The Jiangsu section of the South-to-North Water Diversion Project (J-SNWDP) crosses the Yangtze River and Huai River basins. A simplified map of the J-SNWDP is shown in Fig. 1. The Beijing–Hangzhou Grand Canal runs through the north and south of Jiangsu Province, connecting the Yangtze River basin, the Huai River basin, and the Yishusi River basin. The total length of the Jiangsu section is 404 km, along which there are six cities, namely Yangzhou, Huaian, Yancheng, Suqian, Lianyungang, and Xuzhou. The J-SNWDP consists of three impounded lakes (Hongze Lake, Luoma Lake, Nansi Lake), six sluices (e.g. Erhe sluice, Gaoliangjian sluice), and 13 pumping stations (Huaian station, Jiangdu station), forming a double-route water diversion system including the West Route and the Canal Route. The water supply scope of the whole J-SNWDP covers the three provinces of Jiangsu, Anhui, and Shandong, supplementing the agricultural, industrial, and domestic water supply as well as navigation and ecological water supply in the areas along the water transfer route. The scale parameters of the pumping stations and sluices in J-SNWDP are listed in Table 1. The characteristics of the lakes in the J-SNWDP are shown in Table 2.

When the J-SNWDP is supplying water along the water transfer route, users closer to the water source (such as Jinbaohangdao (JBHD) user, Sihong–Suining (S-S) user, Luoma (LM) user) are generally prioritized. On the other hand, users like the Feihuanghe user, Lianyungang user, and Siyang–Zaohe user, who are located farther from the water source and receive less water from Luoma Lake, often require water to be pumped from the Yangtze River and Hongze Lake for replenishment. Combined with the uneven spatial and temporal distribution of precipitation, these recipient zones are more susceptible to water deficits.

In Fig. 2, the complex relationship of the J-SNWDP is generalized into a schematic diagram according to the geographical location. In order to represent the main components of the system and the connection between the backbone rivers, we regard the lakes, pumping stations, and sluices as the nodes and the backbone river as the connecting line, while the water users are reasonably distributed among the nodes of the water transmission route.

The joint operation process of the J-SNWDP is described as follows. In addition to natural precipitation and lake inflow, the Yangtze River is the main water source of water, which is pumped from the river to the West Route and the Canal Route by the P1 pumping station. The West Route is pumped step by step from P2, P3, and P4 to Hongze Lake through Sanyang River and Jinbao Channel and pumped from P9, P10, and P11 to the north through Xuhong and Fangting rivers. The Canal Route along the Beijing–Hangzhou Grand Canal route is pumped north from P5, P6, P7, and P8. The two water transmission routes supplement the water demand of users along the routes while pumping to the north and merge into one line at the intersection of Fangting River and Zhongyun River. It is then pumped further north until P12 and P13 are transported to Nansi Lake via Bulao and Hanzhuangyun rivers, respectively, thus delivering water outside Jiangsu Province. When the natural inflow is high, Hongze Lake can be discharged via P1 on the West Route or through S2, S3, S4, S5, and S6 to the relevant rivers and users. Luoma and Nansi lakes are discharged through the original water transmission routes.

The joint optimal operating model of the J-SNWDP uses a monthly time step. According to the annual natural inflow data of Hongze (HZ), Luoma (LM), and Nansi (NS) lakes since 2013 from Jiangsu Water Resources Bulletin, 3 hydrological years were selected to represent wet (October 2017–September 2018, annual mean inflow: 465.12×108 m3), normal (October 2019–September 2020, annual mean inflow: 172.80×108 m3), and dry (October 2013–September 2014, annual mean inflow: 103.06×108 m3) years, respectively. The typical annual runoff curves of HZ, LM, and NS lakes are shown in Fig. 3. The water supply area of the J-SNWDP is divided into 13 receiving areas according to geographical location, and water demand data are provided by the Jiangsu Provincial Water Resources Department: wet years (October 2017–September 2018, water demand: 171.29×108 m3), normal years (October 2019–September 2020, water demand: 173.51×108 m3), dry years (October 2013–September 2014, water demand: 181.75×108 m3).

TWD is a measure of how well the operation strategy completion is being implemented. This objective aims to minimize the total amount of water deficit at the end of a given operation period, potentially improving the satisfaction of water demand for users and increasing the operation strategy completion. 2TWD=min∑t=1T∑i=1nQR(i,t), where QR(i,t) is the water deficit of the ith user at time step t, i=1, 2, …, n, with n being the total number of users (n=13 in this study), and t=1, 2, …, T, with T being the whole operating period.

WDEI indicates the degree of concentration of the water deficit and can be used as an indicator of the uniformity of water diversion. The lower the WDEI, the better the strategy is. 3WDEI=min∑t=1nQR(i,t)-∑i=1nQR(i,t)n2n, where QR(i,t) is the water deficit of the ith user at time step t; i=1, 2, …, n, with n being the total number of users (n=13 in this study); and t=1, 2, …, T, with T being the whole operating period.

PW reflects the economy of operation strategy. The lower the PW is, the lower the operating costs. 4PW=min∑t=1T∑p=1PQS(p,t), where QS(p,t) is the water pumped by the pth pumping station at time step t, and p=1, 2, …, P, with P being the total number of pumping stations (P=13 in this study).

Systems operation should obey operating rules and physical constraints, such as water balance, pumping capacity, and lake storage constraints. The mathematical expressions of the constraints are shown below.

The water balance constraint should be satisfied in the water diversion process. 5V(i,t+1)=V(i,t)+Q(i,t)+DI(i,t)+PC(i+1,t)-DO(i,t)W1(i,t)-PR(i,t) at time step t, where V(i,t) is the water storage of the ith lake at time step t, Q(i,t) is the inflow of the ith lake, W1(i,t) is the water demand of the ith lake (water to be supplemented by J-SNWDP after deducting the locally available water), DO(i,t) is water diversion to the north from the ith lake, DI(i,t) is the water pumped into the ith lake, PC(i,t) is the water discharged into the ith lake, and PR(i,t) is the water discharged from the ith lake.

60≤DO(i,t)≤DOmax(i,t)70≤DI(i,t)≤DImax(i,t) at time step t, where DO(i,t) is water diversion to the north from the ith lake, DI(i,t) is the water pumped into the ith lake, DOmax(i,t) is the maximum pumping capacity that is pumped into the ith lake, and DImax(i,t) is the maximum pumping capacity that is diverted north from the ith lake.

80≤PR(i,t)≤PRmax(i,t), where PR(i,t) is the water discharged from the ith lake, and PRmax(i,t) is the maximum sluice capacity at time step t.

9Vmin(i,t)≤V(i,t)≤Vmax(i,t), where V(i,t) is the water storage of the ith lake at time step t, and Vmin(i,t) and Vmax(i,t) are the minimum and maximum water storage capacities at time step t, respectively. When V(i,t)<Vmin(i,t) is the water deficit, QR(i,t)=Vmin(i,t)-V(i,t), ensure that the lake level is above the limit level, and when V(i,t)>Vmax(i,t) is abandoned water, ensure that the water storage of the lakes is within a reasonable range.

10Zmin(i,t)≤Z(i,t)≤Zmax(i,t), where Z(i,t) is the level of the ith lake at time step t, and Zmin(i,t) and Zmax(i,t) are the minimum and maximum level of the ith lake at time step t, respectively.

Table 4 shows the optimal historical operation strategy in typical years. Compared with the optimal operation strategy, the TWD and WDEI of the historical operation strategy are very high, which means that the water deficit concentration in the historical operation strategy leads to the inability to reduce the TWD. Meanwhile, the PW and abandoned water are slightly large, indicating that the historical operation strategy has more disadvantages on economic and ecological benefits. WDEI can directly reflect the difference of water deficit among users. WDEI has sound optimization effects, of which the reduction in the optimal operation strategy was 94.2 % (81.8 %, 76.7 %) of the historical values in the typical normal year (wet year and dry year). The optimized operation strategy can significantly reduce TWD and PW by 82.06 % (37.69 %, 52.36 %) and 45.13 % (3.25 %, 21.51 %) compared with the historical values, respectively, while maintaining a very low WDEI. It has been demonstrated to be an excellent strategy for inter-basin water diversion, which can make multiple users share the risk of water deficit and alleviate the problem of water deficit concentration. The reduction of the abandoned water and the water withdrawn from the Yangtze River represents an increase in the lake storage capacity. The amount of abandoned water of the operation strategy is greatly reduced, and the amount of abandoned water is only 0.051×108 m3 in the dry year. This indicates that the optimal operation can maximize the utilization efficiency of the limited water resources. However, the amount of water withdrawn from the Yangtze River increased by 10.72×08 m3 after optimization in dry years, indicating that the proportion of the water withdrawn from Yangtze River in PW was raised in the optimal operation strategy to ensure uniform distribution of water and higher economic benefits.

The comparison of water deficit in water users between the optimal operation strategy and the historical operation strategy in 3 typical years is illustrated in Table 5, which more intuitively reflects the optimization effect of WDEI. The “variance” and “decrement” represent the difference between the maximum and minimum values and the percentage reduction, respectively. The study found that after optimization, the variance of the users was reduced by 62.93 % (78.07 %, 54.09 %) in different typical years (i.e. bold font in Table 5). The Feihuanghe (FHH) user, Siyang–Zaohe (S-Z) user, and Hongzehu (HZH) user, which have a large water deficit in the historical operation strategy, decrease by 14.63 (14.41, 14.69) × 108 m3, 5.61 (9.33, 1.77) × 108 m3, and 0.98 (7.98, 5.26) × 108 m3 of the optimal operation strategy, respectively (see Table 5). The possible reason for the large water deficit of users is that they are far away from the main lakes and rivers or that there are many users on the water transmission route that need to be supplied. Users jointly share the water deficit risk in the optimal operation strategy in the case of little natural flow. The difference between the maximum and minimum water deficit is controlled within 8×108 m3, where the difference in the normal year is only 3.13×108 m3, and the problem of water deficit concentrated in a particular user is alleviated. For example, in the dry year, the water deficit of users under optimal operation strategy ranges from 0 to 7.64×108 m3. Although the water deficit of the Gaoliangjian–Huaian (GLH-HA) user, Luomahu (LMH) user, and Weishanhu (WSH) user is slightly higher after optimization, the total water deficit is much lower than the historical value and is not concentrated in the FHH user, HZH user, and S-Z user, compared with the results of historical operation. Overall, the optimal operation strategy is more reasonable and may accord with the aspirations of both the government and the general public.

Figure 6 shows the water level variations of HZ Lake and LM Lake (two major storage lakes in Jiangsu Province). The water level of HZ Lake and LM Lake is slightly higher than the historical value after optimization for most of the time. Both lakes show a regular trend of change: the lake remains at a high level during the dry season, releases water before the flood season, then remains at a low level throughout the flood season for storing flood, and returns to a high level at the end of the flood season. Therefore, the water transmission system can simultaneously implement lake water storage in the flood and non-flood seasons to ensure the fulfilment of water supply tasks and improve the utilization efficiency of water resources.

The optimized water level of HZ Lake was significantly lower than the historical value from February to June in Fig. 6a. This is due to the large amount of natural precipitation with uneven spatial–temporal distribution in the wet year, and the discharge time and discharge volume of HZ Lake are unreasonable in historical operating strategy, resulting in a large amount of water deficit and abandonment. Therefore, in the optimized operation strategy, a certain amount of lake water is discharged during the impoundment period for water supply. Similarly, LM Lake level in April–June is below the historical level. In Fig. 6b and c, the optimized HZ Lake level is significantly higher than the historical value before the flood season, while the flood season is decreased dramatically, indicating that the lake stores a certain amount of water while supplying water during the impoundment period and discharges water before the flood season to ensure that the flood season is at a safe level. Similarly, for LM Lake, the water level increased after optimization in typical years and tended to be consistent after the flood season.

Compared with the historical operation strategy, the water level of HZ Lake and LM Lake is slightly higher than the historical value after optimization for most of the time. Combined with the results of Tables 4 and 5, the coordinated water diversion between LM Lake and HZ Lake increased obviously while reducing TWD and PW.

This paper considers that the water distribution of different routes between two adjacent lakes is an important factor affecting the water deficit in the users. The water transmission capacity of the route is mainly affected by the water demand of users and the pumping capacity of the pumping station along the way. Table 6 shows the distribution of different water transmission routes between neighbouring lakes in typical years. The ratio of water pumped from LM Lake into Hanzhuangyun River and Bulao River is decreased from 19:8 to 1:1, which is convenient for the simultaneous double-route water supply outside the province and reduces the water supply pressure of Hanzhuangyun River. The ratio of water from the Zhongyun River and the Xuhong River to the LM Lake increased from 7:9 to about 7:1, gradually shifting the focus of the water transmission route to the Zhongyun River. The reason is that Hongze Lake transports water to the Zhongyun River through the Erhe and Gaoliangjian sluice without pumping, which can reduce the pumped water and save project cost. In addition, there are many users in Zhongyun River, the water demand is higher. When the water supply is sufficient, the drainage through Huaiyin, Yanhe and Yangzhuang sluice significantly reduces the water deficit of FHH and LYG users (see Table 5). Similarly, the ratio of water from HZ Lake to Zhongyun River and Xuhong River increased from 6:1 to 14:1. The ratio of pumping water from Liyun River and Jinbao Waterway to HZ Lake also increased from 19:1 to 25:1, indicating that the water transport efficiency of Zhongyun River is higher, and the operation strategy should be based on water transfer from the central canal, with the western route as the auxiliary support route. Similarly, the ratio of pumping water from the Yangtze River to the Liyun River and the Jinbao Channel has also increased (see Table 6).