Information Gap Decision-Making Theory-Based Medium- and Long-Term Optimal Dispatching of Hydropower-Dominated Power Grids in a Market Environment

Abstract

In the high-proportion hydropower market, the fairness of the execution of traded electricity and clean energy consumption are two issues that need to be considered in medium- and long-term dispatching. Aiming at the fairness of medium- and long-term optimal dispatching of hydropower-dominated grids and the problem of water abandonment in the power market environment, this paper proposes a medium- and long-term optimal dispatching method for hydropower-dominated grids based on the information gap decision-making theory (IGDT). Firstly, IGDT is used to establish a two-layer model of medium- and long-term optimal dispatching that considers runoff uncertainty, in which the lower layer solves the maximum value of the maximum difference in the contract power completion rate of the power stations, and the upper layer solves the maximum fluctuation range of the interval inflow. Then, a mixed-integer linear programming (MILP)-based single-layer optimization model is obtained through a variety of linearization techniques, and the model is solved via the CPLEX solver (version 12.10.0). The medium- and long-term optimal dispatching of 10 thermal power stations and 22 hydropower stations in Yunnan Power Grid, China, is taken as an example to verify the proposed model. The results show that the maximum difference in the contracted electricity completion rate of each power station is 0.412, and the amount of abandoned hydropower is reduced by 81.33% compared to when the abandoned water penalty function is not considered. It is proved that the proposed model can effectively alleviate the problems of excessive power generation, insufficient power generation and large-scale hydropower abandonment, which are of great significance for realizing the fair dispatching of hydropower-dominated power grids and promoting clean energy consumption in the market environment.

1. Introduction

Over the last twenty years, China has seen remarkable growth in hydropower development, a key strategy for reducing carbon emissions while satisfying the rapidly growing electricity demand [1]. By the close of 2020, China’s hydropower capacity had quadrupled from its 2000 levels, reaching a total of 370 GW [2]. Additionally, China has constructed numerous cascade hydropower stations on major rivers in the southwest, creating several high-proportion hydropower systems (HPHSs), notably in Sichuan and Yunnan Provinces [3].

Since the late 1980s, global electricity sectors have undergone significant reforms, introducing competition into the power supply side via electricity markets. On 15 March 2015, China’s Central Committee of the Communist Party and the State Council issued directives to further advance power system reforms [4], marking the start of a new phase in these efforts. As reforms continue to deepen, medium- and long-term electricity markets have been progressively established across various provinces and cities in China. The primary function of these markets is to stabilize prices and minimize trading risks for participants [5].

The Yunnan and Sichuan Power Grids are two provincial grids in China where hydropower constitutes over 70% of the total installed capacity, exemplifying hydropower-dominant systems [6]. The Yunnan Power Grid was the first to pilot the latest phase of China’s power market reforms. So far, it has formed a variety of transactions, such as centralized bidding transactions, listed transactions, and continuous matching transactions. The trading cycle includes the medium- and long-term market trading system with time scales covering the year, month, and day. In the future, the Yunnan electric power market will form a complete system with medium- and long-term markets and a spot market [7]. Although the scale of electricity market transactions is expanding, it is still dominated by medium- and long-term market transactions [8], which maximizes the risk aversion of market participants. According to current energy utilization policy requirements, it is necessary to absorb as much clean energy as possible on the premise of ensuring the safe and stable operation of the power grids [9]. Completely free and decentralized market transactions may obtain economically optimal solutions, but to a large extent, will lead to an increase in power abandonment throughout the power system, resulting in a great waste of resources. In this case, the formation mechanism of the power generation scheduling plan adapted to the market environment is the connection point between the centralized dispatching power systems and the independent competitive power markets. It is a powerful means to realize clean energy consumption, safe and stable operation of the system and the steady implementation of power transactions in the hydropower enrichment areas. The dispatching of a hydropower-dominated power grid is faced with complex problems, such as the large scale of cascade reservoirs, the close connection of water and electricity between cascaded hydropower stations and basins, the large uncertainty of runoff, and the competition for power generation between upstream and downstream plants [10]. The medium- and long-term power generation scheduling strategy currently used by Yunnan Power Grid is still essentially an energy-saving power generation scheduling model under the planned system [11], focusing more on the matching of power generation and power consumption. As a result, hydropower stations frequently face challenges like overproduction, underproduction, and significant water wastage, which disrupts the smooth operation of medium- and long-term power markets and conflicts with China’s clean energy consumption goals. Balancing grid safety and stability and maximizing clean energy utilization poses difficulties for dispatch agencies. Hence, there is an urgent need for research focused on the medium- and long-term scheduling of hydropower-dominated grids within the power market context.

To date, extensive research has been conducted on the mid-to-long-term optimal dispatching of hydropower stations in the electricity market. Lu et al. [12] addressed the uncertainty of daily runoff and market clearing prices using information gap decision theory (IGDT) to devise a mid-term operation plan for cascade hydropower stations, ensuring varied profit expectations. Skjelbred et al. [13] tackled the bidding and scheduling challenges of hydropower systems across different time scales by developing a mixed integer linear programming (MILP) model that integrates hydropower benefits and time-scale coupling, achieving joint optimization of short- and long-term generation plans. Under a multi-stage mixed integer linear stochastic programming framework, Nieta et al. [14] explored optimal bidding strategies for long-term power generation. Antonio et al. [15] focused on the self-scheduling of a hydropower company in a pool-based market to maximize profits in the day-ahead market. Wu et al. [16] examined the interaction between monthly and day-ahead market trading for cascade hydropower stations. Considering the uncertainty of day-ahead market clearing prices, an optimal decomposition model was created for monthly contracted power of cascade hydropower stations, aiming to maximize power generation efficiency, and it notably improved efficiency. Yu et al. [17] developed a multi-objective model for long-term cascade hydropower operations across different markets, with objectives focused on maximizing economic benefits through portfolio theory and CVaR for risk assessment and maximizing year-end energy storage. This model was effectively solved using multi-objective particle swarm optimization (MOPSO). Liu et al. [18] simultaneously considered runoff and market price uncertainties, establishing a medium- and long-term contracted power decomposition model based on stochastic linear programming. However, most studies prioritize maximizing hydropower station benefits in the medium- and long-term markets from the generation side, overlooking the grid dispatch needs, which can lead to significant disparities in contracted power completion rates among stations, harming market stability. Cheng et al. [19] proposed a monthly transaction power decomposition model for large-scale hydropower stations, adopting different optimization objectives for flood and dry seasons, thus aligning monthly transactions with generation scheduling and mitigating issues of over- and underproduction. However, this study did not address scheduling risks due to runoff uncertainty.

To address fairness in power dispatching and mitigate water wastage due to runoff variability, this paper presents a two-layer IGDT-based model for medium- and long-term optimal dispatching in hydropower-dominated grids. The lower layer minimizes the maximum difference in contracted power completion rates across stations, while the upper layer maximizes the fluctuation range of interval runoff. To achieve a global optimal solution, various linearization strategies are employed to transform the original model into an equivalent single-layer MILP model, which is then solved using the CPLEX solver (Version 12.10.0). The effectiveness of the proposed model and method is demonstrated through the medium- and long-term optimal dispatching of 10 thermal power stations and 22 hydropower stations within the Yunnan Power Grid in China.

2. Optimal Scheduling Model

2.1. Scheduling Model Based on Predicted Interval Runoff

2.1.1. Objective Function

In the high-proportion hydropower electricity market, the fairness of transaction power execution and clean energy consumption are two issues that need to be considered in medium- and long-term scheduling. This paper mainly considers the monthly optimal scheduling and takes the fair execution of trading power as the optimization criterion; that is, the contracted power completion rate of each power station is balanced as much as possible. At the same time, in order to ensure that minimal to no water is abandoned during the execution of trading power, the penalty function for abandoned water is introduced into the objective function. When there is no error in the predicted value of the interval runoff of each hydropower station, the objective function is expressed as follows:

$$\min F=\max \left\{X_i\right\}-\min \left\{X_i\right\}+M\cdot {\displaystyle \frac{W-W_{\min }}{W_{\max }-W_{\min }}}$$

(1)

$$X_i={\displaystyle \frac{{\displaystyle \sum _{t=1}^T{e}_{i,t}}}{E_i}}$$

(2)

$$e_{i,t}=\left\{\begin{array}{l}{\displaystyle \frac{3.6Q_{i,t}\Delta t}{{\omega }_{i,t}}},i\le A\\ P_{i,t}\Delta t,A<i\le I\end{array}\right.$$

(3)

$$W={\displaystyle \sum _{i=1}^A{\displaystyle \sum _{t=1}^T\frac{3.6Q_{i,t}^{loss}\Delta t}{{\omega }_{i,t}}}},i\le A$$

(4)

where $X_i$ is the monthly contracted power completion rate of hydropower station/thermal power station i; M denotes the penalty factor of abandoned hydropower; W is the amount of abandoned hydropower of all hydropower stations [MWh]; i is the index of hydropower stations/thermal power stations; $W_{\max }$ is the maximum amount of abandoned hydropower [MWh]; $W_{\min }$ is the minimum amount of abandoned hydropower [MWh]; $e_{i,t}$ is the power generation of hydropower station/thermal power station i in time period t [MWh]; t is the index of time periods; T is the total number of time periods; $E_i$ is the contracted electricity quantity of hydropower station/thermal power station i [MWh]; $Q_{i,t}$ is the power generation flow of hydropower station i in period t [m³/s]; ${\omega }_{i,t}$ is the water consumption rate of hydropower station i in period t [m³/kW·h]; A is thetotal number of hydropower stations; $P_{i,t}$ is the average power generation of thermal power station i [MW]; I is the total number of thermal power stations and hydropower stations; and $Q_{i,t}^{loss}$ is the abandoned water flow of hydropower station i in period t [m³/s].

2.1.2. Constraints

(1)

Power balance constraints

$${\displaystyle \sum _{i=1}^I{e}_{i,t}}=L_t$$

(5)

where I is the total number of thermal power stations and hydropower stations, and $L_t$ is the predicted power consumption in period t [MWh].

(2)

Hydraulic connections between upstream and downstream reservoirs

$$I_{i,t}=Q_{i-1,t}+Q_{i-1,t}^{loss}+I_{i,t}^s$$

(6)

where $I_{i,t}$ is the inflow of hydropower station i in time period t [m³/s], and $I_{i,t}^s$ is the predicted interval runoff of hydropower station i in period t [m³/s].

(3)

Water balance constraints

$$V_{i,t+1}=V_{i,t}+86400\cdot (I_{i,t}-Q_{i,t})\Delta t,i\le A$$

(7)

where $V_{i,t}$ denotes the storage capacity of hydropower station i at the end of period t [m³].

(4)

Water level constraints

$$Z_i^{\min }\le Z_{i,t}\le Z_i^{\max },i\le A$$

(8)

where $Z_{i,t}$ denotes the forebay water lever of hydropower station i at the end of period t [m]; $Z_i^{\max }$ denotes the upper limits of the forebay water level of hydropower station i [m]; and $Z_i^{\min }$ denotes the lower limits of the forebay water level of hydropower station i [m].

(5)

Initial and final water level control

$$Z_{i,0}=Z_{i,begin},i\le A$$

(9)

$$Z_{i,end}-\delta \le Z_{i,T}\le Z_{i,end}+\delta ,i\le A$$

(10)

where $Z_{i,begin}$ denotes the actual water level of hydropower station i at the beginning of the month [m]; $Z_{i,end}$ denotes the control water level at the end of the scheduling period [m]; and $\delta$ denotes the allowable error of water level at the end of the scheduling period.

(6)

Generation flow constraints

$$Q_i^{\min }\le Q_{i,t}\le Q_i^{\max },i\le A$$

(11)

where $Q_i^{\min }$ denotes the lower limits of the power generation flow of hydropower station i [m³/s], and $Q_i^{\max }$ denotes the upper limits of the power generation flow of hydropower station i [m³/s].

(7)

Discharge flow constraints

$$Q_{i,down}^{\min }\le Q_{i,t}+Q_{i,t}^{loss}\le Q_{i,down}^{\max },i\le A$$

(12)

where $Q_{i,down}^{\min }$ denotes the lower limits of discharge flow of hydropower station i [m³/s], and $Q_{i,down}^{\max }$ denotes the upper limits of discharge flow of hydropower station i [m³/s].

(8)

Power generation constraints of the hydropower station

$$e_{i,t}^{\min }\le e_{i,t}\le e_{i,t}^{\max },i\le A$$

(13)

where $e_{i,t}^{\max }$ denotes the upper limits of power generation of hydropower station i [MWh], and $e_{i,t}^{\min }$ denotes the lower limits of power generation of hydropower station i [MWh].

(9)

Constraints of the water level-storage capacity relationship

$$V_{i,t}=f_{i,zv}(Z_{i,t}),i\le A$$

(14)

where $f_{i,zv}(\cdot )$ denotes the relationship function between reservoir storage capacity and water level of hydropower station i.

(10)

Water level–water consumption rate relationship constraints

$${\omega }_{i,t}=f_{i,z\omega }(Z_{i,t}),i\le A$$

(15)

where $f_{i,z\omega }(\cdot )$ denotes the relationship function between water consumption rate and the water level of hydropower station i.

(11)

Thermal power generation constraints

$$e_{i,t}^{\min }\le e_{i,t}\le e_{i,t}^{\max },A<i\le I$$

(16)

where $e_{i,t}^{\max }$ denotes the upper limits of power generation of thermal power station i [MWh], and $e_{i,t}^{\min }$ denotes the lower limits of power generation of thermal power station i [MWh].

Due to the long time scale of medium- and long-term optimal scheduling, this optimization model does not consider the ramping constraints and start-stop constraints of each power station/unit.

2.2. Robust Optimal Scheduling Model Based on IGDT

2.2.1. Description of IGDT

Information gap decision theory (IGDT) is a non-probabilistic, non-fuzzy optimization method designed to handle uncertainty without requiring probability distribution functions of uncertain variables [20]. Unlike other methods, the IGDT assesses the maximum or minimum fluctuation range of uncertain variables while ensuring the optimization objective meets or exceeds a preset goal [21,22]. In essence, it guarantees that as long as variables fluctuate within this range, the expected objectives are achieved. IGDT comprises two strategies: Risk-Averse Strategy (RAS) and Risk Seeker Strategy (RSS) [23]. RAS is a robust model focusing on minimizing the impact of uncertainty, while RSS seeks to maximize benefits from uncertain risks [24]. To prevent the uncertainty of interval runoff from affecting system fairness and causing water wastage, this paper adopts the IGDT robust model, specifically the RAS strategy. For a deterministic optimization model:

$$\left\{\begin{array}{l}\underset{d}{\min } f(x,d)\\ s.t.\left\{\begin{array}{l}h(x,d)=0\\ g(x,d)\le 0\end{array}\right.\end{array}\right.$$

(17)

where x denotes the input parameter; d denotes the decision variable; $f(x,d)$ denotes the optimization goal; and $h(x,d)=0$ and $g(x,d)\le 0$ denote equality and inequality constraints, respectively.

When d is an uncertain variable, assuming that the actual value of d fluctuates around its predicted value $\overline{d}$, the uncertain variable d can be represented by the information gap model [25], as follows:

$$\left\{\begin{array}{l}d\in U(\alpha ,\overline{d})\\ U(\alpha ,\overline{d})=\left\{d:\left|{\displaystyle \frac{d-\overline{d}}{\overline{d}}}\right|\le \alpha \right\}\end{array}\right.$$

(18)

where $U(\alpha ,\overline{d})$ denotes the set of uncertain variables d, and $\alpha$ denotes the fluctuation range of uncertain variables ($\alpha$ > 0).

Assuming that the actual value of the uncertain variable d in the deterministic model is equal to the predicted value, the optimal solution at this time is $f_0$. When the input variable is an uncertain variable, the target value under the maximum tolerance acceptable to the decision maker is set to $f_c$, which is expressed as follows:

$$f_c=(1+\sigma )f_0$$

(19)

where $\sigma$ denotes the deviation factor with a value range [0, 1]. The larger the $\sigma$, the larger the preset target $f_c$, and the greater the tolerance for the deterioration of the preset target.

Then, the robust optimal scheduling model defined by the IGDT method [26] is as follows:

$$\left\{\begin{array}{l}\underset{x}{\max \alpha }\\ s.t.\left\{\begin{array}{l}f(x,d)\le f_c\\ f_c=(1+\sigma )f_0\\ h(x,d)=0\\ g(x,d)\ge 0\\ d\in U(\alpha ,\tilde{d})\end{array}\right.\end{array}\right.$$

(20)

Through the model shown in Equation (20), the maximum fluctuation range $\alpha$ of uncertain parameters can be obtained. When the uncertain parameters fluctuate within the fluctuation range $\alpha$, the optimization target must be smaller than the preset target $f_c$.

2.2.2. Model Formulation

According to Equation (18), assuming that the actual value of interval runoff ($I_{i,t}^s$) fluctuates around its predicted value $\overline{I_{i,t}^s}$ [27], and the uncertainty of interval runoff can be represented by the information gap model:

$$\left\{\begin{array}{l}{I}_{i,t}^s\in U(\alpha ,\overline{I_{i,t}^s})\\ U(\alpha ,\overline{I_{i,t}^s})=\left\{I_{i,t}^s:\left|{\displaystyle \frac{I_{i,t}^s-\overline{I_{i,t}^s}}{\overline{I_{i,t}^s}}}\right|\le \alpha \right\},i\le A\end{array}\right.$$

(21)

If the optimal solution obtained from the original deterministic model is $F_0$, and the scheduler gives the preset target $F_c$ according to risk preference [28], the IGDT-based medium- and long-term optimal scheduling model of the hydropower-dominated power grid in the electricity market environment is shown as follows:

$$\overline{\alpha }=\max \alpha$$

(22)

$$s.t.\hspace{1em}F\le F_c=(1+\sigma )F_0$$

(23)

$$\max F^{*}=\max \left\{X_i\right\}-\min \left\{X_i\right\}+M{\displaystyle \frac{W-W_{\min }}{W_{\max }-W_{\min }}}$$

(24)

$$(1-\alpha )\overline{I_{i,t}^s}\le I_{i,t}^s\le (1+\alpha )\overline{I_{i,t}^s},i\le A$$

(25)

where $\overline{\alpha }$ denotes the maximum value of runoff fluctuation range in the robust model, and $\max F^{\ast }$ denotes the maximum value of the maximum comprehensive difference in the contracted power completion rate.

For other constraints, see Equations (5)–(16).

3. Model Solving Methods

3.1. Overall Solution Idea

In the above IGDT-based model, the maximum expected target depends on the fluctuation range of the interval runoff, which is the optimization objective of the model and is very difficult to solve directly. Therefore, a two-layer optimization model is established to solve the cross-relationship between the solutions of the two optimization problems. The upper model solves the fluctuation range of interval runoff under the premise that the maximum expected target $F^{\ast }$ satisfies the preset target $F_c$, and the lower model solves the maximum expected target $F^{\ast }$ when the fluctuation range of interval runoff is known. At present, the widely used method is to derive the first-order necessary optimality conditions of the lower model, namely KKT (Karush–Kuhne–Tucker) conditions, and use the constraints of the upper model to represent the lower model [29,30]. However, the long-term optimal scheduling problem of a hydropower-enriched power grid is a typical multi-dimensional and nonlinear problem, and it is difficult to obtain its KKT condition.

Starting from the nature of the research problem, this paper makes an equivalent transformation of the two-layer model [31]. The role of the lower model is to calculate the maximum expected target $F^{\ast }$ when the fluctuation range of the interval runoff is $[1-\alpha ,1+\alpha ]$. The role of the upper model is to solve the maximum fluctuation range $\overline{\alpha }$ of the interval runoff when the maximum expected target $F^{\ast }$ satisfies the preset target of robust optimization, and to obtain the scheduling scheme of each power station. It is worth noting that the larger the interval runoff, the more guaranteed the completion of the contracted power of the hydropower stations. In order to reduce the amount of abandoned water, hydropower stations will exceed the contracted power generation. At the same time, because of the balance between power supply and demand, thermal power stations will fall short of the contracted power quantity, which will cause the maximum difference in the contracted power completion rate of the power stations to increase. The larger the interval runoff, the greater the risk of water abandonment by the hydropower stations. In other words, the value of the expected target $F^{\ast }$ is positively correlated with the interval runoff. Therefore, when $I_{i,t}^s=(1+\alpha )\overline{I_{i,t}^s}$, $F^{\ast }$ takes the maximum value, and the comprehensive range of the contract completion rate is the largest. In this case, the robust model can be equivalently transformed as follows:

$$I_{i,t}^s=(1+\alpha )\overline{I_{i,t}^s},i\le A$$

(26)

For other constraints, see Equations (5)–(16).

3.2. MILP Model Formulation

The medium- and long-term optimal scheduling of a hydropower-dominated grid presents a complex, multi-dimensional, nonlinear, and non-convex challenge. Traditional algorithms like dynamic programming and genetic algorithms [32] often lack efficiency and cannot ensure optimal solutions [33]. Due to its efficiency and stable performance, mixed-integer linear programming (MILP) is widely applied in reservoir operations. This section focuses on linearizing the nonlinear elements of the model, transforming it into a standard MILP model, which is then efficiently solved using the commercial CPLEX solver (Version 12.10.0).

3.2.1. Linearization of the Objective Function

The objective function (1) is a nonlinear minimax form. In this paper, the auxiliary variable B is introduced, which represents the real number greater than or equal to the difference between the contract completion rates of any two power stations. It is presented as follows:

$$B\ge \max \left\{X_i\right\}-\min \left\{X_i\right\},\forall i$$

(27)

Formula (1) can be transformed into the following:

$$\min F=B+M{\displaystyle \frac{W-W_{\min }}{W_{\max }-W_{\min }}}$$

(28)

3.2.2. Linearization of the Forebay Water Level-Storage Capacity Relationship

Typically, reservoir capacity is a cubic or quartic function of the forebay water level [34], necessitating the linearization of the water level-storage capacity relationship. For hydropower stations with regulation capabilities better than weekly, the forebay water level fluctuates minimally within a month. In such cases, linear regression using the least squares method can represent the water level-storage capacity relationship near the initial water level. For stations with weekly or poorer regulation performance, where the forebay water level varies significantly, the piecewise linear interpolation method [35] is applied to linearize the water level-capacity curve between the dead water level and the normal high water level. This approach reduces the number of 0–1 integer variables and enhances solution efficiency.

$$V_{i,t}=a_i{Z}_{i,t}+b_i,i\le A$$

(29)

3.2.3. Linearization of the Forebay Water Lever-Water Consumption Rate Relationship

The water consumption rate is influenced by the forebay water level and decreases as the forebay water level rises, creating a nonlinear relationship. To address this, the successive approximation (SA) method [36] is employed to model the relationship between water level and water consumption rate. The main steps of the SA process are illustrated in Figure 1.

3.3. Model Solving Process

Coupled with the above linearization methods and strategies, the model solving process is presented as follows:

(1)

Substitute the initial water level, interval runoff, initial water consumption rate, and contracted power quantity of each hydropower station into the mid- to long-term optimal operation model based on predicted values.

(2)

Linearize the nonlinear factors to construct a standard MILP model, then solve it using the CPLEX solver (Version 12.10.0).

(3)

Output the power generation and water level processes of the hydropower stations to verify if the water consumption rate at the current water level meets accuracy requirements. (If satisfied, output the optimization results; if not, update the water consumption rate and return to Step 2).

(4)

Set $\sigma$ according to the risk preference of the dispatcher and calculate the preset target $F_c=(1+\sigma )F_0$ when the fluctuation degree of the interval runoff is $\alpha$. The feasible region of $\alpha$ in the robust optimal scheduling model is [0, 1], which is solved by dichotomy iterations. For the first time, ${\alpha }_1=0.5$ is substituted, the maximum value of $F_1^{*}$ is solved via the CPLEX solver (Version 12.10.0), and the water level process of each power station is output.

(5)

Determine whether the water consumption rate corresponding to the current water level meets the accuracy requirements. If it does not meet them, the water consumption rate is updated and then $F_1^{*}$ is recalculated. If it does meet them, the result is directly output to obtain the optimal value of $F_1^{*}$. Comparing the relationship between $F_1^{*}$ and $F_c$, if $F_1^{*}>F_c$, ${\alpha }_2=0.25$ is selected in the second iteration; otherwise, ${\alpha }_2=0.75$ is selected, and the accuracy of the water consumption rate is also judged. The next iteration is performed until $F_k^{*}>F_c$, $F_{k+1}^{*}<F_c$ or $F_k^{*}<F_c$, $F_{k+1}^{*}>F_c$, and the fluctuation range is ${\alpha }_k$. Finally, the iteration continues between $[{\alpha }_{k+1},{\alpha }_k]$ or $[{\alpha }_k,{\alpha }_{k+1}]$ until ${\alpha }_k$ satisfies the accuracy requirements.

(6)

Output the maximum fluctuation range of interval runoff, power generation for each station, and the water level process of the hydropower stations. The model solution process is illustrated in Figure 2.

4. Case Study

4.1. Engineering Background

The Yunnan Power Grid, one of China’s two largest hydropower-dominated provincial grids, had an installed hydropower capacity of 75,562 MW by the end of 2020, representing over 73.1% of the system’s total capacity. This study, based on the Yunnan Power Grid, focuses on the medium- and long-term optimal dispatching of 10 thermal power stations and 22 hydropower stations across the Lancang, Jinsha, and Lixian Rivers (see Figure 3 for the hydraulic connection diagram). The hydropower stations have a total installed capacity of 31,473 MW, while the thermal stations total 10,000 MW. This case study uses actual operational data from May of a given year, including predicted runoff, grid power consumption (excluding new energy and other hydropower stations), and monthly trading power for each station. The model is solved using the CPLEX optimization software (Version 12.10.0) on a 16-core, 2.3 GHz Dell workstation.

4.2. Analysis of the Optimal Scheduling Results

(1)

Power balance of the whole power system

The power balance of the power grid without considering the fluctuation of the interval runoff (that is, the risk tolerance factor was 0) is shown in Figure 4. Thermal power generation accounts for 17.5% of the whole power system, and hydropower generation accounts for 82.5% of the whole power system. The hydropower stations and thermal power stations participating in the optimization calculation achieve better load balance when the monthly contracted power is decomposed.

The power generation of the river basins and all thermal power stations, without considering the fluctuation of interval runoff, is shown in Figure 5. It can be seen that the power generation of Lancang River accounts for 58.61% of the total power generation, followed by Jinsha River, and Lixian River, which has the least power generation. The power generation of thermal power stations is less than 18%, which reflects the characteristics of a hydropower enrichment area dominated by hydropower generation.

(2)

Analysis of the impact of the penalty factor on scheduling results

Without considering interval runoff fluctuations and with a water abandonment penalty factor of 0.2, the power completion deviations for the Lancang River, Jinsha River, Lixian River Basin, and thermal power are −0.48%, −11.79%, 17.10%, and 27.73%, respectively. The range of contracted power completion rates across stations is 0.412, with a total abandoned hydropower of 6520 MWh. Due to less runoff, the Jinsha River Basin generates less power, while the Lixian River Basin, with more runoff and limited adjustment capacity, experiences excessive power generation and minor water abandonment. Without the penalty factor (M = 0), deviations shift to 0.51%, −10.95%, 16.40%, and 22.46%, respectively, with a contracted power completion rate range of 0.359 and abandoned hydropower increasing to 34,923 MWh. This shows that including the penalty reduces abandoned hydropower by 28,403 MWh, equivalent to a reduction of 32,663.5 tons of CO₂ emissions. The completion rates and abandoned hydropower for each station are detailed in

Table 1. The electricity completion rate and the amount of hydropower abandonment of each power station.

Basin	Power Station	Contracted Power Quantity (MWh)	Without Considering the Penalty Factor of Hydropower Abandonment			Considering the Penalty Factor of Hydropower Abandonment
			Actual Power Generation (MWh)	Deviation	Abandoned Hydropower (MWh)	Actual Power Generation (MWh)	Deviation	Abandoned Hydropower (MWh)
Lancang River	GGQ	203,906	202,563	−0.66%	0	198,645	−2.58%	0
	XW	1,314,228	1,271,364	−3.26%	0	1,267,584	−3.55%	0
	MW	528,766	578,376	+9.38%	0	557,101	+5.36%	0
	DCS	482,348	556,451	+15.36%	0	556,451	+15.36%	0
	NZD	2,404,990	2,241,677	−6.79%	0	2,240,870	−6.82%	0
	JH	606,429	718,504	+18.48%	0	693,590	+14.37%	0
	SUM	5,540,667	5,568,935	+0.51%	0	5,514,241	−0.48%	0
Jinsha River	LY	392,277	339,730	−13.40%	0	339,730	−13.40%	0
	AH	247,434	221,928	−10.31%	0	221,928	−10.31%	0
	JAQ	511,154	473,891	−7.29%	0	457,512	−10.49%	0
	LKK	302,789	268,182	−11.43%	0	268,182	−11.43%	0
	LDL	364,137	315,061	−13.48%	0	315,061	−13.48%	0
	GYY	112,790	100,479	−10.91%	0	100,579	−10.83%	0
	SUM	1,930,582	1,719,272	−10.95%	0	1,702,993	−11.79%	0
Lixian River	YYS	10,039	9984	−0.54%	0	9797	−2.41%	0
	SMK	12,975	14,780	+13.92%	0	14,177	+9.27%	0
	LM	50,438	61,767	+22.46%	18,405	64,426	+27.73%	2738
	JFD	101,622	115,096	+13.26%	0	111,452	+9.67%	0
	GLT	113,743	139,290	+22.46%	11,216	145,288	+27.73%	337
	TKH	53,405	57,263	+7.22%	0	56,035	+4.92%	0
	MYJ	8949	10,959	+22.46%	4285	11,431	+27.73%	3445
	PXQ	24,129	21,857	−9.42%	0	21,211	−12.09%	0
	SJK	34,377	42,098	+22.46%	91	42,009	+22.20%	0
	SNJ	62,003	75,930	+22.46%	926	76,495	−2.41%	0
	SUM	471,679	549,024	+16.40%	34,924	552,320	+17.10%	6520
Thermal power	QJ	107,136	131,199	+22.46%		136,848	+27.73%
	XW	107,136	131,199	+22.46%		136,848	+27.73%
	YW	81,468	99,766	+22.46%		104,062	+27.73%
	XLT	183,136	224,269	+22.46%		233,926	+27.73%
	DTHH	185,000	226,552	+22.46%		236,306	+27.73%
	XJS	39,283.2	48,106	+22.46%		50,178	+27.73%
	WX	184,600	226,062	+22.46%		235,796	+27.73%
	ZX	184,600	226,062	+22.46%		235,796	+27.73%
	YZH	107,136	131,199	+22.46%		136,848	+27.73%
	KM	104,008	127,369	+22.46%		132,853	+27.73%
	SUM	1,283,503	1,571,783	+22.46%		1,639,460	+27.73%

. While the absence of the penalty yields a smaller range of contracted power completion rates, indicating more balanced and fairer scheduling, it significantly increases abandoned hydropower, contravening China’s clean energy policies. Thus, incorporating a water abandonment penalty in medium- and long-term scheduling of hydropower-dominated grids effectively reduces water wastage and promotes clean energy consumption while maintaining scheduling fairness.

The completion of the contracted power of each power station under different water abandonment penalty factors is shown in Figure 6. It can be seen that after considering the penalty factor for energy abandonment, the contracted completion rates of LM, GLT, MYJ and SNJ hydropower stations in Lixian River Basin are higher than those which do not consider the penalty factor for energy abandonment, whereas the contracted power completion rate of other hydropower stations remains unchanged or decreases slightly. The reason is that there is too much water in the Lixian River Basin, but its overall regulation ability is poor. The abandoned water of the whole power system is mainly concentrated in four hydropower stations, i.e., LM, GLT, MYJ, and SNJ. After considering the penalty for abandoned water, these four power stations increase their power generation in order to reduce abandoned water, and the final water level also reaches the maximum value within the allowable fluctuation range.

The penalty factor was adjusted, and the scheduling results under different factors were analyzed, as shown in Figure 7. When the penalty factor is 0, the abandoned hydropower is at its highest, and the contracted power completion rate range is at its smallest. As the penalty factor increases, the range of the contracted power completion rate widens while the amount of abandoned hydropower decreases. With a penalty factor of 0.2, the abandoned hydropower reduces to 6520 MWh, and the range stabilizes, with the water abandonment concentrated in the Lixian River Basin. Due to the limited regulation capacity of this basin, further increases in the penalty factor do not reduce the amount of abandoned hydropower.

4.3. Analysis of Scheduling Results under Different Risk Factors

According to the different preset target ($F_c$) or risk tolerance ($\sigma$) of decision makers, the curve of the robust model target $\overline{\alpha }$ and the contract completion rate range with risk tolerance $\sigma$ are plotted, as shown in Figure 8.

As can be seen, the maximum fluctuation range $\overline{\alpha }$ of the interval runoff increases with the increase in the risk tolerance $\sigma$ of the hydropower stations. That is, the greater the risk tolerance, the more able the hydropower stations are to withstand large fluctuations in runoff. In addition, the range of the contract completion rate increases with the increase in risk factors. The main reason is that the interval runoff increases with the increase in the risk factor, and the abandoned water of the hydropower stations will gradually increase. In order to reduce the amount of abandoned water, the hydropower stations will generate as much electricity as possible, and the thermal power stations will have to reduce their power generation so that the range of the contracted power completion rate of each power station will increase.

When the risk tolerance factor $\sigma$ is 0.3, the contracted power completion of each power station is illustrated in Figure 9. When $\sigma$ is 0.3, the trading power completion deviations of the Lancang River, Jinsha River, Lixian River Basin, and thermal power stations are +4.44%, −0.02%, +25.21%, and −14.15%, respectively. The range of the monthly contract completion rate of the power stations is 0.453, the highest completion rate is 131.96%, and the lowest is 85.85%. The maximum fluctuation range $\alpha$ of the future actual interval runoff relative to the predicted value is 0.14082; that is to say, when the deviation between the actual interval runoff and the predicted interval runoff does not exceed 14.082%, it can be ensured that the range of the contract completion rate of each power station is not greater than 0.453. The interval runoff fluctuation ranges of a selection of hydropower stations are presented in Figure 10.

5. Conclusions

This paper proposes a medium- and long-term optimal scheduling method for hydropower-dominated power grids using IGDT, considering runoff uncertainty, fairness in contracted power execution, and clean energy consumption. The method is applied to 22 hydropower and 10 thermal power stations in Yunnan Power Grid, southwest China, with the following findings:

(1)

The range of contracted power completion rates is 0.412, ensuring balanced and fair dispatching across stations. Compared to scenarios without considering water abandonment, the method reduces abandoned hydropower by 28,403 MWh, equivalent to 32,663.5 tons of CO₂ emissions.

(2)

As the penalty factor increases, the range of contracted power completion rates widens while abandoned hydropower decreases. At a penalty factor of 0.2, both the abandonment and completion rate range stabilize. In practice, the choice of penalty factor depends on the decision maker’s preference for fair dispatching (smaller or zero value) or maximizing hydropower consumption (larger value).

(3)

The proposed method determines the range of runoff fluctuation that hydropower stations can withstand under different inflow risks, ensuring fair grid scheduling and minimizing water abandonment. This approach offers valuable insights for medium- and long-term optimal scheduling of hydropower-dominated grids.