Optimization and adjustment policy of two-echelon reservoir inventory management with forecast updates

Computers & Industrial Engineering 63 (2012) 890–900

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Optimization and adjustment policy of two-echelon reservoir inventory management with forecast updates q Yingcheng Xu a,b, Li Wang a, Zhisong Chen c, Siqing Shan a,⇑, Guoping Xia a a

School of Economics and Management, Beihang University, Beijing 100191, China China National Institute of Standardization, Beijing 100088, China c Business School of Nanjing Normal University, Nanjing Jiangsu 210097, China b

a r t i c l e

i n f o

Article history: Received 19 January 2012 Received in revised form 10 May 2012 Accepted 9 June 2012 Available online 16 June 2012 Keywords: Reservoir Inflow forecasts MMFE DP Adjustment

a b s t r a c t A finite-horizon, periodic-review inventory model with inflow forecasting updates following the Martingale Model of Forecast Evolution (MMFE) in multiresevoirs is considered. This model introduces a new method of determining an operating policy in which the policy is based on the Dynamic Programming (DP) model with a physical equation and a recursive equation. Some important constraints are described for the multireservoirs operation. We propose two methods to adjust the decision variable during the optimization. The adjustment policy is essential to improve the profit and decrease the shortage in multireservoirs management. Finally, to assess the effectiveness of the policies, the model is compared with other models and is applied to the Chinese South–North Water Diversion project. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Water is a necessary resource for production and everyday life. The amount needed increases with the economics development and improvements in the standard of living. Yet the distribution of water spatially does not usually coincide with human requirements. At a certain stage of water resource development, therefore, inter-basin transfers become inevitable. Water resources problems are bound to become more complex worldwide in the future. Population growth, climate variability, regulatory requirements, project planning horizons, temporal, and spatial scales, social and environmental considerations, transboundary considerations, etc., all contribute to the complexity of water resources planning and management. Systems analysis techniques can be highly valuable tools for solving planning and operating tasks in water resources management based on the systematic and efficient organization and analysis of relevant information. A variety of methods in systems analysis or operations research have been developed for analyzing water resources systems. Optimization and simulation are used conjunctively to derive and assess alternative operating strategies for single and multiple reservoir systems (e.g., Jacoby & Loucks, 1972; Mawer & Thorn, 1974; Gal, 1979; Karamouz & Houck, q

This manuscript was processed by Area Editor (Qiuhong Zhao).

⇑ Corresponding author. Tel.: +86 10 8231 4911. E-mail address: [email protected] (S. Shan).

0360-8352/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2012.06.003

1982, 1987; Stedinger, Sule, & Loucks, 1984; Tejada-Guibert, Johnson, & Stedinger, 1993; Harboe, Gautam, & Onta, 1995; Liang, Johnson, & Yu, 1996). Most of the available research only considers the entire multireservoir systems optimization issue based on group rationality without sufficiently considering the internal relationship among reservoirs. In particular, it needs to consider the following questions: how to allocate scarce water resources to reservoirs considering priority and equitable allocation principles when the total supply of water resources is less than the total demand for water resources? How to transform a flood into a useful resource when a downstream reservoir is faced with a flood and an upstream reservoir is faced with a water shortage? How to ‘‘produce’’ and allocate the water resource to satisfy the enddemand, reduce water shortages, and achieve both economic and social benefits? These are core issues in reservoir operations management, and these will form the main focus of our study. This paper studies the optimal operation of multireservoirs considering the internal relationship among reservoirs through adjusting the release and spilling strategy. The rest of the paper is organized as follows. Section 2 briefly reviews some of the literature on reservoirs based on different optimization methods. Section 3 introduces the Martingale Model of Forecasting Evolution (MMFE). Section 4 presents the model and formulation for two reservoirs. Adjustment policy is proposed in Section 5. Section 6 develops an application of the model to the SNWD (South–North Water Diversion project). Finally, Section 7 concludes the paper and summarizes the results.

Y. Xu et al. / Computers & Industrial Engineering 63 (2012) 890–900

2. Literature review A great deal of research has been conducted on multireservoir optimal operations in inter-basin water diversion projects. This research mainly focuses on the mathematical programming method, the aggregation–decomposition approach, the GRID computing approach, optimal control theory, the minimum norm approach, the discrete maximum principle, and the linear decision rule. Jairaj and Vedula (2000) applied fuzzy mathematical programming to multireservoir system optimization. Halliburton and Sirisena (1984), Sherkat, Campo, Moslehi, and Lo (1985), Li, Yan, and Zhou (1990), Contaxis and Kavatza (1990), Turgeon (2007), Tilmant, Pinte, and Goor (2008) applied the stochastic programming method to multireservoir operations and scheduling. Nayak and Arora (1973) applied the nonlinear programming method to multireservoir system capacity decisions. Mohan and Raipure (1992) developed a linear multiobjective programming model for multiobjective analysis of multireservoir systems. Wei and Hsu (2008) applied the mixed-integer linear programming method to multireservoir realtime operations for flood control. Khaliquzzaman (1997) developed a network flow programming model for multireservoir sizing. Recently, more and more scholars have focused on stochastic dynamic programming and have obtained many positive results in multireservoir systems. The stochastic nature of the inflows can be handled by two approaches (Nandalal & Bogardi, 2007): implicit and explicit. In the implicit approach, a time series model is used to generate a number of synthetic inflow sequences. The system is optimized for each streamflow sequence and the operating rules are found by multiple regression. During optimization, the synthetic data series are considered deterministic series. The explicit approach considers the probability distribution of the inflows rather than specific flow sequences. This approach generates an operation policy comprising storage targets or release decisions for every possible reservoir storage and inflow state in each time step rather than a single schedule of reservoir releases. Young (1967) proposed an implicit stochastic approach to optimize the operation of a single reservoir. Opricovic and Djordjevic (1976) presented an implicit SDP-based algorithm for optimal long-term control of a single multipurpose reservoir with both direct and indirect users. Karamouz and Houck (1987) derived monthly operating rules for a set of 12 different single-reservoir test cases using their iterative DP model (Karamouz & Houck, 1982) and an explicit SDP optimization model. The sampling stochastic dynamic programming approach (SSDP), first used by Araujo and Terry (1974) for the operation of a hydro system, can also be categorized as an implicit stochastic approach. SSDP was used by Dias, Pereira, and Kelman (1985) for the optimization of flood control and power generation requirements in a multipurpose reservoir. Kelman, Stedinger, Cooper, Hsu, and Yuan (1990) included best inflow forecast as a hydrological state variable in the SSDP algorithm. Faber and Stedinger (2001) compared SSDP models employing the National Weather Service’s (NWS) ensemble streamflow prediction (ESP) forecasts to SSDP models based on historical streamflows and snowmelt volume forecasts. Butcher (1971) used explicit SDP to derive an optimal long-term operating strategy for a single multipurpose reservoir. Loucks and van Beek (2005) elaborated the explicit SDP approach for the optimization of single-reservoir operation. Maidment and Chow (1981) developed two SDP optimization models for a single-reservoir operation problem. Stedinger et al. (1984) compared simulation results based on different operation policies derived from the High Aswan Dam on the River Nile by five SDP based optimization models. Goulter and Tai (1985) used SDP to model a small hydroelectric system. Shrestha (1987) applied SDP to derive optimal operation policies for different configurations of a hydropower system during the

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planning stage. Bogardi, Budhakooncharoen, Shrestha, and Nandalal (1988) investigated the impact of varying the number of storage and inflow classes on the operational performance of SDP for both single and multiple reservoir systems. Shrestha, Bogardi, and Paudyal (1990) studied the effect of the number of discrete characteristic states and the impact of varying the definition of these characteristic states on SDP model performance. Huang, Harboe, and Bogardi (1991) compared four explicit SDP optimization models using the operation of the Feitsui Reservoir in Taiwan as a case. Tejada-Guibert et al. (1993, 1995) applied stochastic dynamic programming to multireservoir operating policies and hydrologic information value analysis. Vasiliadis and Karamouz (1994) adopted both the present period inflow and the next period inflow forecast as hydrological state variables. Bogardi, Brorens, Kularathna, Milutin, and Nandalal (1995) developed a model called ‘‘ShellDP’’ based on stochastic dynamic programming and simulation techniques for analyzing multiunit reservoir systems. The model is applicable during both design and operational stages of a reservoir system. Ampitiya (1995) applied the ShellDP package to derive optimal operation policies for reservoirs in the complex Mahaweli water resources scheme in Sri Lanka. Nandalal and Ampitiya (1997), Nandalal (1998), and Nandalal and Sakthivadivel (2002) used this modified model to derive operation policies for reservoirs in several water resources development schemes in Sri Lanka. Archibald, McKinnon, and Thomas (1997, 2006) developed an aggregation and decomposition stochastic dynamic programming model for multireservoir systems operations. Tilmant, Vanclooster, Duckstein, and Persoons (2002) compared reservoir operation policies obtained from fuzzy and nonfuzzy explicit stochastic dynamic programming. Kumar and Baliarsingh (2003) developed a folded dynamic programming for optimal operation of multireservoir system. A fuzzy stochastic dynamic programming model (FSDP) (Mousavi, Karamouz, & Menhadj, 2004) was also developed for a single reservoir to model the errors associated with discretizing the variables using fuzzy set theory. Kim, Hyung, and Lee (2007) applied sampling stochastic dynamic programming to optimizing operational policies for multireservoir system. Goor, Kelman, and Tilmant (2011) applied stochastic dual dynamic programming to multipurpose–multireservoir operations. In existing literatures, there are some limits on the multireservoir inventory management as follows: (1) The interaction among multireservoir is one of the major motivations and it didn’t be discussed in details. (2) Several types of reservoir operating rules have been suggested such as linear decision rule, stochastic dynamic programming, standard operating policy (SOP), hedging rules and so on. But the literatures for the operating rules with forecast updates are rare. (3) In order to save the cost and reduce the spill of water resource, the adjustment policy for the multireservoir from the local and global analysis has not been discussed. In this paper, we make two major contributions to the literature. The first is the development of model and formulation for two-echelon reservoir inventory management with forecast updates, using an approach of Martingale Model of Forecasting Evolution (MMFE). The second contribution is the development of two methods to adjust the decision variable during the optimization. The adjustment policy is essential to improve the profit and decrease the shortage in multireservoirs management. 3. Martingale Model of Forecasting Evolution (MMFE) In hydrology, there are various indices reflecting the magnitude of streamflow forecast uncertainty. However, few models illustrate

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the forecast uncertainty evolution process. This paper adopts MMFE from supply chain management to quantify the evolution of the uncertainty of real-time streamflow forecasts as time progresses. Iida and Zipkin (2006) considered a dynamic forecastinventory model with forecast updates, based on the martingale model of forecast evolution, and proposed two types of updates, additive and multiplicative. They further developed a piecewiselinear approximation of the cost functions and a simulation-based technique to solve the problem approximately. In this paper, we consider additive type update and a T-period periodic-review water resources inventory system with stochastic inflow and zero lead time. In streamflow forecasts, denote T as the length of forecast lead time or forecast horizon, within which the streamflow is predictable with an available forecasting method, as showed in Fig. 1. The streamflow forecasts can be represented by a vector:

IFi;j ¼ fIF i;ðj;jþ1Þ ; F i;ðj;jþ2Þ ; . . . ; F i;ðj;jþTÞ g

ð1Þ

Let IFi,j = (IFi,(j,j+1), . . . , IFi,(j,j+T)) be the inflow forecast vector made at the end of period j, where IFi,0 is the initial forecast vector for reservoir i. We consider additive forecast updates; To update the inflow forecasts, we let IFi,(j,j+k) be the forecast inflow in reservoir i made at the end of period j for inflow in period j + k, k = 0, . . . , T  j, because forecasts are made after the current inflow information is revealed IFi,(j,j) = IFi,j. Define ei,(j,j+k) as the inflow forecast update made in reservoir iat the end of period j for inflow period j + k, and 0 6 k 6 T  j. i = 1, 2, j = 1, 2, . . . , T.

ei;ðj;jþkÞ ¼ IF i;ðj;jþkÞ  IF i;ðj1;jþkÞ

ð2Þ

Denote Var½ei;ðj;jÞ  ¼ r2i;j , and let ei,j be the inflow forecast update vector made in reservoir i at the end of period j, i = 1, 2, j = 1, 2, . . . , T.

ei;j ¼ ðei;ðj;jÞ ; ei;ðj;jþ1Þ ; . . . ; ei;ðj;jþkÞ Þ

ð3Þ

We assume that the forecasts are unbiased, i.e., E[ei,(j,l)] = 0, j 6 l. We also assume that forecast updates {ei,j, i = 1, 2, j = 1, 2, . . . , T} are independent over time. ej,j is the probability expectation, and its value is generally defined as zero. It is worth noting that forecast uncertainty and forecast horizon are two important features of streamflow forecast and can both affect reservoir operation using the forecast, as the forecast can be too uncertain if it is too long (i.e., it cannot reliably reflect inflow conditions) or too short to be applicable in supporting decision making. 4. Model and formulation 4.1. The interaction among multireservoir From structural analysis of the water system, the inter-basin water transfer changed the layout of the regional water conservancy and hydraulic relations. From the analysis the cost of the water transfer system and the general principle of equity for water allocation, inter-basin water transfer is inconsistent with the local water supply sources and the water users, water pricing and the priority order of supply water. Thereby, they increase the complexity of the structure of the water.

We consider an ideal type of two-echelon reservoir in InterBasin Water Diversion System (IBWDS), as shown in Fig. 2. Because the first stage of the reservoir is lower than the second stage in IBWDS, there are several pumping stations along the channel, transferring water resources from the water sources location to reservoir 1 and transferring water resources from reservoir 1 to reservoir 2. In addition, reservoir i(i = 1, 2) is faced with random inflow. Moreover, when the storage of reservoir i exceeds the flood control storage capacity, spilling operations need to be implemented in reservoir i. In order to discuss the interaction among multireservoir in more details, we consider two reservoirs as an example to explain. The inflow of reservoir 1 includes water transferred from the downstream, natural precipitation, and the overflow of water from upstream reservoirs 2; the outflow of reservoir 1 includes local needs and water transferred the water to reservoir 2 from the reservoir 1 through the pump station. The water of inflow in the reservoir 1, first meets the local demand for water, then if reservoir 2 is out of stock, it needs to transfer water from reservoir 1 via pump station. Similarly, the water of reservoir 2 is transferred from the reservoir 1, first meets the local demand for water, then it can be used in other ways. Between reservoir 1 and reservoir 2, if the coordinate configuration can meet the reservoir inventory and demand for water, it does not need to divert water from the downstream through pump station; That is, if at some point, the reservoir 1 capacity is less than the minimum requirements, reservoir 2 at this time capacity has exceeded the maximum capacity constrain, then we can allocate the water that the reservoir 2 water spill to the reservoir 1. Through the coordination operations, it can greatly reduce the cost of diverting water through the pumping station and get the profit maximization. 4.2. System definitions and notations set In order to model the problem, we set variables and parameters as follows: Si,j = storage of reservoir i at beginning of period j; IFi,j = incremental inflow (net precipitation or rainfall) to reservoir i during period j (Random Variable); Di,j = end-user demand in reservoir i during period j; Ri,j = releasing amount of water from upstream reservoir to reservoir i during period j; Qi,j = withdrawing amount of water from reservoir i to end-customer during period j; SPi,j = spilled (abandoned) water from reservoir i during period j (overflow condition);

SPi;j ¼ ðSi;j þ Ri;j þ IF i;j  Q i;j  Si;j Þþ

Here into, Si;j is the upper bound of Si,j, and it is the flood control storage capacity.pi,j = retailing price for unit of water from reservoir i during period j; ci,j = average cost of taking unit of water from reservoir i to customers during period j;

IF1, j R1, j River

IF2, j R2, j

S1, j

SP1, j

Fig. 1. Schematic representation of rolling horizon decision making in a reservoir.

ð4Þ

S 2, j

Q1, j

SP2, j

Fig. 2. Two-echelon water resources inventory.

Q2, j

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Fig. 3. SDP Flow diagram for two-reservoir.

bi,j = penalty factor for unit shortage water at reservoir iduring period j; ji,j = cost of spilling (abandoning) excess water in reservoir i during period j to guarantee that the storage does not exceed the flood control storage capacity; Energyi,j = energy consumed in pumping water from the upstream reservoir to reservoir i during period j:

Energyi;j ¼ q  Ri;j  g  LHi;j

ð5Þ

LHi,j = average lift height between upstream reservoir and reservoir i during period j;

q = density of water resources, g = gravity constant; Ci,j(Ri,j) = cost of transferring unit water from upstream reservoir to reservoir i during period j: C i;j ðRi;j Þ ¼ k  Energyi;j ¼ k  q  Ri;j  g  LHi;j , ki;j Ri;j

ð6Þ

k = average cost coefficient of energy consumed;

i ¼ 1; 2;

j ¼ 1; 2; . . . ; T

In the inventory management of the multireservoir, considering the interaction of withdrawing, storage, releasing and other factors, Ri,j and Qi,j are the decision variables.

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Table 1 Global adjustment policy of the multireservoirs.

Table 2 Local adjustment policy of the multireservoirs.

if R1;j þ IF 1;j þ IF 2;j 6 S1;jþ1  S1;j þ S2;jþ1  S2;j þ Q 1;j þ Q 2;j

if Q 1;j 6 Q 1;j 6 Q 1;j

if R1;j þ IF 1;j P S1;jþ1  S1;j þ Q 1;j þ R2;j    {R1;j ¼ min R1;j ; max R1;j ; S1;jþ1  S1;j þ IF 1;j  SP 1;j þ SP2;j  Q 1;j  R2;j ;

if R2;j 6 R2;j 6 R2;j fif S2;jþ1 6 S2;jþ1 6 S2;jþ1

R2;j ¼ R2;j ;

{R2;j ¼ minðR2;j ; S2;jþ1  S2;j  IF 2;j þ Q 2;j þ SP2;j Þ;

Q 1;j ¼ minðQ 1;j ; S1;jþ1  S1;j þ IF 1;j  SP 1;j þ SP2;j þ R1;j  R2;j Þ;

SP 1;j ¼ S1;j þ R1;j þ IF 1;j  Q 1;j  R2;j þ SP 2;j  SP 1;jþ1 ; g

Q 2;j ¼ minðQ 2;j ; IF 2;j þ R2;j þ S2;j  S2;jþ1 Þ;

else if S2;jþ1 > S2;jþ1

else if R1;j þ IF 1;j < S1;jþ1  S1;j þ Q 1;j þ R2;j

{R2;j ¼ maxðR2;j ; minðR2;j ; S2;jþ1  S2;j  IF 2;j þ Q 2;j þ SP 2;j ÞÞ;

{R1;j ¼ minðR1;j ; S1;jþ1  S1;j þ IF 1;j  SP1;j þ SP 2;j  Q 1;j  R2;j Þ; R2,j = 0;

SP 1;j ¼ S1;j þ R1;j þ IF 1;j  Q 1;j  R2;j þ SP 2;j  SP 1;jþ1 ; g else fR2;j ¼ minðR2;j ; S2;jþ1  S2;j  IF 2;j þ Q 2;j þ SP 2;j Þ;

Q 1;j ¼ minðQ 1;j ; S1;jþ1  S1;j þ IF 1;j  SP 1;j þ SP2;j þ R1;j  R2;j Þ;

SP 1;j ¼ S1;j þ R1;j þ IF 1;j  Q 1;j  R2;j þ SP 2;j  SP 1;jþ1 ; g

Q 2;j ¼ minðQ 2;j ; IF 2;j þ R2;j þ S2;j  S2;jþ1 Þ;

SP1;j ¼ S1;j þ R1;j þ IF 1;j  Q 1;j  R2;j þ SP2;j  SP1;jþ1 ; g

else if R1;j þ IF 1;j þ IF 2;j > S1;jþ1  S1;j þ S2;jþ1  S2;j þ Q 1;j þ Q 2;j if R1;j þ IF 1;j P S1;jþ1  S1;j þ Q 1;j þ R2;j   {R1;j ¼ min R1;j ; maxðR1;j ; S1;jþ1  S1;j þ IF 1;j  SP 1;j þ SP2;j  Q 1;j  R2;j Þ ;

area, while the upper bound is based on the maximum pumping or effusion capacity and flood control downstream. Therefore, the outflow capacity constraint is as follows:

if S2;jþ1 6 S2;jþ1 6 S2;jþ1 {R2;j ¼ minðR2;j ; S2;jþ1  S2;j  IF 2;j þ Q 2;j þ SP2;j Þ;} else if S2;jþ1 > S2;jþ1 {R2;j ¼ maxðR2;j ; minðR2;j ; S2;jþ1  S2;j  IF 2;j þ Q 2;j þ SP 2;j ÞÞ;}

Ri;j 6 Ri;j 6 Ri;j

else fR2;j ¼ minðR2;j ; S2;jþ1  S2;j  IF 2;j þ Q 2;j þ SP2;j Þ} Q 1;j ¼ minðQ 1;j ; S1;jþ1  S1;j þ IF 1;j  SP1;j þ SP 2;j þ R1;j  R2;j Þ;

4.3.1.4. Water withdrawing capacity constraints. For reservoir i during period j, there exist a low bound and an upper bound for the water withdrawing capacity, the low bound being based on the need for ecological protection, water pollution control, and shipping depth for the channel near the reservoir area, while the upper bound is based on the maximum pumping or effusion capacity. Therefore, the water withdrawing capacity constraints are as follows:

Q 2;j ¼ minðQ 2;j ; IF 2;j þ R2;j þ S2;j  S2;jþ1 Þ; if R1;j þ IF 1;j < S1;jþ1  S1;j þ Q 1;j þ R2;j {R2,j = 0; Q 2;j ¼ minðQ 2;j ; IF 2;j þ R2;j þ S2;j  S2;jþ1 Þ; SP2;j ¼ S2;j þ R2;j þ IF 2;j  Q 2;j  S2;jþ1 ; if S1;jþ1 6 S1;jþ1 6 S1;jþ1 fR1;j ¼ minðR1;j ; S1;jþ1  S1;j þ IF 1;j  SP 1;j þ SP 2;j  Q 1;j  R2;j Þ; g else if S1;jþ1 > S1;jþ1 {R1,j = 0;}

Q i;j 6 Q i;j 6 Q i;j

else fR1;j ¼ minðR1;j ; S1;jþ1  S1;j þ IF 1;j  SP 1;j þ SP 2;j  Q 1;j  R2;j Þ;} Q 1;j ¼ minðQ 1;j ; S1;jþ1  S1;j þ IF 1;j  SP 1;j þ SP2;j þ R1;j  R2;j Þ; g

4.3. Model description and solutions 4.3.1. Constraints 4.3.1.1. Water balance equations. There exists a water balance equation for the reservoir and the channel. Water storage in reservoir i at beginning of period j + 1 is equal to the water storage at beginning of period j plus incremental net inflows and rainfall minus outflows, withdrawing, evaporation, and spilled (or abandoned) water in reservoir i during period j. Therefore, the water balance equation for the reservoir is as follows:

S1;jþ1 ¼ S1;j þ IF 1;j þ R1;j  R2;j  Q 1;j  SP1;j þ SP2;j

ð7Þ

S2;jþ1 ¼ S2;j þ IF 2;j þ R2;j  Q 2;j  SP2;j

ð8Þ

Here into, IFi,j = IFi,(j,j) = IFi,(j1,j) + ei,(j,j), i = 1, 2, j = 1, 2, . . . , T. 4.3.1.2. Reservoir storage capacity constraint. For reservoir i during period j, there exist a low bound and an upper bound for storage, the low bound being the dead storage capacity while the upper bound is the flood control storage capacity. Therefore, the reservoir storage capacity constraint is as follows:

Si;j 6 Si;j 6 Si;j

4.3.1.3. Release capacity constraint. For reservoir i during period j, there also exist a low bound and an upper bound for the outflow capacity, the low bound being based on the need for ecological protection, water pollution control, and shipping depth for the channel

4.3.1.5. Pumping output capacity constraint. In the process of pumping water from the upstream reservoir to reservoir i during period j, there exists an upper bound for the pumping output capacity, which is based on the maximum pumping ability of the pumping station. Then, the pumping output capacity is as follows:

Energyi;j ðRi;j Þ 6 Energyi;j Here into, Energyi,j = q  Ri,j  g  LHi,j 4.3.1.6. Supply–demand constraints. The supply amount of water withdrawn from reservoir i and Channel (i, i + 1) to end-users should not exceed the demand amount of water during Period j. Therefore, the supply–demand constraint is as follows:

Q i;j 6 Di;j Obviously, we can combine the pumping output capacity constraints with the flow capacity constraints, giving, Ri;j 6 Ri;j n o Energy 6 R0i;j  min Ri;j ; qgLHi;ji;j We can also combine the supply–demand constraints with the water taking capacity constraints, giving,

Q i;j 6 Q i;j 6 Q 0i;j  minfQ i;j ; Di;j g 4.3.2. Objective function Equality-based water resources allocation principle is to allocate water resources among all demands in the sense that no shortage ratio can be decreased further without either violating a constraint or increasing an already equal or worse-off shortage ratio value that is associated with another demand in inter-basin water diversion project. For the inter-basin water transferring project, the objective should include two aspects: one is economic

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4.3.3. Recursive equation The recursive equation for SDP optimization is the following (Nandalal & Bogardi, 2007):

Table 3 The initiate allocation of reservoir 1 and reservoir 2. Parameters (Unit:106 m3)

IF SP S D Q R

Reservoir 1 j1

j

40 0 80 50 50 30

50 10 60 60 50 40

Reservoir 2 j+1 50 10 90 80 70 60

j1

j

30 20 100 40 40 20

50 30 90 50 40 60

50 20 110 60 50 50

IF-inflow, SP-spill, R-release, D-demand of terminal users, Q-withdraw, and Sstorage.

Table 4 Upper bound and low bound of storage, release and withdrawing for reservoir 1 and reservoir 2. Bound

Reservoir 1

Upper bound (Unit:106 m3) Low bound (Unit:106 m3)

Reservoir 2

S

R

Q

SP

S

R

Q

SP

150 40

110 0

110 0

70 0

120 30

90 0

90 0

50 0

R-release, Q-withdraw, S-storage, SP-Spill.

Table 5 The allocation results with adjustments policy for reservoir 1 and reservoir 2. Parameters (Unit:106 m3)

IF SP S D Q R

Reservoir 1

(

j+1

Reservoir 2

j1

j

j+1

j1

j

j+1

40 0 80 50 50 30

50 0 60 60 60 30

50 10 90 80 70 55

30 20 100 40 40 20

50 40 90 50 50 50

50 40 110 60 55 45

IF-inflow, SP-spill, R-release, D-demand of terminal users, Q-withdraw, and Sstorage.

profit and efficiency, and the other is social benefit. Specifically, economic profit and efficiency mainly require profit maximization while the social benefit mainly focuses on water shortage punishment, flood control, and fairness in water allocation. Therefore, we try to combine these objectives to formulate a single-period operations objective function, as follows:

U i;j ðRi;j ;Q i;j ;IF i;ðj1;jÞ Þ ¼ ðpi;j  ci;j ÞQ i;j  ki;j Ri;j  PSi;j ðQ i;j Þ  CSi;j ðSPi;j Þ

ð9Þ

F nj ðk; pÞ

) X j n1 ¼ max Bk;p;l;j þ Prp;q  F jþ1 ðl; qÞ l

ð11Þ

q

where k = storage state space consisting of representative values of joint storage states of reservoirs at beginning of period j, l = decision space consisting of representative values of joint storage states of reservoirs at beginning of period j + 1, p = inflow state space consisting of representative values of joint inflow states during period j, q = inflow state space consisting of representative values of joint inflow states during period j + 1, F nj ðk; pÞ ¼accumulated expected cost generation by optimal operation of system over the last n stages (when storage class at beginning of period j is k and inflow class during period j is p), Bk,p,l,j = cost generation when the system changes from state k (reservoir 1 and reservoir 2 at states k1 and k2 to state l (reservoir 1 and reservoir 2 at states l1 and l2) when inflow class is p(p1 to reservoir 1 and p2 to reservoir 2) in period j, and Prjp;q ¼joint transition probabilities of inflows as defined by Eq. (12). The joint transition probability Prjp;q that the probability that the inflows to reservoir 1 and reservoir 2 at period j + 1 will fall in states q1 and q2 (represented by state vector q) given that at period j, the streamflows to reservoirs 1 and 2 were in states p1 and p2 (represented by state vector p), respectively. This can be expressed as:

Prjp;q ¼ probðI1;jþ1 ¼ q1 ; I2;jþ1 ¼ q2 jI1;j ¼ p1 ; I2;j ¼ p2 Þ

ð12Þ

also

0 6 Prjp;q 6 1:0; for all p and q; j ¼ 1; 2; . . . ; 12; X j Prp;q ¼ 1:0; for all p; j ¼ 1; 2; . . . ; 12; q

where Ii,j = inflow to reservoir i during period j (106 m3), i = 1, 2; j = 1, 2, . . . , T. The flow chart of the SDP procedure of two reservoirs is displayed in Fig. 3. 5. Adjustments policy

Here into

PSi;j ðQ i;j Þ ¼ bi;j

Di;j  Q i;j ; Di;j

CSi;j ðSPi;j Þ ¼ ji;j SPi;j ;

5.1. Global adjustment and local adjustment

i ¼ 1; 2;

j ¼ 1; 2; . . . ; T Q i;j 6 Di;j : Therefore, the Dynamic Programming (DP) for the water resources inventory is as follows:

" # T X 2 X maximize E U i;j ðRi;j ; Q i;j ; IF i;ðj1;jÞ Þ Ri;j ;Q i;j

j¼1 i¼1

8 S1;jþ1 ¼ S1;j þ IF 1;ðj1;jÞ þ e1;ðj;jÞ þ R1;j  R2;j  Q 1;j  SP 1;j þ SP 2;j > > > > > S2;jþ1 ¼ S2;j þ IF 2;ðj1;jÞ þ e2;ðj;jÞ þ R2;j  Q 2;j  SP 2;j > > < i ¼ 1; 2 ; s:t: Si;j 6 Si;j 6 Si;j > j ¼ 1; 2; ... ; T > 0 > Ri;j 6 Ri;j 6 Ri;j > > > > :Q 6 Q 6 Q0 i;j i;j i;j ð10Þ

During the optimization for multireservoirs, in order to achieve the maximization of profit, we need to aggregately analyze the process of optimization. There are two ways to actually adjust the policy from the SDP. One is global adjustment, we consider the total flow, the storage, the withdrawing and the release of reservoir 1 and reservoir 2, and then analyze the relationship among them. It tries water resources to be used sufficiently, to avoid the water to spill and reduce the pump stations used. Because spill will waste the water resource and using the pump stations need some certain cost. In order to account for the relationship among the total flow, the storage, the withdrawing and the release for reservoir 1 and reservoir 2, we point out two points: (1) If the inflow flow is no more than the total flow of the storage, the release and the withdrawing for reservoir 1 and reservoir 2, we will analyze the relationship among them for

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Y. Xu et al. / Computers & Industrial Engineering 63 (2012) 890–900

Pumping Station

Fig. 4. Structure of Lakes Hongze and Luoma.

Table 6 Inflow class discretization of the operation policy (Unit 106 m3). Inflow Class

January

February

March

April

May

June

July

August

September

October

November

December

1

89.5 82.3 88.5 127.4 89.5 198.9 88.5 275 220.8 82.3 225.8 139.3 221.8 198.9 208.8 277 336.3 82.3 338.3 139.3 340.3 198.9 334.3 277 489.7 82.3 512.4 139.3 508.5 198.9 506.4 277

53.4 43.4 53.4 108.6 53.4 176.2 53.4 266.7 128.9 43.4 125.9 108.6 123.9 176.2 129.9 266.7 188.6 43.4 190.6 108.6 185.6 176.2 183.6 266.7 305.9 43.4 305.9 108.6 305.9 176.2 305.9 266.7

52.3 40.2 52.3 63 52.3 97.8 52.3 132.8 75.1 40.2 75.1 63 75.1 97.8 75.1 132.8 106.3 40.2 112.3 63 106.3 97.8 111.3 132.8 137.6 40.2 139.6 63 153.6 97.8 137.6 132.8

70.6 42.1 71.6 56.2 76.6 72.4 75.9 89.1 114.3 42.1 112.6 57.2 99.7 72.4 110.3 89.1 158.4 42.1 160.4 56.2 148.7 78.4 160.4 89.1 232.1 42.1 225.6 56.2 225.1 72.4 230.1 89.1

78.7 45.7 78.7 65.6 78.7 97.8 78.7 128 199.4 45.7 177.8 65.6 201.5 97.8 198.4 128 297.2 45.7 294.7 65.6 288.2 97.8 266.2 128 409.9 45.7 410.6 65.6 407 97.8 399.7 128

117.7 40.8 121.7 54.4 123.6 70 117.7 97.8 311 40.8 308 54.4 317 70 318.5 97.8 456.3 40.8 475.3 54.4 488.7 70 489.1 97.8 756.1 40.8 753.6 54.4 765.5 70 777.1 97.8

108.5 37.5 108.5 51 108.5 67.2 108.5 84.1 252.6 37.5 252.6 51 252.6 67.2 252.6 84.1 341.3 37.5 341.3 51 341.3 67.2 341.3 84.1 529.5 37.5 529.5 51 529.5 67.2 529.5 84.1

128.2 36.2 121.2 48 125.2 60 129.8 83.7 234.1 36.2 256.1 48 267.4 60 268.8 83.7 393.2 36.2 396.4 48 396.2 60 400.2 83.7 563.6 36.2 569.1 48 581.3 60 584.5 83.7

127.8 37.1 124.8 52.2 131.4 71.6 134.8 85.6 286.7 37.1 289.7 52.2 293.5 71.6 293.7 85.6 453.9 37.1 461.1 52.2 465.9 71.6 468.4 85.6 719.8 37.1 730.7 52.2 732.1 71.6 728.8 85.6

123.6 49.2 107.8 67.4 118.7 89.6 121.6 114 201.5 49.2 204.3 67.4 211.5 89.6 210.5 114 318.8 49.2 317.6 67.4 331.4 89.6 329.1 114 421.5 49.2 432.1 67.4 429.7 89.6 435.8 114

173 49.9 172 86.1 181 116.1 181.7 159.1 293.6 49.9 289.6 86.1 275.7 116.1 297.1 159.1 453.8 49.9 431.9 86.1 459.3 116.1 432.7 159.1 598.4 49.9 585.6 86.1 653.6 116.1 685.3 159.1

165.1 98.7 178.5 183.8 171.7 299.5 175.1 457.4 405.7 98.7 413.4 183.8 418.9 317.1 432.7 468.9 737.2 98.7 742.3 183.8 753.8 317.1 762.7 468.9 875.8 98.7 911.6 183.8 932.5 317.1 967.5 468.9

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Hongze Luoma Hongze Luoma Hongze Luoma Hongze Luoma Hongze Luoma Hongze Luoma Hongze Luoma Hongze Luoma Hongze Luoma Hongze Luoma Hongze Luoma Hongze Luoma Hongze Luoma Hongze Luoma Hongze Luoma Hongze Luoma

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Y. Xu et al. / Computers & Industrial Engineering 63 (2012) 890–900 Table 7 Storage classes of the operation policy (Unit 106 m3). Class

Hongze

Luoma

Class

Hongze

Luoma

Class

Hongze

Luoma

1 2 3

38 68 148

286 478 390

4 5 6

158 266 328

465 585 620

7 8 9

490 605 724

682 778 896

Table 8 Serial correlation coefficients of Lakes Hongze and Luoma. Month

January

February

March

April

May

June

July

August

September

October

November

December

Hongze Luoma

0.22 0.22

0.29 0.07

0.5 0.36

0.39 0.21

0.51 0.53

0.68 0.55

0.31 0.44

0.55 0.49

0.19 0.36

0.07 0.11

0.01 0.05

0.54 0.33

Table 9 Value of some parameters in the simulation. Parameters

c_1j

c_2j

p_1j

p_2j

b_1j

b_2j

j_1j

j_2j

LH_1j

LH_2j

k_1j

k_2j

q

g

k

Value

0.2

0.3

1.99

3.01

0.5

0.5

0.01

0.01

10

10

98

98

1000

9.8

0.001

Table 10 Comparison of results of SDP_A and SDP. Policy no.

1 2 3 4 5

Total max annual profit ($)

Average annual storage (106 m3)

Average annual release (106 m3)

Average annual withdrawing (106 m3)

SDP_A

SDP

SDP_A

SDP

SDP_A

SDP

SDP_A

SDP

894,000 885,000 940,000 921,000 918,000

859,000 847,000 903,000 887,000 894,000

1.78 1.13 0.6 1.46 0.45

1.74 1.08 0.57 1.39 0.43

891.3 912.7 976.4 864.8 895.1

875.5 870.1 935.6 816.4 866.9

564.7 765.5 975.3 875.1 910.8

544.8 744.3 946.5 832.6 880.2

Ratio of average annual shortage

0.06 0.03 0.01 0.03 0.07

reservoir 1 and reservoir 2 respectively, and then decide to adjust the release and the withdrawing according to the related constraints. (2) If the inflow flow is greater than the total flow of the storage, the release and the withdrawing for reservoir 1 and reservoir 2, the current status of the storage, release and the withdrawing needs to be considered for every reservoir. Through the analysis and comparison, we can decide to adjust the scale of the release and the storage.

Table 4, there is the upper bound and low bound of the storage, release, withdrawing and spill for reservoir 1 and reservoir 2. The adjustment steps are as follows:

The steps of adjustment are shown as Table 1. Another is called local adjustment. During the calculation the decision variable may occur to overflow and not meet the some constraints, we need to adjust the value of decision variable. There are three aspects to consider during the adjustment. First, if the release ðR1j 6 R1j 6 R1j Þ of the reservoir 1 is feasible when it is between the lower and upper bounds, the storage of the reservoir 1 is analyzed from two different conditions: S1;jþ1 > S1;jþ1 and S1;jþ1 6 S1;jþ1 . If the storage capacity of reservoir 1 exceeds the upper bound constraint during period j þ 1ðS1;jþ1 > S1;jþ1 Þ, it requires to compare whether the withdrawing satisfies the low bound and the upper bound, then we further need to analyze the current release and storage of the reservoir 2, finally the adjustment policies are decided. As an example, we consider the adjustment under the condition of R1j 6 R1j 6 R1j and Q 1;j 6 Q 1;j 6 Q 1;j . The process of optimization is shown as Table 2.

S1;jþ1  S1;j þ Q 1;j þ R2;j , according to the policy, the decision variables need to be adjusted.

5.2. Example In order to illustrate the operating process of the adjustment policy, we consider the simple case where the parameters of inflow, terminal users demand, release, withdrawing and storage for two reservoirs are shown in Table 3, here T = 3. As shown in

Step 1. we apply the global policy to synthetically analyze the reservoir 1 and reservoir 2. At the time of j, because R1,j + IF1,j + IF2,j

is

less

than

S1;jþ1  S1;j þ S2;jþ1  S2;j

þQ 1;j þ Q 2;j , we need to analyze the relationship R1,j + IF1,j and

S1;jþ1  S1;j þ Q 1;j þ R2;j ,

for

R1,j + IF1,j

less

than

R1;j ¼ minðR1;j ;S1;jþ1  S1;j þ IF 1;j  SP1;j þ SP 2;j  Q 1;j  R2;j Þ ¼ 30; R2;j ¼ 0; Q 1;j ¼ minðQ 1;j ;S1;jþ1  S1;j þ IF 1;j  SP1;j þ SP 2;j þ R1;j  R2;j Þ ¼ 60; Q 2;j ¼ minðQ 2;j ;IF 2;j þ R2;j þ S2;j  S2;jþ1 Þ ¼ 50;

Similarly, we can adjust the decision variables of R1,j1, R1,j+1, Q1,j1, Q1,j+1, R2,j1, R2,j+1, Q2,j1, Q2,j+1 at the time of j  1 and j + 1 for reservoir 1 and reservoir 2. Step 2. the local policy is applied to adjust the inventory of reservoir 1 and reservoir 2 respectively. First, we consider the relationship of Q1,j, R2,j, S2,j+1, upper bound and low bound. Because they can meet the constraints of Q 1;j 6 Q 1;j 6 Q 1;j ; R2;j 6 R2;j 6 R2;j and S2;jþ1 6 S2;jþ1 6 S2;jþ1 , according the policy, R2,j and SP1,j need to be changed.

R2;j ¼ minðR2;j ; S2;jþ1  S2;j  IF 2;j þ Q 2;j þ SP2;j Þ ¼ 50; SP1;j ¼ S1;j þ R1;j þ IF 1;j  Q 1;j  R2;j þ SP2;j  SP1;jþ1 ¼ 0;

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Y. Xu et al. / Computers & Industrial Engineering 63 (2012) 890–900

SDP_A

SDP

SDP_A

960,000

Average annual withdrawing (106m3)

940,000

Total max annual profit ($)

SDP

1200

920,000 900,000 880,000 860,000 840,000 820,000

1000 800 600 400 200

800,000 1

2

3

4

0

5

1

2

3

4

5

Policy No

Policy No Fig. 5. Comparison the differences between SDP_A and SDP.

Cost

Storage

Release

Withdrawing

7.00%

The Rate of Enhance

6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00%

1

2

3

4

5

Policy No Fig. 6. Improve rate from four different indexes based on our proposed model.

Similarly, the variables of R2,j1, R2,j+1, SP1,j1, SP1,j+1 at the time of j  1 and j + 1 can be adjusted for reservoir 1 and reservoir 2. Step 3. we can carry on the adjustment policy for several times, until the results of decision variables meet the requirements for allocation configuration of reservoir 1 and reservoir 2. According to analyze the results via adjustment policy, we find the total release reduces from 260 to 230, the ratio is 11%, this will save the cost of the transfer water in certain ways. The withdrawing is increased from 300 to 325; the ratio is about 10%, in other words, the demand for water of terminal users can be meted more. The results are shown in Table 5 with the adjustments policy. 6. Application of the model to the SNWD project The South to North water diversion project is the largest water resources engineering project involving the largest water resources allocation aiming to resolve the serious water shortage problem in Northern China. The Eastern route of the SNWD project is a highly complex inter-basin system with multi-source, multiobject, and multi-project which pumps, stores, and supplies water to achieve rational water distribution in Northern China. The project was constructed and extended with a total investment of US10 billion based on the Northern water transfer project in Jiangsu Province. The

Eastern route project transfers water resources from the Yangzi River in Jiangdu through the Beijing-Hangzhou Grand Canal and its parallel river, linking Lake Hongze, Lake Luoma, Lake Nansi, and Lake Dongping, with 13 pumping stations and a total lift of 65 m. To verify the model, we consider as an example the Yangzi River and the plan to transfer water to Lake Hongze and Lake Luoma, as showed in Fig. 4. Lake Hongze has a total storage capacity of 4.25 billion m3 and a non-flood season adjustment capacity of 3.15 billion m3, while Lake Luoma has a total storage capacity of 0.91 billion m3 and a non-flood season adjustment capacity of 0.59 billion m3). The operation policy designated for a reservoir by the model is a set of rules specifying the storage level at the beginning of the following month for each combination of storage levels at the beginning of the current month and the inflow during the current month. As an example, the operation policy is designed for a month using four inflow classes and three storage classes for each reservoir. The numerical values used to identify the different inflow and storage levels are presented in Tables 6 and 7, respectively. Table 8 reports the mean, standard deviation, and historical maximum and minimum of the monthly flows for the Lakes Hongze and Luoma for the past 50 years (1958–2007). Some parameters are initiated, as shown in Table 9. To verify the feasibility of our model, we consider applying SDP based on adjustment (SDP_A) and SDP model (Nandalal, 1986; Bogardi et al., 1988) to the SNWD project, then analyzing and comparison the simulation results.

Y. Xu et al. / Computers & Industrial Engineering 63 (2012) 890–900

The simulation results are summarized in Table 10. Four different indexes are compared: total annual profit, average annual storage, average annual release and average annual withdrawing. As shown in Fig. 5, we compare the simulation results between SDP based on adjustment and SDP, the total annual cost has an average increase by 3.84%; the average annual withdrawing is improved by 3.62%. According to the horizontal comparison, policy no. 3 is observed to be the best policy for this water resources system when considering the total annual cost and average annual storage, release, and withdrawing. Moreover, policy no. 3 has some certain improvement in four indexes based on our proposed model, the improve rate are 4.10%, 5.26%, 4.36% and 3.04% respectively. The ratio of average annual shortage is a very important index in an operation policy for water resources management. In terms of average water shortage, this policy is negligibly inferior when compared to policy nos. 1 and 5. In Fig. 6, it shows the curve for specific improve rates of policy nos. 1–5 from four different indexes, the average improve rates are 3.84%, 4.38%, 4.05% and 3.62%. Generally speaking, those indicate that our model based on adjustment can solve the water resource operation problem more effectively and get a better operation policy. 7. Conclusions Although streamflow forecast uncertainty plays an important role in reservoir operation, the effects of forecast uncertainty on reservoir operation have yet to be thoroughly addressed in a unifying framework. The Martingale Model of Forecast Evolution (MMFE) was introduced to synthetically generate deterministic and probabilistic streamflow forecasts through explicit representations of forecast uncertainty under various scenarios. A stochastic dynamic programming model with a physical recursive equation was developed to determine the optimum operation of the multireservoir operation systems. An adjustment policy with two ways was introduced to optimize and maximize the profit. The experience reported in this paper and the resulting conclusions provide additional guidance in real-world reservoir operations. Acknowledgments Funding for this research is supported by the National Science Foundation of China under Grants Nos. 90924020 and 70971005, the Societal Science Foundation of China under Grants No. 11AZD096, the PhD Program Foundation of the Education Ministry of China under Contract No. 200800060005, and Quality Inspection Projects No. 200910088 and No. 201010268. References Ampitiya, H. K. (1995). Stochastic dynamic programming based approaches for the operation of a multi-reservoir system. Master’s thesis, Netherlands: Wageningen Agricultural University. Araujo, A. R., & Terry, L. A. (1974). Operation of a hydrothermal system. Brazil Journal of Electrical Engineering. (in Portuguese). Archibald, T. W., McKinnon, K. I. M., & Thomas, L. C. (1997). An aggregate stochastic dynamic programming model of multireservoir systems. Water Resources Research, 33(2), 333–340. Archibald, T. W., McKinnon, K. I. M., & Thomas, L. C. (2006). Modeling the operation of multireservoir systems using decomposition and stochastic dynamic programming. Naval Research Logistics, 53(3), 217–225. Bogardi, J. J., Brorens, B. A. H. V., Kularathna, M. D. U. P., Milutin, D., & Nandalal, K. D. W. (1995). Long-term assessment of a multi-unit reservoir system operation: The ShellDP Program package manual. Report 59, Internal Publication Series, Department of Water Resources, Netherlands: Wageningen Agricultural University. Bogardi, J. J., Budhakooncharoen, S., Shrestha, D. L., & Nandalal, K. D. W. (1988). Effect of state space and inflow discretization on stochastic dynamic programming based reservoir operation rules and system performance. In Proceedings, 6th congress, asian and pacific regional division, IAHR (Vol. 1, pp. 429–436). Kyoto, Japan. Butcher, W. S. (1971). Stochastic dynamic programming for optimum reservoir operation. Water Resources Bulletin, 7(1), 115–123.

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