Overexploitation of groundwater reserves is a major environmental problem
around the world. In many river basins, groundwater and surface water are
used conjunctively and joint optimization strategies are required. A
hydroeconomic modeling approach is used to find cost-optimal sustainable
surface water and groundwater allocation strategies for a river basin, given
an arbitrary initial groundwater level in the aquifer. A simplified
management problem with conjunctive use of scarce surface water and
groundwater under inflow and recharge uncertainty is presented. Because of
head-dependent groundwater pumping costs the optimization problem is
nonlinear and non-convex, and a genetic algorithm is used to solve the
one-step-ahead subproblems with the objective of minimizing the sum of
immediate and expected future costs. A real-world application in the
water-scarce Ziya River basin in northern China is used to demonstrate the
model capabilities. Persistent overdraft from the groundwater aquifers on
the North China Plain has caused declining groundwater levels. The model
maps the marginal cost of water in different scenarios, and the minimum cost
of ending groundwater overdraft in the basin is estimated to be CNY 5.58 billion yr

Groundwater aquifers are of high economic importance around the world and often act as buffers in the water supply system during droughts (Tsur and Graham-Tomasi, 1991; Tsur, 1990). On the North China Plain, persistent groundwater overexploitation over the past decades has caused decline of the shallow and deep groundwater tables (Liu et al., 2001). The immediate benefits of satisfying the water demands greatly exceed the costs of pumping, which highlights the problem of the present self-regulating management. As the groundwater resource is overexploited, the immediate benefits of the increased unsustainable supply have to be traded off against the long-term increase in pumping costs and reduced buffering capacity. Optimal allocation of the water resources should address coordinated use of the water resources by considering the long-term total costs, while utilizing the groundwater as a buffer. This is in line with the 2011 Chinese No. 1 Policy Document, which targets improvement of water use efficiency and reduction of water scarcity (CPC Central Committee and State Council, 2010).

Optimal management of conjunctive use of surface water and groundwater has been addressed widely in the literature (e.g Booker et al., 2012; Burt, 1964; Knapp and Olson, 1995; Labadie, 2004; Noel and Howitt, 1982). While control-based methods, such as model predictive control (e.g., Morari and Lee, 1999; Mayne et al., 2000) and reinforcement learning (Lee and Labadie, 2007), focus on deriving real-time optimal control policies, this study will focus on planning oriented optimization techniques. Deterministic optimization problems for a given time horizon allow a detailed representation of the groundwater system using spatially distributed groundwater models (Andreu et al., 1996; Harou and Lund, 2008; Marques et al., 2006; Pulido-Velázquez et al., 2006). Stochasticity is commonly represented in scenarios where a regression analysis is used to formulate operation rules; see e.g., the implicit stochastic optimization approaches reviewed by Labadie (2004). Singh (2014) reviewed the use of simulation–optimization (SO) modeling for conjunctive groundwater and surface water use. In SO-based studies, efficient groundwater simulation models are used to answer “what if” questions, while an optimization model is wrapped around the simulation model to find “what is best”. Groundwater aquifers have been represented as simple deterministic box or “bathtub” models (e.g., Cai et al., 2001; Riegels et al., 2013) and as spatially distributed models (e.g., Maddock, 1972; Siegfried et al., 2009) with stochasticity (Reichard, 1995; Siegfried and Kinzelbach, 2006). While the results obtained from these methods are rich in detail, they yield only a single solution to the optimization problem.

Methods based on dynamic programming (DP, Bellman, 1957) have been used extensively to demonstrate the dynamics of conjunctive groundwater–surface water use for both deterministic (e.g., Buras, 1963; Provencher and Burt, 1994; Yang et al., 2008) and stochastic DP (SDP, e.g., Burt, 1964; Philbrick and Kitanidis, 1998; Provencher and Burt, 1994; Tsur and Graham-Tomasi, 1991) optimization problems. In DP-based methods, the original optimization problem is decomposed into subproblems, which are solved sequentially over time. The entire decision space is thereby mapped, enabling use of the results as dynamic decision rules. However, the number of subproblems grows exponentially with the number of state variables and this curse of dimensionality has frequently limited the use of DP and SDP (Labadie, 2004; Provencher and Burt, 1994; Saad and Turgeon, 1988). Although it causes loss of detail and inability to disaggregate the results, reservoir aggregation has been suggested as a possible solution strategy (Saad and Turgeon, 1988).

This study aims to answer the following two macroscale decision support questions for conjunctive groundwater and surface water management for the Ziya River basin in North China. (1) What are the minimum costs of ending groundwater overdraft? (2) What is the cost-efficient recovery strategy of the overpumped aquifer? A hydroeconomic modeling approach is used to identify the least-cost strategy to achieve sustainable groundwater abstraction, defined as the long-term average abstraction that does not exceed the long-term average recharge. To overcome the management problem similar to Harou and Lund (2008) with increased complexity caused by uncertain surface water runoff and groundwater recharge, the surface water reservoirs are aggregated. This is adequate at a macroscale (Davidsen et al., 2015) and allows the use of DP-based approaches. The cost minimization problem is solved with the water value method, a variant of SDP (Stage and Larsson, 1961; Stedinger et al., 1984) which produces dynamic tables of marginal costs linked to states, stages, and water source. Head- and rate-dependent pumping costs introduce nonlinearity in the discrete subproblems. This nonlinearity is handled with a hybrid genetic algorithm (GA) and linear programming (LP) method similar to that used by Cai et al. (2001), here applied in a coupled groundwater–surface water management problem within an SDP framework.

Northern China and particularly the North China Plain (NCP) have experienced
increasing water scarcity problems over the past 50 years due to population
growth, economic development, and reduced precipitation (Liu and
Xia, 2004). The deficit in the water balance has historically been covered
by overexploitation of the groundwater aquifer, causing a regional lowering
of the groundwater table by up to 1 m yr

The Ziya River basin, a part of the Hai River basin, was selected as a case
study area (see Fig. 1). The upper basin is
located in the Shanxi province, while the lower basin is located in the
Hebei province on the NCP. The 52 300 km

A previous hydroeconomic study of the Ziya River basin was a traditional implementation of SDP on a single-reservoir system (surface water reservoir) and showed optimal water management, while disregarding dynamic groundwater storage and head-dependent groundwater pumping costs (Davidsen et al., 2015). Instead, the groundwater resource was included as a simple monthly upper allocation constraint.

In the present study, the groundwater resource is included as a dynamic
aquifer box model with a storage capacity of 275 km

The Ziya River basin. Watershed and rivers automatically delineated from a digital elevation map (USGS, 2004) and manually verified and corrected with Google Earth (Google Inc., 2013). The SNWTP routes (central and eastern) were sketched in Google Earth and verified with field observations. Provincial boundaries from NGCC (2009).

Annual water demands and curtailment costs for the users in the Ziya River basin. Based on the data set from Davidsen et al. (2015).

Conceptual sketch of the simplified water management problem.
Water users are located upstream (

A conceptual sketch of the management problem is shown in
Fig. 2. The water users are divided into groups
of economic activities; irrigation agriculture, industrial users, and domestic
water users. Ideally, each water user group should be characterized by
flexible demand curves, but due to poor data availability a constant water
demand (m

An SDP formulation is used to find the expected value of storing an
incremental amount of surface water or groundwater, given the month of the
year, the available storage in surface and groundwater reservoirs, and the
inflow scenarios. The backward recursive equation calculates the sum of
immediate and expected future costs for all combinations of discrete
reservoir storage levels (states) and monthly time steps (stages). The
immediate management cost (IC) arises from water supply and water
curtailment, whereas the expected future cost (EFC) is the optimal value
function in

Equation (

A rainfall–runoff model based on the Budyko Framework
(Budyko, 1958; Zhang et al., 2008) has been used in a previous study to estimate the near-natural daily surface water
runoff into reservoirs (Davidsen et al., 2015). The
resulting 51 years (1958–2008) of simulated daily runoff was aggregated to
monthly runoff and normalized. A Markov chain, which describes the runoff
serial correlation between three flow classes defined as dry (0–20th
percentile), normal (20–80th percentile), and wet (80–100th percentile), was established and validated to ensure
second-order stationarity (Davidsen et al., 2015; Loucks
and van Beek, 2005). The groundwater recharge is estimated from the
precipitation data also used in the rainfall–runoff model. The average
monthly precipitation (mm month

Nomenclature.

The SDP loop is initiated with EFC set to zero and will propagate backward
in time through all the discrete system states as described in the objective
function. For each discrete combination of states, a cost minimization
subproblem will be solved. A subproblem will have the discrete reservoir
storage levels (

The sets of equilibrium shadow prices, referred to as the water value tables, can subsequently be used to guide optimal water resources management forward in time with unknown future runoff. In this study, the available historic runoff time series is used to demonstrate how the derived water value tables should be used in real-time operation. The simulation will be initiated from different initial groundwater aquifer storage levels, thereby demonstrating which pricing policy should be used to bring the NCP back into a sustainable state.

The groundwater aquifer is represented as a simple box model (see
Fig. 2) with recharge and groundwater pumping
determining the change in the stored volume of the aquifer (Eq.

In Eq. (

With two reservoir state variables and a climate state variable, the number
of discrete states is quickly limited by the curse of dimensionality. A very fine discretization of
the groundwater aquifer to allow discrete storage levels and decisions is
computationally infeasible. A low number of discrete states increases the
discretization error, particularly if both the initial and the end storages

Nonlinear optimization problems can be solved with evolutionary search methods, a subdivision of global optimizers. A widely used group of evolutionary search methods are genetic algorithms (GAs), which are found to be efficient tools for getting approximate solutions to complex nonlinear optimization problems (see, e.g., Goldberg, 1989; Reeves, 1997). GAs use a random search approach inspired by natural evolution and have been applied to the field of water resources management by, e.g., Cai et al. (2001), McKinney and Lin (1994), and Nicklow et al. (2010). Cai et al. (2001) used a combined GA and LP approach to solve a highly nonlinear surface water management problem. By fixing some of the complicating decision variables, the remaining objective function became linear and thereby solvable with LP. The GA was used to test combinations of the fixed parameters, while looking for the optimal solution. The combination yielded faster computation time than if the GA was used to estimate all the parameters.

A GA implemented in MATLAB is used to solve the cost minimization subproblems. This GA function will initially generate a set of candidate solutions known as the population. Each of the candidate solutions contains a set of decision variables (sampled within the decision space), which will yield a feasible solution to the optimization problem. In MATLAB, a set of options specifies the population size, the stopping criteria (fitness limit, stall limit, function tolerance, and others), the crossover fraction, the elite count (number of top parents to be guaranteed survival), and the generation function (how the initial population is generated). The options were adjusted to achieve maximum efficiency of the GA for the present optimization problem.

The computation time for one single subproblem is orders of magnitude larger
than solving a simple LP. As the optimization problem became computationally
heavier with increasing number of decision variables, a hybrid version of GA
and LP, similar to the method used by Cai
et al. (2001), was developed (see Fig. 3).
Decision variables that cause nonlinearity are identified and chosen by the
GA. Once these complicating decision variables are chosen, the remaining
objective function becomes linear and thereby solvable with LP. In the
optimization problem presented in Eq. (

For a given combination of stages, discrete states and flow classes, the
objective of the GA is to minimize the total cost, TC, with the free
states

The performance of the GA–SDP model is compared to a fully deterministic DP,
which finds the optimal solution given perfect knowledge about future
inflows and groundwater recharge. The DP model uses the same algorithm as
the SDP model and one-dimensional state transition matrices with

SDP optimization algorithm design.

Temporal changes of the water values (CNY m

Without any regulation or consideration of the expected future costs arising
from overexploitation of the groundwater aquifer, the water users will
continue maximizing immediate profits (producers) or utility (consumers).
Because there are only electricity costs for groundwater, the users will
continue pumping groundwater until the marginal groundwater cost exceeds the
curtailment cost. At CNY 1 kWh

The backward recursive SDP algorithm was run with a looped annual data set until equilibrium water values, i.e., no interannual changes, were obtained. The water values increase fastest during the first years, and after approximately 100 years, the annual increases become small. Due to the large storage capacity of the groundwater aquifer, equilibrium is however not achieved until after 150–180 years. These marginal water values represent the true values of storing a unit volume of water for later use, and vary with reservoir storage levels, runoff flow class, and time of the year. A sample of the resulting equilibrium water value tables is presented in Fig. 4. This figure shows the temporal variations of water values as a function of one state variable, keeping the other state variables at a fixed value. The state variables are fixed at empty, half full, and full storage respectively. During the rainy season from June to August, high precipitation rates reduce water scarcity, resulting in lower surface water values. Because the groundwater storage capacity is much larger, increased recharge can easily be stored for later use, and groundwater values are therefore not affected. Addition of stream–aquifer interactions to the model is expected to affect this behavior, but since the flow in rivers/canals in the case study area is small most of the year, and since most areas are far from a river, it is reasonable to ignore these dynamics. The water values after 1980 are clearly higher than in the period before 1980 due to increased water scarcity caused by a reduction in the regional precipitation. In contrast, the groundwater value tables are uniform, with variation only in groundwater storage. The detailed water value tables are included in the Supplement.

Simulated groundwater aquifer storage levels for 51 years of
historical runoff with different initial groundwater tables (0, 100, 200, 258,
and 275 km

Simulated storage levels in the surface water reservoir and the groundwater aquifer at equilibrium groundwater storage.

User's price for groundwater and surface water through for a 51-year simulation based on simulated historical runoff for two initial
groundwater storages.

We simulate management using the equilibrium water value tables as a pricing
policy and force the system with 51 years of simulated historical runoff.
Time series of the simulated groundwater storage can be seen in
Fig. 5 for different initial storage scenarios.
The groundwater aquifer approaches an equilibrium storage level around 260 km

Composition of allocations and curtailments to wheat agriculture
in the Hebei province for the months March, April, May, and June through 51-year
simulation from an initial groundwater storage at equilibrium
(258 km

In the simulated management runs, water will be allocated to the users up to
a point where reductions in immediate cost are compensated by increases in
expected future costs. The user's price, which can be applied in a marginal
cost pricing (MCP) scheme, is the marginal value of the last unit of water
allocated to the users. The user's price is the sum of the actual pumping
cost (electricity used) and the additional marginal cost given by the
equilibrium water value tables. In Fig. 7, the
user's prices for groundwater and surface water are shown for the 51-year
simulation at and below the long-term sustainable groundwater storage level.
When the groundwater storage level is close to equilibrium, the user's
prices of groundwater and surface water are equal during periods with water
scarcity. In wet months with reduced water scarcity, the model switches to
surface water allocation only, and the groundwater user's price is undefined
(gaps in the time series in Fig. 7). If the
groundwater storage level is below equilibrium, the groundwater user's price
will be higher, causing an increase in water curtailments and an increase in
storage level, as shown in Fig. 5. Under these
circumstances the surface water user's price increases up to a point where
the two prices meet. With an initial aquifer storage at one-third of the
aquifer capacity (100 km

At the equilibrium groundwater storage level, the user's prices for
groundwater is stable at around CNY 2.15 m

Average minimum total cost (TC) and hydropower benefits (HP) over
the 51-year planning period for different scenario runs. SNWTP scenarios:
pre-2008 refers to the period before the central route of the SNWTP was built; 2008–2014 refers to a
situation with a partly completed SNWTP central route from Shijiazhuang to
Beijing; post-2014 refers to the fully operational SNWTP central route from the Yangtze River to Beijing. Scenarios: LGW
is initial groundwater storage at 100 km

The average total costs of the 51-year simulation for different scenarios
can be seen in Table 3. The average reduction in the total costs, associated
with the introduction of the SNWTP canal, can be used to estimate the
expected marginal economic impact of the SNWTP water. The minimum total
costs after the SNWTP is put in operation are compared to the scenario
without the SNWTP (pre-2008) and divided by the allocated SNWTP water. The
resulting marginal value of the SNWTP water delivered from Shijiazhuang to
Beijing (2008–2014 scenario) is CNY 3.2 m

A local sensitivity analysis focused on the water demands and curtailment
costs used directly in the objective function (Eq.

The minimum total costs were lowered from CNY 10.50

From any initial groundwater reservoir storage level, the model brings the groundwater table to an equilibrium storage level at approximately 95 % of the aquifer storage capacity. Only small variations in the aquifer storage level are observed after the storage level reaches equilibrium as shown in Fig. 6. While addition of the Thiem stationary drawdown has only a small effect on total costs and total allocated water, it is clear from Fig. 8 that the additional Thiem drawdown highly impacts the allocation pattern for some of the water users. High groundwater pumping rates result in larger local drawdown and thus in higher pumping costs. This mechanism leads to a more uniform groundwater pumping strategy, which is clearly seen in Fig. 8, and results in a much more realistic management policy.

This study presents a hydroeconomic optimization approach that provides a macroscale economic pricing policy in terms of water values for conjunctive surface water–groundwater management. The method was used to demonstrate how the water resources in the Ziya River basin should be priced over time, to reach a sustainable situation at minimum cost. We believe that the presented modeling framework has great potential use as a robust decision support tool in real-time water management. However, a number of limitations and simplifications need to be discussed.

A first limitation of the approach is the high level of simplification needed. There are two main reasons for the high level of simplification: limited data availability and the limitations of the SDP method. The curse of dimensionality limits the approach to 2–3 interlinked storage facilities and higher dimensional management problems will not be computationally feasible with SDP today. This limit on the number of surface water reservoirs and groundwater aquifers requires a strongly simplified representation of the real-world situation in the optimization model. The simulation phase following the optimization is not limited to the same extent, since only a single subproblem is solved at each stage. The water values determined by the SDP scheme can thus be used to simulate management using a much more spatially resolved model with a high number of users; this was not demonstrated in this study. The advantage of SDP is that it provides a complete set of pricing policies that can be applied in adaptive management, provided that the system can be simplified to a computationally feasible level. An alternative approach known as stochastic dual dynamic programming (SDDP, Pereira and Pinto, 1991; Pereira et al., 1998) has shown great potential for multi-reservoir river basin water management problems. Instead of sampling the entire decision space with the same accuracy level, SDDP samples with a variable accuracy not predefined in a grid, focusing the highest accuracy around the optimal solution. This variable accuracy makes SDDP less suitable for adaptive management. Despite the highly simplified system representation, we believe that the modeling framework provides interesting and nontrivial insights, which are extremely valuable for water resources management on the NCP.

Computation time was a limitation in this study. Three factors increased the computational load of the optimization model. (1) Inclusion of the groundwater state variable resulted in an exponential growth of the number of subproblems; (2) the non-convexity handled by the slower GA–LP formulation caused an increase in the computation time of 10–100 times a single LP; and (3) the SDP algorithm needed to run through more than 200 years to reach a steady state. A single scenario run required 4000 CPU hours and was solved in 2 weeks, using 12 cores at the high-performance computing facilities at the Technical University of Denmark. This is 50 000 times more CPU hours than a single-reservoir SDP model (Davidsen et al., 2015). Since the water value tables can be used offline in the decision making, this long computation time can be accepted.

The long computation time made the use of, e.g., Monte Carlo-based
uncertainty analysis infeasible. The local sensitivity analysis showed that
a 10 % increase in the curtailment costs is returned as a 6.0 % increase
in the total costs, while a similar increase of the demands generates a
2.1 % increase in costs. The transmissivity can vary over many orders of
magnitude because it is a log-normally distributed variable. The sensitivity
of

Intuitively, one would expect the equilibrium groundwater storage level to be as close as possible to full capacity, while still ensuring that any incoming groundwater recharge can be stored. Finding the exact equilibrium groundwater storage level would require a very fine storage discretization, which, given the size of the groundwater storage, is computationally infeasible. Therefore the equilibrium groundwater storage level is subject to significant discretization errors. The long time steps (monthly) make the stationarity required for using the Thiem stationary drawdown method a realistic assumption.

The difference between total cost with SDP and with DP (perfect foresight) is small (1.3 %). Apart from Beijing, which has access to the SNWTP water, the remaining downstream users have unlimited access to groundwater. The large downstream groundwater aquifer serves as a buffer to the system and eliminates the economic consequences of a wrong decision. The model almost empties the reservoir every year as shown in Fig. 6, and wrong decisions are not punished with curtailment of expensive users as observed by Davidsen et al. (2015). The groundwater aquifer reduces the effect of wrong decisions by allowing the model to minimize spills from the reservoir without significant economic impact of facing a dry period with an empty reservoir. A dynamic groundwater aquifer thereby makes the decision support more robust, since it is the timing and not the amount of curtailment that is being affected.

The derived equilibrium groundwater value tables in Fig. 4 (and detailed water value tables in the Supplement) show that the groundwater values vary with groundwater storage alone and are independent of time of the year, the inflow and recharge scenario, and the storage in the surface water reservoir. This finding is important for future work, as a substitution of the groundwater values with a simpler cost function could greatly reduce the number of states and thereby the computation time. The equilibrium groundwater price, i.e., the groundwater values around the long-term equilibrium groundwater storage, can possibly be estimated from the total renewable water and the water demands ahead of the optimization, but further work is required to test this. Further work should also address the effect of discounting of the future costs on the equilibrium water value tables and the long-term steady-state groundwater table. In the present model setup, the large groundwater aquifer storage capacity forces the backward-moving SDP algorithm to run through 200–250 model years, until the water values converge to the long-term equilibrium. Another great improvement, given the availability of the required data, would be to replace the constant water demands with elastic demand curves in the highly flexible GA–LP setup.

A significant impact of including groundwater as a dynamic aquifer is the
more stable user's prices shown in Fig. 7. The
user's price of groundwater consists of two parts: the immediate groundwater
pumping costs (electricity costs) and the expected future costs represented
by the groundwater value for the last allocated unit of water. As the model
is run to equilibrium, the user's prices converge towards the long-term
equilibrium at approximately CNY 2.2 m

This study describes development and application of a hydroeconomic approach
to optimally manage conjunctive use of groundwater and surface water. The
model determines the water allocation, reservoir operation, and groundwater
pumping that minimizes the long-term sum of head- and rate-dependent
groundwater pumping costs and water curtailment costs. The model is used to
quantify potential savings of joint water management of the Ziya River basin
in northern China, but the model can be applied to other basins as well.
Estimates of natural runoff, groundwater recharge, water demands, and
marginal user curtailment costs are cast into an SDP-based optimization
framework. Regional and Thiem stationary drawdown is used to estimate rate-
and head-dependent marginal groundwater pumping costs. The resulting
optimization subproblems become nonlinear and non-convex and are solved with
a hybrid GA–LP setup. A central outcome from the SDP framework is tables of
shadow prices of surface and groundwater for any combination of time, inflow
class, and reservoir storage. These tables represent a complete set of
pricing policies for any combination of system states and can be used to
guide real-time water management. Despite a significant computational demand
to extract the water value tables, the method provides a suitable approach
for basin-scale decision support for conjunctive groundwater and surface
water management.
The model provides useful insight to basin-scale scarcity-driven tradeoffs.
The model outputs time series of optimal reservoir storage, groundwater
pumping, water allocation, and the marginal economic value of the water
resources at each time step. The model is used to derive a pricing policy to
bring the overexploited groundwater aquifer back to a long-term sustainable
state. The economic efficient recovery policy is found by trading off the
immediate costs of water scarcity with the long-term additional costs of a
large groundwater head. From an initial storage at one-third of the aquifer
capacity, the average costs of ending groundwater overdraft are estimated to
be CNY 13.32

S. Liu and X. Mo were supported by grants of the Natural Science Foundation of China (31171451, 41471026). The authors thank the numerous farmers and water managers in the Ziya River basin for sharing their experiences; L. S. Andersen from the China Europe Water Platform for sharing his strong willingness to assist with his expert insight on China; and K. N. Marker and L. B. Erlendsson for their extensive work on a related approach early in the development of the presented optimization framework. Edited by: F. Pappenberger