Forecast-based analysis for regional water supply and demand relationship by hybrid Markov chain models: a case study of Urumqi, China

Water resources are increasingly being viewed as a severely stressed and scarce natural resource, especially for arid and semi-arid regions in China (Babel & Shinde 2011; Shen et al. 2013). Moreover, the regional population growth and economic development has so far led to high water consumption. This would put more and more pressure on regional water supply. Taking Urumqi City, a core city in an arid area of China as an example, the daily capacity of water supply has increased from 0.74 × 10⁶ m³ in 2011 to 1.50 × 10⁶ m³ in 2015, resulting in the continuous decline of groundwater level. In addition, the water utilization structure has been significantly changed since water saving policies were implemented in recent years. Given these complex and dynamic situations, efforts should be made to help policy-makers make a reliable estimate of future water supply and demand in order to identify desired management alternatives.

Mathematical models can be a useful technique to predict and analyze regional water supply and demand problems, and previous studies made valuable attempts in developing forecasting models, such as regression analysis, artificial neural networks, time series analysis, and system dynamics models (Miaou 1990; Jain et al. 2001; Altunkaynak et al. 2005; Mohamed & Al-Mualla 2010; Tiwari & Adamowski 2013; Kim & Seo 2015). For example, Babel et al. (2007) developed a multivariate econometric approach to forecast and manage the domestic water use/demand of Kathmandu Valley, Nepal. Cheng & Chang (2011) proposed a novel system dynamics model to reflect the interactive relationship between water demand and macroeconomic environment and forecast long-term municipal water demand in Manatee County. Zhang et al. (2013) adopted the Cobb–Douglas model to predict regional water demand and calculated the contribution rates of the regional water demand influencing factors. Nourani et al. (2014) applied genetic fuzzy system (GFS) model and multivariate wavelet-GFS (WGFS) model to predict the runoff discharge of two distinct watersheds. The obtained results showed that the runoff could be better forecast through the proposed WGFS model.

In general, most of the aforementioned models have been successfully applied to water demand predication and other water resources management issues; however, these models generally need large numbers of historical data and complicated input factors to make reasonable predictions. In fact, regional water supply and demand is affected by various climatic elements, socio-economic factors, and related government policies. Under the influence of water saving policies and climate changes, the developing trends of regional water demand and water utilization structure would significantly change and the available observed data may not satisfy the requirements during those modeling processes. Moreover, the forecasted results from the traditional models are typically expressed as a deterministic value rather than interval value. This would result in a false or excessive confidence in the model and inaccuracy of the forecasting results. In contrast, an interval forecasting model of water resources management has the advantage of taking the variability and uncertainties into consideration in order to reduce the forecasting errors and risk when making water resources planning and operational decisions (Alvisi & Franchini 2012; Alvisi et al. 2012; Xiong et al. 2014). Besides, although various forecasting models have been successfully adopted for the predication of water demand and annual streamflow, the integrated predication and analysis for regional water supply and demand, the water utilization structure still needs to be investigated.

Therefore, this study attempts to develop hybrid models for forecasting of streamflow, water demand, and water utilization structure. The traditional Markov chain model was selected to forecast the annual streamflow and related available water resources while the unbiased Grey-Markov model (UGMM) was applied for regional water demand prediction. As an extension of the traditional Grey model (1, 1) and Markov chain model, the UGMM can be used to forecast the water demand with large random fluctuations. The forecasted results would be expressed as interval values with corresponding probability aiming to reduce the forecasting risk. In addition, the Markov approach based on quadratic programming model (QP-Markov model) was selected for water utilization structure prediction. The QP-Markov model minimizes the total difference between predicted value and actual value and provides a more precise one-step transition probability than the Markov chain model. The proposed models are applicable to water resources planning and management in Urumqi, China. Analysis of water supply and demand in Urumqi under different reuse ratios was conducted based on the forecasting results. The results would help managers and policy-makers obtain a clear understanding of regional water supply and demand trend as well as the water utilization structure in the future.

Markov chain is a particular type of stochastic process, and it has been used extensively to establish stochastic models and predict future data by occurred events (Mao & Sun 2011). Regional streamflow and related available water resources are dynamic and stochastic variables due to changes in average climatic parameters, particularly temperature and rainfall. The Markov chain forecasting model can be used to forecast the regional available water resources and the associated probability distributions. Procedure of the special model is summarized as follows.

• Step 1: Calculate the average value of annual streamflow and the relative anomaly as

  

  1a

• Step 2: The relative anomaly is classified into three continuous interval values, which correspond to different water inflow levels (low, normal, and high flow years) according to the standard for hydrological information and hydrological forecasting (GB/T 22482-2008).

• Step 3: Calculate the transition probability as

  

  1b

  

  1c

  where denotes the transition probability that the proportion of the system belongs to state i at time t−1 is transferred to state j at time t. is the number of observation pairs and with in state i and in state j; is the number of in state i.
• Step 4: Define the transition probability matrix between different kinds of as

  

  1d
• Step 5: Calculate the probability that the state of system at time t+k:

  

  1e

  where is the initial probability distribution of the state at time t. Once the probability distribution of different states is obtained, the associated streamflow and the available water resources at time t + k can be calculated through Equation (1e).

Although the UGM (1, 1) is a simple and relatively accurate method to forecast the regional water demand with an exponential increasing trend, the accuracy may decrease when the historical data sequence fluctuates with time. The Markov chain forecasting model can also be used to forecast the water demand with randomly varying time series. Therefore, an UGMM can be obtained through the incorporation of UGM (1, 1) and Markov chain model. The relative error series obtained through the former UGM (1, 1) can be used to predict the relative error in future by the Markov chain model. Thus, the revised water demand can be obtained through the integration of UGM and Markov chain model. The steps of the Markov chain model in UGMM are similar to the former steps of a typical Markov model. In order to maintain the integrity of the UGMM, is introduced and the specific process is described below.

• Step 1: Establish a Markov model based on relative error series⁠: divide the relative error series into n states. Any state of relative error can be denoted as an interval:

  

  2m
• Step 2: Assume is the data number of historical sequence, the transition probability from to can be established:

  

  2n

  where is the transition probability of state transferred from state for m steps, m is the number of transition steps each time, is the number of data in state ⁠, is the number of historical data of state transferred from state for m steps, its transition probability matrix can be expressed as follows:

  

  2o

  Generally, it is necessary to observe one-step transition matrix, ⁠. Suppose the object to be forecasted is in state ⁠, row r in matrix should be considered. If ⁠, then what will most probably happen in the next moment is the transition from state to state ⁠.

• Step 3: Calculate the predicted probability of the future relative error. The relative error zone in the future is predicted by studying the transition probability matrix ⁠. Therefore, the forecasting result of water demand can be presented as a single value through the medium of relative error zone or an interval according to Equation (2k).

Water resources consumption is generally composed of agriculture, industry, municipality, and eco-environment. In Urumqi, recent years have seen not only increasing water consumption but also the varying structure of water consumption. Under the influence of a series of policies, the proportion of agricultural water consumption is decreasing while that of municipality and environment is increasing year by year. The water utilization structure in the future depends mainly on the status in recent years rather than in past states. Therefore, a novel stochastic model, incorporating the quadratic programming model into Markov approach, was proposed to predict the utilization structure of water resources in Urumqi. The developed model is summarized as follows.

• Step 1: Collect the annual data of water utilization structure (i = 1, 2, 3, 4; t = 1, 2, …n).

• Step 2: Define the transition probability matrix between different water users as

  

  3a

  where means one-step transition probability that the proportion belongs to user i at t– 1 period is transferred to user j at t period.
• Step 3: Calculate the one-step transition probability using the quadratic programming model. The predicted proportions of each user at time t is expressed as

  

  3b

  To minimize the total difference between the real value and simulated value of ⁠, the quadratic programming can be established as follows:

  

  3c

  

  3d

  Then we can get the transition probability value.

• Step 4: The absolute weighted error (AE) of each year and mean absolute weighted error (MAE) is defined as indicators for accuracy assessment.

  

  3e

  

  3f

The real water utilization structure can be forecasted based on the solved transition probability matrix and the water utilization structure in the initial year.

In recent years, Urumqi has entered into the fast lane of economic growth since the strategies of the Great Western Development and Silk Road Economic Belt were promulgated. The process of industrialization and urbanization is accompanied by massive water consumption. For example, the annual average available water resources are 11.20 × 10⁸ m³; however, in 2015 the estimated water demand reached 12.20 × 10⁸ m³. Therefore, water shortage becomes an important restrictive factor for economic and social sustainable development. In Urumqi, agricultural water demand represent about 60% of the total water consumption, followed by regional domestic and industrial sectors. It is unsustainable in the long term due to the limited water resources and irrational water utilization structure. Therefore, effective measures are needed for the sustainable utilization of all the water sources.

It is obvious that the observed data for 2000 lie in state 3. The state in 2001 and 2002 can be calculated through Equation (1d). The forecasting state of 2001 and 2002 would be normal water inflow-level year and the associated annual streamflow would be [9.18, 11.22] × 10⁸m³. The actual streamflow in 2001 and 2002 were 10.50 × 10⁸m³ and 9.20 × 10⁸m³, respectively. Results indicated that the Markov chain model can be used for forecasting the annual streamflow and the associated probabilities. The streamflow in 2012 was selected as initial point and the probabilities of water inflow states for 2015–2020 are represented in Table 2. The corresponding available water resources would be obtained when the water inflow state is recognized.

The forecasting trend curves for water demand were built by GM (1, 1) and the associated variant UGM (1, 1). These curves are shown as solid lines in Figure 3. It should be noted that the time series of water consumption in Urumqi vacillated around the trend curves. The fluctuations in actual water consumption would reduce the forecasting accuracy and the associated variant. Following Equation (2l), the values of MAPE in GM (1, 1) and UGM (1, 1) are 8.12% and 5.97%, respectively. The relative errors of prediction from GM (1, 1) are [−20.46, 11.91] %, and there have been 12 results whose relative errors exceed the range [−5.00, 5.00] %. The relative errors from UGM (1, 1) are [−19.85, 13.65] % with nine results beyond the range [−5.00, 5.00] %. The results indicated that neither of the two forecasting models have performed well for random time series prediction.

The relative error zone in the future would be predicated using the one-step or m-steps transition matrix. Correspondingly, the water demand and the related probabilities would be obtained. In this study, the water consumption in 2014 was chosen as the historical point. The forecasting results of water demand during 2015–2020 are presented in Table 3. It is obvious that the forecasted values of GM (1, 1) would always lie in the intervals of the highest water demand with lowest probabilities. However, the results of UGM (1, 1) and the modified values would remain at the medium demand level with maximum probability. A comparison of the predicted values form GM (1, 1) and UGM (1, 1) indicated that the forecasted results of the hybrid model would be presented as a single value or interval values. The interval values of water demand, taking the variability or uncertainties into account, have the advantage of reduction in risk when making water resources planning and operation decisions. The results obtained in this study can also be compared with the predicated values in urban comprehensive planning. The forecasted water demand of Urumqi in 2015 and 2020 would be 12.20 × 10⁸ m³ and 15.00 × 10⁸ m³, respectively. It is clear that the values lie in the forecasted intervals with maximum probability. The comparison clearly points to the enormous potential that the model possesses in water demand forecasting and can be considered as a viable alternative.

In this study, three hybrid Markov chain models have been proposed for predicting regional streamflow, water demand, and water utilization structure in Urumqi. The hybrid models required less computational data than other traditional models and took into account water supply and demand, and water utilization structure trend. Results show that 2015–2020 would correspond to the normal flow year and the associated streamflows were [9.18, 11.22] × 10⁸ m³ with maximized probability. The water demand of Urumqi during 2015–2020 would keep increasing slightly. For the water utilization structure, the proportions of agriculture would continue to decrease while the proportions of industry, municipality, and eco-environment would increase slowly. Analyses of water supply and demand in Urumqi under different reuse ratios were conducted, and results indicated that water deficit would not be dramatically relieved despite the reuse ratio increases. The reduction of water consumption and an increase of diverted water would be the applicable measures for alleviating the water shortage in Urumqi. In general, the results presented in this paper would help managers and policy-makers to have a clear understanding of the regional water supply and demand trend as well as the water utilization structure.

This research was supported by the Fundamental Research Funds for the Central Universities (2015XS99), National Natural Science Foundation of China (61471171). The authors are extremely grateful to the editor and the anonymous reviewers for their insightful comments and suggestions.