Identifying Explicit Formulation of Operating Rulesfor Multi-Reservoir Systems Using Genetic Programming

Liping Li & Pan Liu & David E. Rheinheimer &Chao Deng & Yanlai Zhou

Received: 15 June 2013 /Accepted: 18 February 2014 /Published online: 16 March 2014# Springer Science+Business Media Dordrecht 2014

Abstract Operating rules have been widely used to handle the inflows uncertainty forreservoir long-term operations. Such rules are often expressed in implicit formulations noteasily used by other operators and/or reservoirs directly. This study presented genetic pro-gramming (GP) to derive the explicit nonlinear formulation of operating rules for multi-reservoir systems. Steps in the proposed method include: (1) determining the optimal operationtrajectory of the multi-reservoir system using the dynamic programming to solve a determin-istic long-term operation model, (2) selecting the input variables of operating rules using GPbased on the optimal operation trajectory, (3) identifying the formulation of operating rulesusing GP again to fit the optimal operation trajectory, (4) refining the key parameters ofoperating rules using the parameterization-simulation-optimization method. The method wasapplied to multi-reservoir system in China that includes the Three Gorges cascade hydropowerreservoirs (Three Gorges and Gezhouba reservoirs) and the Qing River cascade hydropowerreservoirs (Shuibuya, Geheyan and Gaobazhou reservoirs). The inflow and storage energyterms were selected as input variables for total output of the aggregated reservoir and fordecomposition. It was shown that power energy term could more effectively reflect theoperating rules than water quantity for the hydropower systems; the derived operating ruleswere easier to implement for practical use and more efficient and reliable than the conventionaloperating rule curves and artificial neural network (ANN) rules, increasing both averageannual hydropower generation and generation assurance rate, indicating that the proposedGP formulation had potential for improving the operating rules of multi-reservoir system.

Keywords Reservoir operation . Genetic programming . Operating rules .

Aggregation-decomposition

Water Resour Manage (2014) 28:15451565DOI 10.1007/s11269-014-0563-9

L. Li : P. Liu : D. E. Rheinheimer : C. Deng : Y. ZhouState Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University,Wuhan 430072, China

L. Li : P. Liu (*)Hubei Provincial Collaborative Innovation Center for Water Resources Security, Wuhan 430072, Chinae-mail: liupan@whu.edu.cn

D. E. RheinheimerUniversity of California, Merced, CA 95343, USA

1 Introduction

Reservoirs are often key infrastructure components in integrated water resources develop-ment and management schemes (Simonovic 1987; Guo et al. 2004). The effectiveoperation of reservoirs, which has been widely studied and summarized (Yeh 1985;Simonovic 1992; Wurbs 1993; Labadie 2004; Rani and Moreira 2010), is still a difficulttask due to the complexity such as curse of dimensionality (especial for dynamicprogramming (DP)), inflows uncertainty (Yeh 1985). To track inflows uncertainty, oper-ating rules have been widely used in reservoir long-term operations (e.g., Young 1967;Stedinger et al. 1984). It is ease of implementation since reservoir operators can followoperating rules to decide how much water should be released or retained at the beginningof the current period.

Implicit stochastic optimization (Young 1967) is an efficient alternative to explicitstochastic optimization (Stedinger et al. 1984), and has often been utilized to deriveoptimal reservoir operating rules (e.g., Wei and Hsu 2008; Celeste and Billib 2009;Rani and Moreira 2010). Within an implicit stochastic optimization framework, reservoiroperating rules are defined by their form and by parameters that define the form.Operating rule parameters can be found from deterministic optimization models usingeither fitting or parameterization-simulation-optimization (PSO) methods (Koutsoyiannisand Economou 2003). The fitting method finds the optimal reservoir trajectory (e.g.,releases) under several different inflow ensembles (observed or simulated hydrologicinflows (e.g., Lu et al. 2012)) using a deterministic optimization model, and the operatingrules are set up by fitting the optimal reservoir trajectory (Young 1967). In a ratherdifferent way, PSO starts with the pre-determined form of the operating rules and someassociated key parameters (Celeste and Billib 2009; Rani and Moreira 2010). Theparameters are changed and the reservoir is operated again and then improved withnonlinear optimization (Celeste and Billib 2009; Rani and Moreira 2010).

In developing operating rules, the selection of the operating rule form is the mostimportant task. Many operating rules, including linear operating rules (Karamouz et al.1992; Nalbantis and Koutsoyiannis 1997), operating rules curve (Huang and Yuan2004; Chang et al. 2005a; Chen et al. 2007; Ngo et al. 2007; Suiadee andTingsanchali 2007; Liu et al. 2011b) and nonlinear operating rules, such as artificialneural networks (Saad et al. 1994; Saad et al. 1996; Chandramouli and Raman 2001;Liu et al. 2006), the fuzzy method (Chaves and kojiri 2007) and decision trees (Weiand Hsu 2008; Kumar et al. 2013), have been used to derive the operating rules for asingle and/or multi-reservoir system in recent years. Nonlinear operating rules consis-tently outperform linear operating rules (Bhaskar and Whitlatch 1980; Celeste andBillib 2009). However, these types of operating approaches have been often expressedwith implicit formulations that could not be easily used by other reservoirs and/oroperators directly.

This study uses genetic programming (GP) to derive the explicit formulation of operatingrules for a multi-reservoir system. Extended from genetic algorithm (GA) (Oliveira andLoucks 1997; Wardlaw and Sharif 1999; Kim et al. 2006; Chen et al. 2007; Chiu et al.2007; Chang 2008; Chang and Chang 2009; Wang et al. 2011; Ahamadi et al. 2014), GPassists in identifying relationships that may not be evident from other analytical techniques,and has been used in a wide range of problems, including the field of water resourcesengineering (e.g., Koza 1992; Makkeasorn et al. 2008; Kisi and Shiri 2011; Kisi et al.2012). Aytek and Kisi (2008) used the GP approach for the explicit formulation of dailysuspended sediment-discharge relationship. Aytek et al. (2008) proved the proposed GP

1546 L. Li et al.

formulation performed quite well compared to the results obtained by artificial neuralnetwork (ANN) in rainfall-runoff modeling. GP has been used to develop help deriveoptimal pumping strategies for coastal aquifer management (Sreekanth and Datta 2010), totest the structure of hydrologic models (Selle and Muttil 2011), and to estimate dailyreference evapotranspiration at weather stations (Shiri et al. 2012). This study deals withidentifying the explicit nonlinear formulation of reservoir operating rules, which hasseldom been addressed in the literature.

In the rest of this paper, we present the procedures and methodology in Section 2. Themethod is then applied in Section 3 to the case study of multi-reservoir systems, includingChinas Three Gorges cascade hydropower reservoirs and the nearby Qing River cascadehydropower reservoirs. Finally, conclusions are given in Section 4.

2 Methodology

The following steps were used to derive the operating rules of the multi-reservoir system (Fig. 1).

(1) A deterministic operation model was set up and used to produce the optimal operationtrajectory using DP (Section 2.1).

(2) The GP was used to select the input variables for the operating rules in an aggregation-decomposition framework (Section 2.2).

(3) The explicit formulation of the operating rules was identified using a separate GP basedon the optimal operation trajectory and selected input variables obtained from theprevious steps (Section 2.3).

(4) Some parameters of the derived GP formulation were refined using PSO to obtain thefinal operating rules.

Deterministic reservoir operation model

Optimal operation trajectory

Selecting input variableswith GP

Identifying formulationwith GP

Simulating the total output

Selecting input variableswith GP

Identifying formulation withGP

Allocating output into eachcascade reservoirs

Refining the operating rules

Decom

posion

Agg

re ga tio

n

Operating rules

Determining output of individualreservoir with linear relationship

Fig. 1 Sketch of the methodology and procedure

Identifying Explicit Formulation of Operating Rules for Multi-Reservoir 1547

2.1 Deterministic Optimal Operation Model

2.1.1 Objective Function

Supposing the main purpose of the multi-reservoir system is hydropower generation with thegoal of maximizing both of the average annual hydropower generation and assurance rate, theobjective function has two parts, as follows.

(1) Average annual hydropower generation is given by:

max1

K

Xi1

nK Xj1

m

Pi; j t 1

where n is the number of time periods within a year, m is the number of reservoirs, t isthe time interval, Pi, j is the hydropower generation of reservoir j during time period i, andK is the number of the operation years. Hydropower generation is defined as (Liu et al.2011a):

Pi; j min k j Qi; j Hi; j; f max Hi; j 2

where kj denotes the simple conversion coefficient for reservoir j, Qi, j and Hi, j denote therelease discharge and hydraulic head for reservoir j during time period i, respectively. Thefunction fmax(Hi, j) shows that the output is limited by the generation capacity and is afunction of the hydraulic head.

(2) The assurance rate of hydropower generation is given by:

max

#Xj1

m

Pi; jPmin

!

K3

where# j1

mPi; jPmin

!counts the number of times that the total output is greater

thanor equal to the multi-reservoir systems firm output Pmin.

Equations (1) and (3) were integrated into a single objective optimization model that waseasier to solve (Liu et al. 2011b).

max1

K

Xi1

nK Xj1

m

Pi; j gXj1

m

Pi: j

! !t 4

where g j1

mPi; j

!is a penalty function defined as follows:

gXj1

m

Pi; j

!

kXj1

m

Pi; jPmin

!;Xj1

m

Pi; j < Pmin

0;Xj1

m

Pi; j < Pmin

8>>>>>>>>>:

5

1548 L. Li et al.

where k and are both adjustable positive constants that enable the assurance rate ofhydropower generation (Eq. (3)) to meet the demand.

2.1.2 Constraints

The following constraints were used in the model:

(1) The water balance equation.

V i1; j V i; j I i; jQi; j

t 6

where Vi, j and Vi+1, j are the initial and final water storages of reservoir j at time period i,respectively. Ii, j is the inflow of reservoir j during time period i. Note that the currentstudy did not consider water loss to groundwater or evaporation.

(2) Water storage capacity constraints.

VLi; jV i; jVUi; j 7

where VLi, j and VUi, j are the minimum and maximum allowable water storages forreservoir j during the time period i, respectively. VLi, j is usually considered as deadstorage. VUi, j is associated with the flood control water level during the flood season andthe normal water level during non-flood season.

(3) Release constraints.

QLi; jQi; jQUi; j 8

whereQLi, j andQUi, j denote the minimum and maximum reservoir release for reservoir jduring time period i, respectively. The minimum reservoir release is subject to down-stream irrigation, water supply, navigation and ecology requirements (Suen and Eheart2006; Li et al. 2012). The maximum reservoir release depends upon flood control andspillway release capacities.

(4) Local inflow accretions between reservoirs.

I i; j1 Qi; j qi; j 9

where qi, j denotes the inflow from the zone between the upstream reservoir j and thedownstream reservoir j+1 during time period i. Note that the flow time delay from theupstream reservoir to the downstream reservoir was neglected in this long-term operationmodel.

(5) Hydropower output limits.

PLi; jPi; jPUi; j 10

where PLi, j and PUi, j denote the minimum and maximum outputs for reservoir j duringtime period i, respectively.

Identifying Explicit Formulation of Operating Rules for Multi-Reservoir 1549

(6) Initial and final conditions.

ZI ;1 zbi 11

Zi;K1 Z ei 12where z i

b and Zie are the reservoir water elevation at the initial and final time periods,

respectively.

2.1.3 Optimization Method

Since the above deterministic optimization model for a multi-reservoir system is a multi-stageproblem, it can be easily solved with a DP algorithm. As originally developed in its generalform by Bellman (1957), DP decomposes the original problem into sub-problems that aresolved sequentially over each stage. To overcome the curse of dimensionality (Yeh 1985),modified dynamic programming algorithms such as discrete differential dynamic program-ming (DDDP) and the progressive optimality algorithm (POA) have been used to identifynear-optimal solutions (Yeh 1985; Labadie 2004; Liu et al. 2011a). In this study, DDDP andPOA were used together to solve the deterministic optimal operation model to obtain theoptimal long-term operation trajectory (Guo et al. 2011).

2.2 Potential Input Variables

The water storage and inflow were selected as the potential input variables for the operatingrules. Two other potential input variables, namely inflow energy and storage energy, wereselected to describe the available energy or output to reflect the main operating objective ofhydropower generation.

2.2.1 Aggregation-Disaggregation

The general framework for the multi-reservoir system is to determine the total output, then toallocate it into each cascade reservoirs and individual reservoirs, i.e., an aggregation-decomposition method (Hall and Dracup 1970; Archibald et al. 1997).

Aggregation transforms a multiple-reservoir system into an equivalent single reser-voir, which preserves some features of the original system and reduces the computa-tion or data storage burden of large simulation and optimization models (Hall andDracup 1970; Terry et al. 1986; Saad et al. 1994). Valdes et al. (1992) applied thismethod to hydropower systems with energy units rather than water units. Oliveira andLoucks (1997) showed that the aggregation approach was efficient for making deci-sions in water supply systems. Lund and Guzman (1999) proposed the aggregationmethod for reservoirs in series and in parallel, and concluded that the aggregation forreservoirs in series was easier than reservoirs in parallel.

In contrast, decomposition decentralizes the whole system into individual reservoirs (Terryet al. 1986; Saad et al. 1994). Turgeon (1981) decomposed a system of M reservoirs into Msubproblems and noted that the computation time linearly increases with the number of thereservoirs in the system. Valdes et al. (1992) showed that the decomposition method was notonly spatially, but also temporally by deriving daily operating rules from monthly equivalentrules. Lund and Guzman (1999) summarized and proposed analytical decomposition methodsfor reservoirs in series and in parallel. Joint operating rule curves have been proposed in Qing

1550 L. Li et al.

River cascade hydropower reservoirs using aggregation-decomposition to overcome the dis-advantage of applying in a single reservoir (Liu et al. 2011b).

2.2.2 Power Energy Term

For a single reservoir, the inflow energy term is described by:

ri; j k j I i; j Hi; j t 13where ri, j denotes the inflow energy for reservoir j during time period i.

The storage energy is calculated by:

xi; j k jZVLi; j

V i; j

H j V dV 14

where xi, j denotes the storage energy for reservoir j during time period i, Hj(V) denotes thehydraulic head for reservoir j with the storage of V.

Based on the above definitions, the inflow and storage energy terms of the aggregatedreservoir in parallel, including reservoir j and reservoir j+1 during time period i, can bedescribed as follows:

ri; j ri; j1 k j I i; j Hi; j t k j1 I i; j1 Hi; j1 t 15

xi; j xi; j1 k jZVLi; j

V i; j

H j V dV k j1ZVLi; j1

V i; j1

H j1 V dV 16

The inflow and storage energy terms for the multi-reservoir in series must take into accountthat water released from an upstream reservoir can be reused by all downstream reservoirs.Assuming that the water levels of the downstream reservoir j+1 stay unchanged until the waterin upstream reservoir j is used in its entirety, the inflow energy and storage energy ofaggregation cascade reservoirs in series can be calculated as follows:

ri; j ri; j1 k j I i; j Hi; j t k j1 I i; j I i; j1

Hi; j1 t 17

xi; j xi; j1 kiZVLi; j

V i; j

H j V dV k j1 V i; j Hi; j1 k j1ZVLi; j1

V i; j1

H j1 V dV 18

Based on the above definition, a system including reservoirs in both series and parallel cancalculate the energy first in series then in parallel to obtain the total inflow energy Ri andstorage energy Xi during time period i, respectively.

2.3 Genetic Programming

Derived from GA, GP can identify relationships that may not be evident from other analyticaltechniques (Koza 1992; Yang et al. 2008). Whereas traditional GA is useful for identifying

Identifying Explicit Formulation of Operating Rules for Multi-Reservoir 1551

operational parameters, GP is used to develop a description of the operation form. It has beenwidely applied in forecasting and classification (Aytek et al. 2008), data mining (Hejazi andCai 2011; Selle and Muttil 2011), information searching (Shiri et al. 2012), and so on. Thebasic steps of GP are as follows (Fig. 2):

(1) Determine the set of functions and terminals. The set of functions is composed of thestatements, operators, and so on. For example, Boolean functions (AND, OR, NOT),arithmetic functions {+, , , ,

p}. The set of terminals is comprised of the input

variables and the constants.(2) Generate the random initial population after the GP parameters have been set.(3) Calculate and evaluate fitness of individuals based on goodness-of-fit metrics.(4) Generate new offspring or individuals from parents by reproduction, crossover and

mutation.(5) Repeat steps (3) and (4) until generations reach the given number.

We used three measurement indexes, coefficient of determination (CoD), root mean squareerror (RMSE) and average error (AE), to evaluate the derived formulations.

CoD X

xyXx2X

y2q264

3752

19

RMSE X

XY 2N

s20

Gen=0

Create initial random population

Termination criterion satisfied?

Evaluate fitness of each individual inpopulation

Individuals=0

Individuals=M?

Select genetic operation probably

Select one individual or two individuals based onfitness

Perform reproduction or crossover or mutationto create newpopulation

Gen=Gen+1

Designate result

End

Y

N

Gen=Gen+1

Y

N

Crossover or mutation

Fig. 2 Genetic programmingflowchart

1552 L. Li et al.

AE XXY

X 100

N21

where X and Y are the target and the simulated values, respectively; X and Y are the mean ofX and Y, respectively; N is the number of the samples; and x XX ; y YY .

GP yields different formulations in different simulations even with the same data set, andhence, this attribution can be used to select the input variables (Yang et al. 2008). Once theinput variables have been determined, GP can be reused to identify the formulations.

It should be noted that GP is not very powerful in finding coefficients and is particularlycomputationally intensive and thus time consuming for complex problems (Davidson et al.1999). Furthermore, the maximum goodness-of-fit criterion for selection of the operating rulesmay not always be appropriate and simulations of actual operations have revealed that thismethod does not always produce the best rule (Bhaskar and Whitlatch 1980). Therefore, a PSOmethod was used to refine the parameters in the last step.

2.4 Artificial Neural Network

AS ANN can efficiently handle complex nonlinear relationships, it has been suggested andwidely used in deriving reservoir operating rules (Liu et al. 2006; Aytek et al. 2008) and real-time reservoir operations (Chang et al. 2005b). Within the back-propagation neural network, athree-layer construction including input layer, hidden layer and output layer network, can meetthe requirements of general function mapping with high accuracy.

3 Case Study and Results

3.1 TGR Cascade and Qing River Cascade Hydropower Reservoirs

To multi-reservoir hydropower systems, including the TGR cascade hydropower reservoirs onthe Yangtze River, China, and the nearby Qing River cascade hydropower reservoirs, wereselected for this study. With a length of more than 6,300 km, the Yangtze River is the longestriver in China and the third longest in the world. As shown in Fig. 3, the Three GorgesReservoir (TGR) is a backbone project in the development of the Yangtze River. TheGezhouba reservoir, 38 km downstream from the TGR, is a low dam used to regulate thenavigation and also generate hydropower. The TGR and Gezhouba reservoirs form the TGRcascade reservoirs. The Qing River is the first large tributary of the Yangtze River downstreamof the TGR. Three reservoirs on the Qing Riverthe Shuibuya, Geheyan and Gaobazhoureservoirsform the Qing River cascade reservoirs. With the completion of the TGR and theShuibuya reservoir in 2009, the TGR and Qing cascade systems together form one of theworlds largest multi-reservoir hydropower systems, with a total installed capacity of24,200 MW and an average annual hydropower generation of over 100 billion kWh.

These basic reservoir parameters are listed in Table 1, with further detail in Guo et al.(2011). The entire 55-year-long (from 1951 to 2005) observed inflow data from two flow gagestations were used for optimization and simulation in this study, as 55 years limited the abilityto use as training and validation. Furthermore, for comparison purposes this study was anextension of our previous work (Liu et al. 2011b), which used the same time period.Streamflow records from the Yichang flow gage station, about 40 km downstream of theTGR, were used as inflow for the TGR. Streamflow records from the Changyang flow gage

Identifying Explicit Formulation of Operating Rules for Multi-Reservoir 1553

station, about 9 km downstream of the Gezhouba reservoir, were used as the basic streamflowfor the Qing River. The time step used was ten days, a traditional Chinese measure of time.

For long-term operations, flood control is simplified in the by using the maximum waterlevel as a constraint rather than a separate objective function. Only the operating rules of theTGR, Shuibuya and Geheyan reservoirs were studied, because the Gezhouba and Gaobazhoureservoirs can only be used for daily regulation.

3.2 Conventional Operating Rule Curves

Each reservoir uses its own conventional operating rule curves. For example, the conventionaloperating rule curves for the TGR divide the entire storage space is into four operational zones,as shown in Fig. 4. Zone I is the flood control zone. If the water level is above 145 m (floodcontrol limited level) during the flood period, the reservoir operators lower the water level to145 m to create flood storage space. If the water level lies in Zone II, the reservoir can generateas much hydropower as possible, i.e., at maximum capacity. If the water level lies in Zone III,

Table 1 List of the characteristic parameters of the studied multi-reservoir system

Reservoir Three Gorges Gezhouba Shuibuya Geheyan Gaobazhou

Total storage(billion m3) 393 15.8 42 34 5.4

Flood control storage(billion m3) 221.5 N/A 5 5 N/A

Crest elevation(m) 185 70 409 206 83

Normal pool level(m) 175 66 400 200 80

Flood control limited level(m) 145 N/A 391.8 193.6 N/A

Installed capacity(MW) 22400 2930 1840 1212 270

Firm output(MW) 4990 1040 310 241.5 77.3

Regulation ability Seasonal Daily Multiyear Annual Daily

Fig. 3 Location of the Three Gorges cascade reservoirs and the Qing River cascade reservoirs in China

1554 L. Li et al.

the hydropower station generates at firm capacity (4,990 MW). Otherwise, if the waterlevel lies in Zone IV, the hydropower station must reduce output to keep the water levelas high as possible to increase hydraulic head for generating efficiency. The conventionaloperating rule curves of the Shuibuya and Geheyan reservoirs are described by Liu et al.(2011b).

The TGR cascade and Qing River cascade reservoirs were simulated using their existingoperating rule curves with the 55-year inflow observations. Simulated average annual hydro-power generation was 103.17 billion kWh and hydropower generation assurance rate was95.36 % for the simulation period (Table 2).

The above operating rule curves have been applied in practice due to their ease ofimplementation. However, the conventional rule curves were developed independently foreach reservoir without considering the complex relationship between the upstream anddownstream reservoirs in series or the reservoirs in parallel. The optimization method devel-oped in this study was used to increase both the average annual hydropower generation and theassurance rate of hydropower generation simultaneously.

3.3 Deterministic Optimization

A deterministic optimization model was developed for the multi-reservoir system with theobjective function of Eq. (4) and constraints from Eqs. (6) to (12). Using the initial solutionobtained with DDDP, an additional optimization model was developed using the POAmethod,which can provide a near-optimal solution.

Month Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr MayUpper boundary curve 145 145 145 145 175 175 175 175 175 175 175 155Lower boundary curve 145 145 145 145 145 156.3 169.6 166.6 160.9 155 155 145

145

150

155

160

165

170

175

wat

er le

vel(m

) upper boundarylower boundary

Fig. 4 Conventional operating rule curves of the Three Gorges Reservoir

Table 2 Comparison of operation results with different schemes

Schemes Average annual hydropower generation (billion kWh) AssuranceRate (%)

Qing River Cascade Three Gorges Cascade Total

Shuibuya Geheyan Gaobazhou Three Gorges Gezhouba

Rule curves 3.57 2.84 0.86 81.36 14.54 103.17 95.36

ANN 3.50 2.91 0.95 81.06 14.97 103.39 96.77

GP 3.57 2.91 0.95 81.57 14.99 103.99 96.83

Refined 3.66 3.02 0.95 81.84 15.04 104.51 97.26

Deterministic optimization 3.75 3.17 0.95 83.74 15.04 106.65 99.04

Identifying Explicit Formulation of Operating Rules for Multi-Reservoir 1555

Optimization results show that the average annual hydropower generation is 106.65billion kWh and the assurance rate of the hydropower generation is 99.04 % (Table 2).The near-optimal solution was a potential maxima with perfect operation, and alsoused as the optimal operation trajectory for analyzing the operating rules using the GPmethod.

3.4 GP Operating Rules

3.4.1 General Operating Rules Form

Based on the aggregation-decomposition method, the operating rules operate a three-stage procedure: (1) determine the total output of the system (aggregation formula-tion); (2) decompose it into each cascade reservoirs (decomposition formulation); and(3) decide the output of the individual reservoir. For the third stage, the output of theupstream reservoir is linear related to the output of cascade reservoirs (Liu et al.2011b). These relationships are plotted for the Qing River and the TGR cascadereservoirs in Figs. 5a and b, respectively. These relationships were then used toallocate the output of the cascade reservoirs into individual reservoirs.

In this study, the purpose of the formulation identification is to determine the total output ofthe multi-reservoir system and how to subsequently decompose total output into each cascadereservoirs. For the latter (decomposition formulation), an allocation hydropower Pi,k was usedto describe the output of each cascade reservoir, namely P1,k and P2,k denote the output for theQing River and TGR cascade output for time period i, respectively.

3.4.2 Input Variables Selection

Many formulations could be identified using GP after one calculation. The frequencyis ratio of the number of each input variables occurrence to the total number of theformulations. The higher the frequency is, the more likely the input variable will beselected. The final input variables can be chosen as follows: (1) Identify the primaryinput variables according to the frequency of each input variable with GP. (2)Calculate the CoD and RMSE of each combination of the primary variable with GP

Fig. 5 Relationship between the output of the upstream reservoir and the total output of cascade reservoirs. aShuibuya Reservoir vs. the Qing River cascade reservoirs, b Three Gorges Reservoir vs. Three Gorges cascadereservoirs

1556 L. Li et al.

again; the combination with highest CoD and lowest RMSE would be selected as thefinal input variables.

In order to determine the total output of the multi-reservoir system and decomposeit into each cascade reservoirs, four potential input variables (water storage, inflow,storage energy and inflow energy) were selected as the potential input variables forthe operating rules, as discussed above.

Based on the optimal operation trajectory, the total inflow energy Ri j1

mri; j and total

storage energy X i j1

mxi; j were calculated using Eqs. (15) to (18).

GP was employed to select the best input variables to use from among the potential inputvariables. Table 3 shows the parameters used for the GP implementation. Based on the methodproposed by Yang et al. (2008), GP simulations were repeated 100 times (a high number),which resulted in similar fitness values but with different input variables among the variousruns. The frequency at which a potential input variable appeared in those simulation outputswas recorded. Table 4 shows that the total inflow energy Ri and total storage energy Xi had thehighest frequency for both total system output and allocation output. That is, the multi-reservoir system was best aggregated in hydropower units (energy) rather than water units(flow).

Several input combinations were employed to fit the optimal operation trajectory with GP(Table 5). Since the combination of Ri and Xi had the highest CoD and the lowest RMSE, thesewere chosen as the final input variables.

3.4.3 Formulation Identification

The explicit formulation of the operating rules was identified using GP based on the selectedinput variables. To determine the total output of the multi-reservoir system (the first stage ofthe operating rules), the following formulation was derived:

Pi Ri aiX i bi X iRiciX 1:5i

R2i X 2:5i X iRiX 1:5ii 1; 2;; n 22

where Pi is the predicted total output of the multi-reservoirs, ai, bi and ci are parameters for thetime period i. Note that each coefficient for different time periods was the same in Eq. (22),though they varied when the refinement was employed, as discussed below.

Table 3 Genetic programmingparameters Parameter Value

population size 250

number of children to produce 500

crossover probability 0.9

mutation probability 0.05

function set {+, , , ,p}

maximum size at initialization 15

maximum size 45

Identifying Explicit Formulation of Operating Rules for Multi-Reservoir 1557

In the optimal form identified by the GP (Eq. 22), the GP also estimated initial parameters(a=0.29, b=5.66, c=0.083). With Xi as a constant, the following formulation can be derived:

PiRi

1 bX i R2i X

2:5i X 1:5i 2cX 0:5i RicX 1:5i

R2i RiX i X 2:5i X 1:5i 2

R2i RiX i X 2:5i X 1:5i 2 bX iR2i bX 3:5i bX 2:5i 2bcX 1:5i RibcX 2:5i

R2i RiX i X 2:5i X 1:5i 2

>R2i RiX i X 2:5i X 1:5i 2bX 3:5i 2bcX 1:5i

R2i RiX i X 2:5i X 1:5i 2 > X 2:5i X 1:5i

2bX 3:5i 2bcX 1:5iR2i RiX i X 2:5i X 1:5i 2

X5i 2X

4i X 3i bX 3:5i 2bcX 1:5i

R2i RiX i X 2:5i X 1:5i 2 > X 5i 2X 4i bX 3:5i

R2i RiX i X 2:5i X 1:5i 2

X3:5i X

1:5i 2X

0:5i b

R2i RiX i X 2:5i X 1:5i 2 > 0 23

In this case, the Eq. (23) is always greater than 0 and therefore the total output is amonotonic increasing function of inflow energy, which is suitable for our knowledge.

For the second stage in determining the optimal operating rules, the GP method was againused, with the following formulation obtained to decompose the total output into the TGRcascade reservoirs:

Pi;2 Ri

Ri

Ri diR5i

X 4:25i R

2i

eiX i f iRi X i

vuuut i 1; 2;; n 24

Table 5 Selection the input variables with GP model

Input variables Aggregation Decomposition

CoD RMSE AE CoD RMSE AE

Ri 0.79 2662.26 41.35 % 0.80 2149.57 37.65 %Xi 0.32 4545.54 50.73 % 0.29 4022.90 49.83 %Ri+Xi 0.73 2901.45 45.68 % 0.76 2759.13 42.58 %Ri,Xi 0.98 690.52 10.94 % 0.96 967.94 13.91 %

Table 4 Input metrics identified byGP and frequency analysis Input variables Probability of GP results (%)

Aggregation Decomposition

Qi 52 46

Vi 46 53

Ri 96 93

Xi 90 85

1558 L. Li et al.

where di, e

iand f

idenote the parameters for the time period i. Note that

k1

2

Pi;k Pi i 1; 2;; n .

The attribution of the decomposition formulation is similar to the aggregation formulation(Eq. 22). Figs. 6 and 7 show the strong relationship among the output, inflow energy andstorage energy in the aggregation formulation (Eq. 22) and decomposition formulation(Eq. 24), respectively. As shown in the 3D graphs, the formulations were suitable to fit optimaloperation trajectories. Aggregation and decomposition outputs both increase with increasinginflow energy and/or storage energy (as discussed in Eq. 23).

The coefficients ai, bi, ci, di, ei and fi for each time period were identified using the Simplexand Powell algorithm, a nonlinear optimization method (Simonovic 1987; Liu et al. 2006,2011b). Figures 8 and 9 plot simulated total output versus optimal output for each of the totalmulti-reservoir system and only the TGR cascade system, respectively, for different represen-tative time periods. In all cases, the aggregation and decomposition formulations have high R2

values, ranging from 0.91 to 0.97. Therefore, the derived GP formulations were efficient to fitthe optimal operation trajectory.

The derived multi-reservoir system operating rules, with input variables, form, and initialoperating rule parameters identified, were then used in a simulation model to guide thereservoirs operation. Simulation results showed that average annual hydropower generationof 103.99 billion kWh and an assurance rate of the hydropower generation is 96.83 % can beachieved (Table 2).

3.4.4 Operating Rules Refinement

Since the derived multi-reservoir system operating rules are multivariable, highly nonlinear,and unconstrained, they can be refined using nonlinear programming (Simonovic 1987). Thenonlinear Simplex (e.g., Liu et al. 2006, 2011b) and Powell algorithm (e.g., Simonovic 1987)

Fig. 6 Aggregation rules for the multi-reservoir system of the Three Gorges cascade hydropower reservoirs andQing River cascade hydropower reservoirs

Identifying Explicit Formulation of Operating Rules for Multi-Reservoir 1559

was used together to maximize the objective function (Eq. (4)) to adjust and improve theoperating rule parameters.

With this further refinement of the operating rule parameters, average annual hydropowergeneration increased from 103.99 billion kWh to 104.51 billion kWh, while the assurance rateof hydropower generation increased from 96.83 % to 97.26 % (Table 2).

3.5 ANN Operating Rules

The current and previous inflows and storage volumes of Shuibuya, Geheyan and ThreeGorgeswereselected as the input variables, while the releases of the three reservoirs were selected as the targetvariables. The three-layer networkswere used to simulate the relationship between input variables andtarget variables. ANN operating rules show that the average annual hydropower generation is 103.39billion kWh and the assurance rate of the hydropower generation is 96.77 % (Table 2).

Fig. 7 Decomposition rules for the Three Gorges cascade hydropower reservoirs based upon the total storageand inflow energy

Fig. 8 Comparison of the simulated total output and the optimal one for the multi-reservoir system. a the first10 days in November, b the middle 10 days in November, c the last 10 days in November

1560 L. Li et al.

3.6 Comparison and Discussions

Table 2 summarized the above four schemes, including the conventional operating rulecurves, deterministic optimization, the ANN, the original GP, and the GP withrefinement for both average annual hydropower generation and hydropower generationassurance. It can be obviously seen from Table 2 that the GP method performs betterthan the conventional operating rule curves and the ANN approach. Compared withthe conventional operating rule curves and ANN approach, the average annual hydro-power generation increases from 103.17 billion kWh, 103.39 billion kWh to 104.51

Fig. 9 Comparison of simulated TGR cascade output and the optimal one for the Three Gorges cascadereservoirs. a the first 10 days in October, b the middle 10 days in October, c the last 10 days in October

(a)

Shuibuya Reservoir(b)

Three Gorges Reservoir

Fig. 10 Comparison of water levels with different operation schemes in 1985

Identifying Explicit Formulation of Operating Rules for Multi-Reservoir 1561

billion kWh, while the firm output assurance rate increases from 95.36 %, 96.77 % to97.26 %.

Table 4 shows the identified input variable by GP frequency method. The totalinflow energy Ri and total storage energy Xi have higher frequency (greater than85 %). Table 5 shows the combination of the input variables selected with GP model.The combination Ri,Xi have the highest CoD and lowest RMSE and AE and peformbetter than the combination Ri+Xi (available energy) both in agrregation and decom-position. It can be concluded that: (1) the power energy term could more effectivelyreflect the operating rules than water quantity for the hydropower systems, and (2) therelationship between the target variable (output or release) and input variable (inflowenergy and storage energy) is nonlinear other than linear.

Figure 10 shows the water levels for the four schemes in year 1985. The conventionaloperating rule curves operate the reservoir based on the single current storage and withoutconsidering states of other reservoirs in the multi-reservoir system. By contrast, GPcontrols the water levels of Shuibuya reservoir based upon the water levels of TGR.As shown in Fig. 10a, GP usually generated higher water levels than the conventionaloperating rule curves during non-flood seasons and lower water levels during floodseasons. For example, the Shuibuya reservoir kept more water to raise the water headfor hydropower generation and reduced the spilled water to make full use of waterresources, with its output was shown in (Fig. 11a). Similarly, as shown in Fig. 10band Fig. 11b, the TGR remained higher water levels during non-flood seasons to improvethe hydropower output.

Shuibuya Reservoir output

Three Gorges Reservoir output

(a)

(b)

Fig. 11 Comparison of hydropower output with different operation schemes in 1985

1562 L. Li et al.

4 Conclusions

In this paper, the GP method was employed to derive the explicit formulation ofmulti-reservoir system operating rules. With a case study of Chinas Three Gorgescascade hydropower reservoirs and Qing River cascade hydropower reservoirs,following conclusions can be drawn:

(1) The multi-reservoir system can be transformed into an equivalent reservoir and theoperating rules can be simplified with aggregation-decomposition techniques. For ahydropower system, the total output and its decomposition were both related to theinflow and storage energy terms.

(2) The proposed GP formulation increased the average annual hydropower generation from103.17 billion kWh, 103.39 billion kWh to 104.51 billion kWh, as well as the assurancerate of hydropower generation from 95.36 %, 96.77 % to 97.26 % compared with theconventional operating rule curves and ANN rules using the observed inflow from 1951to 2005, indicating that the proposed GP formulation had potential implications for theoperating rules of multi-reservoir systems.

However, it should be noted that the physical explanation of the identified GP formulationis not explicit, which needs further research.

Acknowledgments This study was supported by the Program for New Century Excellent Talents in University(NCET-11-0401), the National Natural Science Foundation of China (51190094) and Non-Profit IndustryFinancial Program of Ministry of Water Resources (201201051).

References

Ahamadi M, Bozorg Haddad O, Mario MA (2014) Extraction of flexible multi-objective real-time reservoiroperation rules. Water Resour Manag 28(1):131147

Archibald TW, McKinnon KIM, Thomas LC (1997) An aggregate stochastic dynamic programming model ofmulti-reservoir systems. Water Resour Res 33(2):333340

Aytek A, Kisi O (2008) A genetic programming approach to suspended sediment modeling. J Hydrol 351:288298

Aytek A, Asce M, Alp M (2008) An application of artificial intelligence for rainfall-runoff modeling. J Earth SystSci 117(2):145155

Bellman R (1957) Dynamic programming. Princeton University Press, PrincetonBhaskar NR, Whitlatch EE (1980) Derivation of monthly reservoir release policies. Water Resour Res 16(6):

987993Celeste AB, Billib M (2009) Evaluation of stochastic reservoir operation optimization models. AdvWater Resour

32(9):14291443Chandramouli V, Raman H (2001) Multi-reservoir modeling with dynamic programming and neural networks. J

Water Resour Plan Manag ASCE 127(2):8998Chang LC (2008) Guiding rational reservoir flood operation using penalty-type genetic algorithm. J Hydrol 354:

6574Chang LC, Chang FJ (2009) Multi-objective evolutionary algorithm for operating parallel reservoir system. J

Hydrol 337:1220Chang FJ, Chen L, Chang LC (2005a) Optimizing the reservoir operation rule curves by genetic algorithms.

Hydrol Process 19(11):22772289Chang YT, Chang LC, Chang FJ (2005b) Intelligent control for modeling of real time reservoir operation: part II

ANN with operating curves. Hydrol Process 19(7):14311444Chaves P, Kojiri T (2007) Deriving reservoir operational strategies considering water quantity and quality

objectives by stochastic fuzzy neural networks. Adv Water Resour 30(5):13291341

Identifying Explicit Formulation of Operating Rules for Multi-Reservoir 1563

Chen L, McPhee J, Yeh WW-G (2007) A diversified multi-objective GA for optimizing reservoir rule curves.Adv Water Resour 30(5):10821093

Chiu YC, Chang LC, Chang FJ (2007) Using a hybrid genetic algorithmsimulated annealing algorithm forfuzzy programming of reservoir operation. Hydrol Process 21(23):31623172

Davidson JW, Savic D, Walters GA (1999) Method for the identification of explicit polynomial formulae for thefriction in turbulent pipe flow. J Hydroinformatics 1(2):115126

Guo S, Zhang H, Chen H, Peng D, Liu P, Pang B (2004) Reservoir flood forecasting and control system in China.Hydrol Sci J 49(6):959972

Guo S, Chen J, Li Y, Liu P, Li T (2011) Joint operation of the multi-reservoir system of the Three Gorges and theQingjiang cascade reservoirs. Energies 4(7):10361050. doi:10.3390/en4071036

Hall WA, Dracup JA (1970) Water resources systems engineering. McGraw-Hill, New YorkHejazi MI, Cai X (2011) Building more realistic reservoir optimization models using data mining - a case study

of Shelbyville reservoir. Adv Water Resour 34(6):701717Huang WC, Yuan LC (2004) A drought early warning system on real-time multi-reservoir operations. Water

Resour Res 40(6), W06401. doi:10.1029/2003WR002910KaramouzM, HouckMH, Delleur JW (1992) Optimization and simulation of multiple reservoir systems. J Water

Resour Plan Manag ASCE 118(1):7181Kim T, Heo JH, Jeong CS (2006) Multi-reservoir system optimization in the Han River basin using multi-

objective genetic algorithms. Hydrol Process 20(9):20572075Kisi O, Shiri J (2011) Precipitation forecasting using wavelet-genetic programming and wavelet-neuro-fuzzy

conjunction models. Water Resour Manag 25(13):31353152Kisi O, Dailr AH, Cimen M, Shiri J (2012) Suspended sediment modeling using genetic programming and soft

computing techniques. J Hydrol 450451:4858Koutsoyiannis D, Economou A (2003) Evaluation of the parameterization simulation - optimization approach

for the control of reservoir systems. Water Resour Res 39(6):1170. doi:10.1029/2003WR002148Koza J (1992) Genetic programming: on the programming of computers by means of natural selection. The MIT

Press, CambridgeKumar ARS, Goyal MK, Ojha CSP, Singh RD, Swamee PK, Nema R (2013) Application of ANN, fuzzy logic and

decision tree algorithms for the development of reservoir operating rules. Water Resour Manag 27(3):911925Labadie J (2004) Optimal operation of multi-reservoir systems: state-of-the-art review. J Water Resour Plan

Manag ASCE 130(2):93111Li Q, Yu M, Zhao J, Cai T, Lu G, Xie W, Bai X (2012) Impact of the Three Gorges reservoir operation on

downstream ecological water requirements. Hydrol Res 43(12):4853Liu P, Guo S, Xiong L, Li W, Zhang H (2006) Deriving reservoir refill operating rules by using the proposed

DPNS model. Water Resour Manag 20(3):337357Liu P, Cai X, Guo S (2011a) Deriving multiple near-optimal solutions to deterministic reservoir operation

problems. Water Resour Res 47(8), W08506. doi:10.1029/2011WR010998Liu P, Guo S, Xu X, Chen J (2011b) Derivation of aggregation-based joint operating rule curves for cascade

hydropower reservoirs. Water Resour Manag 25(13):31773200Lu B, Gu H, Xie Z, Liu J, Ma L, Lu W (2012) Stochastic simulation for determining the design flood of cascade

reservoir systems. Hydrol Res 43(12):5463Lund JR, Guzman J (1999) Derived operating rules for reservoirs in series or in parallel. J Water Resour Plan

Manag ASCE 125(3):143153Makkeasorn A, Chang NB, ZhouX (2008) Short-term streamflow forecasting with global climate change implications -

a comparative study between genetic programming and neural network models. J Hydrol 352:336354Nalbantis I, Koutsoyiannis D (1997) A parametric rule for planning and management of multiple - reservoir

systems. Water Resour Res 33(9):21652177Ngo LL, Madsen H, Rosbjerg D (2007) Simulation and optimization modeling approach for operation of the Hoa

Binh reservoir, Vietnam. J Hydrol 336:269281Oliveira R, Loucks DP (1997) Operating rules for multireservoir systems. Water Resour Res 33(4):839852Rani D, Moreira MM (2010) Simulationoptimization modeling: a survey and potential application in reservoir

systems operation. Water Resour Manag 24(6):11071138Saad M, Turgeon A, Bigras P, Duquette R (1994) Learning disaggregation technique for the operation of long-

term hydroelectric power systems. Water Resour Res 30(11):31953202Saad M, Bigras P, Turgeon A, Duquette R (1996) Fuzzy learning decomposition for the scheduling of

hydroelectric power systems. Water Resour Res 32(1):179186Selle B, Muttil N (2011) Testing the structure of a hydrological model using Genetic Programming. J Hydrol 397:19Shiri J, Kisi O, Landeras G, Lopez JJ, Nazemi AH, Stuyt LC (2012) Daily reference evapotranspiration modeling by

using genetic programming approach in the Basque Country (Northern Spain). J Hydrol 414415:302316Simonovic SP (1987) The implicit stochasticmodel for reservoir yield optimization.Water Resour Res 23(12):21592165

1564 L. Li et al.

Simonovic SP (1992) Reservoir systems analysis: closing gap between theory and practice. J Water Resour PlanManag ASCE 118(3):262280

Sreekanth J, Datta B (2010) Multi-objective management of saltwater intrusion in coastal aquifers using geneticprogramming and modular neural network based surrogate models. J Hydrol 393:245256

Stedinger JR, Sule BF, Loucks DP (1984) Stochastic dynamic programming models for reservoir operationoptimization. Water Resour Res 20(11):14991505

Suen JP, Eheart JW (2006) Reservoir management to balance ecosystem and human needs: incorporating theparadigm of the ecological flow regime. Water Resour Res 42(3), W03417. doi:10.1029/2005WR004314

Suiadee W, Tingsanchali T (2007) A combined simulation-genetic algorithm optimization model for optimal rulecurves of a reservoir: a case study of Nam Oon Irrigation Project, Thailand. Hydrol Process 21(23):32113225

Terry LA, Pereira MVF, Araripe NTA, Silva LFCA, Sales PRH (1986) Coordinating the energy generation of theBrazilian national hydrothermal electrical generating system. Interfaces 16(1):1638

Turgeon A (1981) A decomposition method for the long term scheduling of reservoirs in series. Water ResourRes 17(6):15651570

Valdes J, Montbrun-Di FJ, Strzepek K, Restrepo P (1992) Aggregation - disaggregation approach to multi -reservoir operation. J Water Resour Plan Manag ASCE 118(4):423444

Wang KW, Chang LC, Chang FJ (2011) Multi-tier interactive genetic algorithms for the optimization of long-term reservoir operation. Adv Water Resour 34:13431351

Wardlaw R, Sharif M (1999) Evaluation of genetic algorithms for optimal reservoir system operation. J WaterResour Plan Manag ASCE 125(1):2533

Wei CC, Hsu NS (2008) Derived operating rules for a reservoir operation system: comparison of decision trees,neural decision trees and fuzzy decision trees. Water Resour Res 44(2), W02428. doi:10.1029/2006WR005792

Wurbs RA (1993) Reservoir-system simulation and optimization models. J Water Resour Plan Manag ASCE119(4):445472

Yang YE, Cai X, Herricks EE (2008) Identification of hydrologic indicators related to fish diversity andabundance: a data mining approach for fish community analysis. Water Resour Res 44(4), W04412. doi:10.1029/2006WR005764

Yeh WW-G (1985) Reservoir management and operations models: a state-of-the-art review. Water Resour Res21(12):17971818

Young GK (1967) Finding reservoir operating rules. J Hydraul Div 93(6):297321

Identifying Explicit Formulation of Operating Rules for Multi-Reservoir 1565

Identifying Explicit Formulation of Operating Rules for Multi-Reservoir Systems Using Genetic ProgrammingAbstractIntroductionMethodologyDeterministic Optimal Operation ModelObjective FunctionConstraintsOptimization Method

Potential Input VariablesAggregation-DisaggregationPower Energy Term

Genetic ProgrammingArtificial Neural Network

Case Study and ResultsTGR Cascade and Qing River Cascade Hydropower ReservoirsConventional Operating Rule CurvesDeterministic OptimizationGP Operating RulesGeneral Operating Rules FormInput Variables SelectionFormulation IdentificationOperating Rules Refinement

ANN Operating RulesComparison and Discussions

ConclusionsReferences