analysis (LCSBSA) to address this problem. To deal with the uncertainty of natural inflows, Latin hypercube sampling is implemented to provide an adequate number of sampling scenarios efficiently, and Cholesky decomposition is introduced to describe the correlated natural inflows among cascaded stations. In addition, to overcome the difficulties in solving the objectives of all the scenarios, scenario bundling and sensitivity analysis algorithms are developed to improve the computational efficiency. Simulation results from both twostation and tenstation systems indicate that the proposed method has the merits in accuracy as well as calculation speed for the midterm cascaded hydro generation scheduling. The consideration of natural inflow correlation makes the formulated problem more realistic.
Keywords
Cascaded hydro systems Midterm scheduling Stochastic optimization algorithm Correlation Sensitivity1 Introduction
Midterm hydro scheduling (MTHS) aims to manage hydropower generation as well as water release with maximum profit while satisfying various system constraints over a yearly horizon [1, 2]. This problem plays a crucial role in power system operation and a good solution can provide significant economic benefit. Due to uncertainty and probability distribution factors (e.g., natural inflow [3]), it is intractable to formulate and solve MTHS problems without approximation or simplification. Historically, the mean of annual natural inflow is formulated as input and the problem is described by deterministic models [4, 5, 6]. It is difficult to use these models to investigate the impact of natural inflow fluctuation on system operation and potential revenue.
To study the impact of uncertainty of natural flow on the MTHS, stochastic models are proposed in the literature. In [7], a stochastic optimization problem with Markov decision process models was developed allowing for the constraints of governmental regulations. In [8], a novel stochastic riskconstrained model was discussed. To consider the seasonal character and uncertainty of reservoir inflow, a composite optimization model was presented in [9]. A stochastic model with consideration of random water inflow and price errors was described by scenarios in [10]. And an energy equivalent reservoir model [11] was applied to the stochastic optimization of hydropower production decisions in the Brazilian power system. For cascaded hydropower systems, there are correlations among stations due to their dependence on a common river basin [12]. Typically, natural inflows of stations in the same river basin will increase or decrease synchronously under the influence of geographic factors. However, most reported literature focuses on the relationship between upstream and downstream reservoirs [13, 14], and considerations of correlation among natural inflows are limited. Therefore, it is necessary to consider the correlation of random natural inflows in MTHS problems.
The stochastic model can describe the midterm hydro scheduling problem more accurately, while bringing great challenges of computation due to the fluctuation of variables and boundary conditions. Currently, stochastic dynamic programming (SDP) and the multiscenario method are two major approaches to cope with this difficult issue.
SDP adopts discrete water inflow data and a transition probability matrix of adjacent periods to represent the randomness of water inflow [15, 16, 17], and maximizes the expected hydropower generation by finding the optimal solution for reservoir volume levels in each period. To reduce the computation time, stochastic dual dynamic programming (SDDP) [18, 19, 20] decomposes the multistage stochastic optimization problem into several onestage subproblems.
With the development of sampling and cutting technology, the multiscenario method [21] is used to solve the MTHS problem. In this approach, uncertainty is formulated by random variables with known forecasting information, and many possible scenarios are generated. However, such a large number of scenarios will increase the computational burden. Scenario reduction methods, e.g., fast forward reduction [22], moment based reduction [23], and particle swarm algorithms [24], are employed to select representative discrete scenarios and bundle corresponding probabilities with the purpose of decreasing the computation load. However, most of these approaches can only obtain the mean value of the optimization objective, and cannot directly obtain other statistical information such as the standard deviation and probability distribution curves. One way to provide comprehensive statistical data is Monte Carlo simulation with a large enough sampling scale [21], but the computation time is too long. To overcome above mentioned deficiencies, it is imperative to develop an accurate and computationally efficient stochastic optimization algorithm to estimate the objective completely.
 1)
Latin hypercube sampling combined with Cholesky decomposition, enabling the detailed modeling of stochastic natural inflows with the consideration of correlations among cascaded hydro stations. Cholesky decomposition has been successfully applied to solve the correlation of wind farms [25]. In this paper, for the first time, it is introduced to deal with correlations in the MTHS problem.
 2)
Scenario bundling and sensitivity analysis to obtain rapidly and accurately the optimal midterm scheduling decisions for a large number of sampling scenarios. The mean, standard deviation, and probability distribution of the objective are captured, which allows the probability analysis of generation profits.
The rest of this paper is structured as follows. Section 2 describes MTHS mathematical formulation for a cascaded hydro system. Section 3 presents the stochastic optimization approach. Numerical results from case studies are provided in Section 4, and conclusions are summarized in Section 5.
2 Mathematical model
2.1 Objective function
In the deregulated electricity market, the energy price during period t (C_{t}) is varying according to supply and demand [8]. However, this paper focuses on the vertically integrated utility in China, and the energy price is regulated and set by the government [7], hence, energy price parameters in the studied time horizon are set to be 1, and equivalently the objective of this problem is to maximize the total generation over the planning horizon.
2.2 Natural inflow probability model
2.3 Constraints on reservoirs and hydro units
The above discussion, apart from probability expression of (2), establishes the mixedinteger linear programming (MILP) formulation for the MTHS problem. Since water inflows in (2) are modeled as random variables, the objective in (1) is stochastic. In order to estimate the profit and risk for the cascaded hydro system in the scheduling horizon, objectives for all of the possible natural inflow scenarios have to be calculated by the stochastic dispatching method.
3 Stochastic dispatching method
In order to solve MTHS problem with stochastic and correlated natural inflows, a new algorithm is developed. It consists of three parts, which are sampling, bundling and sensitivity analysis.
3.1 Sampling with correlated random variables
 Step 1

Initialize t = 1, define the sample scale K, where K is a sufficiently large number of scenarios, such as 3000.
 Step 2

Use Latin hypercube sampling to obtain the independent standardized normal distribution sampling matrix \(\varvec{R}_{0}^{t}\) with dimension N_{H}×K.
 Step 3

Based on analysis of historical statistics, calculate the correlation matrix of natural inflows \(\varvec{\rho}_{H,t}\). The matrix is symmetric and positive definite as:
$$\varvec{\rho}_{H,t} { = }\left[ {\begin{array}{*{20}l} 1 \hfill & {\rho_{1.2}^{t} } \hfill & \cdots \hfill & {\rho_{{1,N_{H} }}^{t} } \hfill \\ {\rho_{2,1}^{t} } \hfill & 1 \hfill & \cdots \hfill & {\rho_{{2,N_{H} }}^{t} } \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ {\rho_{{N_{H} ,1}}^{t} } \hfill & \cdots \hfill & {\rho_{{N_{H} ,N_{H}  1}}^{t} } \hfill & 1 \hfill \\ \end{array} } \right]$$(16)  Step 4

Decompose the correlation matrix using the Cholesky method:
where \(\varvec{H}\) is a lower triangular matrix.$$\varvec{\rho}_{H,t} { = }\varvec{HH}^{\text{T}}$$(17)  Step 5

Construct a new matrix \(\varvec{R}_{u}^{t}\) including standard normal distribution variables with correlation:
$$\varvec{R}_{u}^{t} = \varvec{HR}_{0}^{t}$$(18)  Step 6

Update the matrix \(\varvec{R}_{u}^{t}\) to \(\varvec{R}_{\text{p}}^{t}\) with variables following the normal distribution [μ_{hp,t}, σ_{hp,t}], and save \(\varvec{R}_{\text{p}}^{t}\) to the matrix \(\varvec{R}_{\text{s}}\) from rows N_{H}(t − 1) + 1 to N_{H}t.
 Step 7

If t < T, set t = t + 1, and go to Step 2, otherwise stop and output the natural inflow sampling matrix \(\varvec{R}_{\text{s}}\) with dimension (N_{H}T)K.
3.2 Natural inflow scenario bundling
 Step 1

Determine the bundling distance parameter \(\Delta d\) based on the actual requirement.
 Step 2

Initiate the index and number of clusters as j = J = 1, set the cluster core Q_{j} to be the first scenario \(R_{1}\), let the number of scenarios contained in cluster N_{j}= 0, and set the scenario index k = 1.
 Step 3

For scenario k, find out the minimum Euclidean distance d_{k} from the scenario to all cluster cores Q_{j} (\(j \in \{ 1,2, \ldots ,J\}\)), which may be expressed as:
$$d_{k} = \mathop {\hbox{min} }\limits_{{j \in \{ 1,2, \ldots ,J\} }} d(R_{k} ,Q_{j} )$$(19)  Step 4

If d_{k} is less than or equal to \(\Delta d\), bundle scenario k to cluster j (where \(j = \arg \mathop {\hbox{min} }\limits_{{j \in \{ 1,2, \ldots ,J\} }} d(R_{k} ,Q_{j} )\)), increase \(N_{j} = N_{j} + 1\), update the cluster core Q_{j} as the average of all the included scenarios and go to Step 6; otherwise, go to Step 5.
 Step 5

If d_{k} is greater than \(\Delta d\), create a new cluster, set J = J + 1, assign the scenario k to cluster J, save the scenario R_{k} as the core of the new cluster Q_{J}, and set N_{J}= 1.
 Step 6

If k is less than K, update k = k + 1 and go to Step 3; otherwise, finish the bundling because all of the scenarios are distributed to clusters, and Q_{j} (\(j \in \{ 1,2, \ldots ,J\}\)) are the corresponding clustering cores.
3.3 Fast stochastic scheduling based on sensitivity analysis
In the framework of stochastic optimization, the objective of all the scenarios should be solved. Firstly, we can take one of the scenario clustering cores Q_{j} as the natural inflow input, hence the MTHS problem becomes singlescenario hydropower scheduling.
It can be seen that the physical interpretation of λ is the dual solution of the constraints (21). According to the dual principle, the sensitivity can be reached, and on the basis of (25) the objective functions for the scenarios included in cluster j may be obtained with high calculation speed.
Finally, if all of the clusters have been calculated by this method, the objectives for the original sampling scenarios (K scenarios) are captured, and then the expectation and probability distribution of the scheduling objective are able to be analyzed.
Consequently, by considering random and correlated natural inflows, the proposed approach can provide an operator a probability analysis of generation profits for cascaded hydro systems.
4 Case studies
The models and methods presented are applied to two actual cases in China, a twostation system and a tenstation system. The systems have been implemented on the GAMS and MATLAB platform. The tests are executed on an IBM dual processors computer operating at 2.5 GHz with 4 GB RAM under 32bit Windows 7. The time horizon of the midterm scheduling is 1 year, and the time step is 1 month.
4.1 Twostation system
Characteristics of hydro stations in twostation system
Station No.  Capacity (MW)  \(V_{{\hbox{max} } ,h_{p}}\) (10^{6}m^{3})  \(V_{{\hbox{min} } ,h_{p}}\) (10^{6}m^{3})  \(QS_{{\hbox{max} } ,h_{p}}\) (m^{3}/s)  \(V_{{{\text{ini}} },h_{p}}\) (10^{6}m^{3})  \(V_{{{\text{term}}},h_{p} }\) (10^{6}m^{3})  Hydro units No. 

1  320  455  133  8386  364  364  1–3 
2  84  348  100  5260  278  278  4–6 
Predicted values of natural inflow in twostation system
Period (month)  Station 1 (m^{3}/s)  Station 2 (m^{3}/s)  

Mean  Standard deviation  Mean  Standard deviation  
1  8.16  0.41  20.83  1.04 
2  7.68  0.38  19.04  0.95 
3  7.04  0.35  16.58  0.83 
4  11.36  0.57  25.50  1.28 
5  34.72  1.74  85.76  4.29 
6  68.16  6.07  233.57  20.79 
7  70.40  6.27  261.26  23.25 
8  55.04  4.90  179.99  16.02 
9  43.52  3.87  138.46  12.32 
10  33.28  2.96  101.78  9.06 
11  19.36  0.97  52.38  2.62 
12  10.88  0.54  27.04  1.35 
Comparisons of optimization results for total generation output with different methods in twostation system
Optimization results  LC  LCSB  LCSBSA  

Y (MWh)  Mean  1220626  1220429  1220429 
Standard deviation  23577  15755  23709  
Maximum  1292018  1282422  1293962  
Minimum  1148110  1172157  1147966  
ε_{Y} (%)  Mean  /  0.02  0.02 
Standard deviation  /  33.18  0.56  
Maximum  /  0.743  0.15  
Minimum  /  2.09  0.01 
The cumulative probability distribution curves show the likelihood of achieving different levels of total generation. If only considering the predicted mean value of natural inflow, the objective function is 1220800.29 MWh, and the probability of achieving less than or equal to this target is 51.27%. This indicates the high risk of focusing only on the scenario with predicted mean value. Thus, being able to obtain the cumulative distribution curve quickly can help operators to evaluate the profit and risks of the cascaded hydropower system.
Comparison of computing time with different methods for the twostation system
Methods  Sampling time (s)  Bundling time (s)  Optimizing time (s)  Restoring time (s)  Total computing time (s) 

LC  0.678  0  2269.601  0  2270.279 
LCSB  0.678  0.807  55.212  0  56.696 
LCSBSA  0.678  0.807  55.212  0.361  57.057 
Analysis of distance parameters for LCSBSA
\(\Delta d\)  Size of scenario clusters  Relative error ε_{Y} (%)  Total computing time (s)  

Mean  Standard deviation  
10  2462  0.006  0.083  2854.88 
20  243  0.013  0.258  415.05 
30  46  0.016  0.560  57.06 
40  15  0.122  0.954  18.16 
50  8  0.217  1.267  10.17 
4.2 Tenstation system
Predicted mean values of natural inflow in tenstation system test
Period  Station No. (m^{3}/s)  

1  2  3  4  5  6  7  8  9  10  
1  4  21  11  40  42  2  21  12  37  21 
2  4  21  11  40  41  2  19  12  40  21 
3  4  21  11  38  43  3  17  10  44  23 
4  6  40  17  52  84  10  26  19  90  41 
5  17  93  42  108  189  25  86  63  158  87 
6  34  145  87  234  326  30  234  134  204  164 
7  35  130  76  269  292  19  261  110  158  161 
8  28  83  48  218  183  18  180  72  102  100 
9  22  63  37  172  122  12  138  46  73  79 
10  17  50  32  122  121  8  102  38  78  61 
11  10  38  20  70  87  6  52  24  67  43 
12  5  26  14  47  52  3  27  15  43  28 
Water inflow correlation parameters among stations
Station No.  Station No.  

1  2  3  4  5  6  7  8  9  10  
1  1  0.58  0.33  0.02  0.19  0.11  0.46  0.36  0.01  0.05 
2  0.58  1  0.7  0.33  0.33  0.2  0.24  0.54  0.05  0.1 
3  0.33  0.7  1  0.73  0.36  0.25  0.06  0.25  0.14  0.15 
4  0.02  0.33  0.73  1  0.61  0.41  0.19  0.23  0.18  0.26 
5  0.19  0.33  0.36  0.61  1  0.65  0.07  0.03  0.24  0.6 
6  0.11  0.2  0.25  0.41  0.65  1  0.03  0  0.16  0.21 
7  0.46  0.24  0.06  0.19  0.07  0.03  1  0.82  0.14  0.14 
8  0.36  0.54  0.25  0.23  0.03  0  0.82  1  0.23  0.11 
9  0.01  0.05  0.14  0.18  0.24  0.16  0.14  0.23  1  0.69 
10  0.05  0.1  0.15  0.26  0.6  0.21  0.14  0.11  0.69  1 
Comparison of optimization results using different methods in tenstation system
Methods  Optimization results (MWh)  

Mean value  Standard deviation  Maximum  Minimum  
LC  22745954.94  316343.84  23916060.70  21541920.97 
LCSBSA  22751224.08  318074.77  23933036.68  21538074.23 
Comparisons of computing time with different methods in tenstation system
Methods  Sampling time (s)  Bundling time (s)  Optimizing time (s)  Restoring time (s)  Total computing time (s) 

LC  2.09  0  45375.15  0  45377.24 
LCSBSA  2.09  2.32  585.03  1.49  590.93 
5 Conclusion
 1)
The proposed Latin hypercube sampling with Cholesky decomposition combined with SB and LCSBSA to solve the stochastic MTHS problem has the merits both in calculation speed and precision compared with LC alone and LCSB. Thus LCSBSA is able to yield the probability distribution of the objective function solution rapidly and reliably, and operators can make decisions in a better way to reduce risks.
 2)
The appropriate bundling distance parameter can be selected according to the requirements of actual operating conditions. Generally, if the parameter is smaller, it is possible to achieve higher solution accuracy with the expense of longer computing time.
 3)
Natural inflow correlation has a significant impact on the estimated generation. Scheduling without consideration of natural inflow correlation significantly misjudges the fluctuation of total energy production. Consequently, the solutions of MTHS problem become more realistic by adding such correlation to simulations.
 4)
While coping with many random and correlated variables as in a tenstation system, this approach can still provide an accurate probability curve of total power generation for assessment of midterm hydro scheduling.
Further work may include pumped hydro storage facilities to adapt to grids with high penetration of variable renewable generation.
Notes
Acknowledgements
This work was supported in part by National Natural Science Foundation of China (No. 51507100), in part by Shanghai Sailing Program (No. 15YF1404600), and in part by “Chen Guang” project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation (No. 14CG55).
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