An Optimal Operation Method for Parallel Hydropower Systems Combining Reservoir Level Control and Power Distribution

Parallel hydropower systems encounter a high risk of spillage in flood periods. Therefore, controlling spillage should be considered when hydropower systems increase power generation and satisfy power grid demand. To meet multiple operation targets, an effective operation method needs to precisely control the reservoir level and quickly adjust the power schedule based on real-time load changes. In this paper, an optimal operation method for parallel hydropower systems combining reservoir level control and power distribution is proposed. The method generates reservoir level control rules using a multi-objective simulation method for no gird demands operation. It obtains a power distribution schedule considering spill risk and non-storage losses for gird demands. Taking a parallel system in southwest China as an example, the results show that the method can produce an annual power generation of 55.2 billion kWh, which is a 6.1% improvement compared to regular method. Moreover, it can precisely control the daily water level and quickly adjust power distribution. It can be applied to parallel systems and has favorable results in power generation, peak shaving, and especially spillage reduction.


Introduction
Hydropower operation rules are usually used to determine the water level and output of hydropower plants to meet the operation targets of reservoirs and power grids. Various operation rules have been derived, but they are more widely used for a single reservoir (Vedula et al. 2005;Wang et al. 2019;Lund and Guzman 1999;Zhou and Guo 2013). With the increasing objectives and requirements of operation, operation rules applicable to cascaded hydropower systems and parallel hydropower systems began to appear (Lund and Guzman 1999), which provides research directions for many scholars (Ji et al. 2014;Wu et al. 2016;Zhou et al.2016;Ahmadianfar et al. 2020). In China, hydropower plants face complex operating environments such as joint reservoir group operation and multi-energy complementary operation. Spillage control has also become an important indicator in the operation work (Cheng et al. 2012;Feng et al. 2019). Against this background, the use of reservoir water level control to coordinate current and long-term operation benefits, power generation, and control of spillage risk has become an important operation method for many hydropower plants (Labadie 2004). To explore reservoir operation rules, Turgeon (2007) and Wang (2010) used the optimal reservoir trajectory (ORT) and a new stochastic control approach (NSCA) to solve the optimal water level operation trajectory of multi-reservoirs, respectively. He et al. (2020) derived optimality conditions for the two-stage problem of long-term generation scheduling, based on which the fast water level optimal control (FWLOC) method is proposed. In order to utilize flood resources as much as possible, Xu et al. (2020) developed a multi-objective stochastic programming (MOSP) model to address the conflict between flood risk and water shortage risk. However, there are relatively few studies on the risk of spillage in regions other than China (Moraga et al. 2006). Combining water level control rules and spillage control needs more research and exploration.
Operation rules provide the theoretical basis for long-term reservoir operation. It is equally important to determine the reservoir system's optimal storage and release sequence to cope with short-term load variations when developing a generation plan based on these rules. Common storage and release discriminative methods include the discriminative coefficient method ) and the reservoir capacity efficiency method (Wang et al. 2015), etc. Based on the discriminant coefficient method, some new load distribution methods for cascade hydropower plants have been developed (Jiang et al. 2018;Zhu et al. 2022). Zhang et al. (2019) defined an energy function that derives an α-discriminant formula that can be used to explicitly infer the optimal reservoir storage allocation scheme for maximum hydroelectric generation. Zeng et al. (2015Zeng et al. ( , 2019) derived a set of storage and release rules for a parallel reservoir system based on a two-stage model. They analyzed the relationship between the amount of water released and the amount available in each reservoir. Then obtained optimality conditions, followed by further research to propose improved dynamic programming (RIDP) and stochastic dynamic programming (RISDP) algorithms to optimize the dimensional disaster problem. Hui and Lund (2015) derived flood storage allocation rules for parallel reservoirs system to deal with the flood risk problem. Chang and Chang (2009) introduced the water scarcity index. They used the non-dominated ranking genetic algorithm (NSGA-II) to find the optimal Pareto front solution to guide the storage and release of two parallel reservoir systems in Taiwan. The above-mentioned reservoir level control rules and reservoir storage and release rules both solve only a single problem and have different application scenarios. In contrast, the actual operation problems faced are usually more complex and require more factors to be considered. To overcome this drawback, this paper proposes an optimal operation method combining daily water level control (DWLC) and power distribution. In this method, the optimal water level operation mode of each parallel hydropower station under different inflow is obtained based on the principle of more power generation and less spillage, referring to the cross-validation method (Wong 2015). Moreover, the optimal solution meeting the operation objectives is screened according to the fuzzy preference method (Guo et al. 2008). In order to meet the total load demand of the power grid, the hedging rule is extended to the parallel hydropower system, and the reservoir operation is dynamically adjusted with the goal of minimum spillage and storage loss. The method can precisely adjust the water level operation mode of the parallel system on a daily scale and reduces spillage while providing favorable power generation efficiency. The case study analyzes its applicability and feasibility in the Chongqing power grid in southwest China.

Problem Description
The drought limited water level (DLWL) rules are studied to benefit operation during dry seasons (Zhang et al. 2022). The DWLC proposed in this paper aims to improve the utilization of water resources and power generation efficiency in flood season. The two methods derived reservoir water level rules but have different application environment. In addition, although both methods can obtain the water level control of one year, DLWL can obtain the monthly water level change of large reservoirs. In contrast, DWLC can obtain the daily water level change of parallel small reservoirs. Generally, medium and long-term hydropower optimal operation problems are mainly discussed monthly or decadal. However, for parallel hydropower systems, some reservoirs with small storage may lose a lot of water resources because they can not afford concentrated inflow in a short period of time (Barros et al. 2003). Therefore, it is necessary to reduce the step size to a daily basis. Meanwhile, for specific physical and human characteristics, the operation results guided by the reservoir level control rules may not correspond to the grid demands. Further operation work based on this is required.

Solving Framework
The method proposed in this paper can optimize the parallel reservoirs operation problems in the following main steps: (1) generate multiple daily water level control rules (DWL-CRs) for each reservoir considering different operation needs; (2) determine operation targets and select a group of optimal DWLCRs for all reservoirs from simulation results of each feasible solution set; (3) control water levels guided by obtained DWLCRs; and (4) if there are grid demands, distribute power for all hydropower plants based on the real-time storage and release sequence.
The overall framework of this study is shown as Fig. 1.

Objective Functions
The hydraulic calculations of each hydropower plant in parallel system are performed independently and are no longer distinguished in the following equations.
For all hydropower plants, maximizing power generation is the most significant target, which can be described as: where N t k = the output of the hydropower plant k at period t; T = the calculation period, and one year is used as the control period in this study.
In some areas particular China, the target of minimizing spillage is also of great concern. The mathematical expression is formulated as: (1) where QS t k = the spill flow of the hydropower plant k at period t. To respond grid demands, the garget of minimizing variance of remaining load is formulated as: where C t = the system load at period t.

Water level constraints:
where Z t k = the water level of hydropower plant k at period t; Z t k , Z t k = the upper and lower bounds of water level of hydropower plant k at period t, respectively. 2. Hydropower plant output constraints: where p t k = the output of hydropower plant k at period t; p t k , p t k = the upper and lower bounds of output of hydropower plant k at period t, respectively. 3. Storage constraints: where V t k = the storage of reservoir k at period t; V t k , V t k = the upper and lower bounds of storage of reservoir k at period t, respectively. 4. Turbine discharge constraints: where q t k = the turbine discharge of hydropower plant k at period t; q t k , q t k = the upper and lower bounds of turbine discharge of hydropower plant k at period t, respectively. 5. Reservoir discharge constraints: where Q ′t k = the discharge of reservoir k at period t and Q �t = the upper and lower bounds of discharge of reservoir k at period t, respectively. 6. Reservoir water balance constraints: where Q t k = the inflow of reservoir k at period t. (2)

DWLCR Generation
The DWLCR is manifested as a set of daily-scale water levels over a one-year time horizon, whose applicability is determined through simulation verification. According to L inflow scenarios and M operation rules, the adaptable operation rule can be obtained by comparing L × M × L simulation results . In this study, when the inflow conditions to be solved are determined, it is not necessary to add L × M rules into various inflow conditions, which are relatively simple and feasible. To focus on the risk of spillage, describe DWLCR from two operation needs: the non-spillage duration and the non-spillage probability. The non-spillage duration means that the operation demands how long in a row no spillage occurs. If the non-spillage duration is too short, the spillage in the subsequent period cannot be effectively utilized. The flood control safety of the reservoir and the downstream reservoir area will be affected. If the non-spillage duration is too long, too much power generation head will be unnecessarily sacrificed, and the power generation benefit will be lost.
The non-spillage probability means that the operation demands how likely is the maximum probability that no spillage occurs for different inflow scenarios. This indicator is introduced to evaluate the fault tolerance of DWLCR for different inflow scenarios, which is linked to the inflow frequency. The richer the inflow conditions, the lower the corresponding water level in DWLCR, and the smaller the non-spillage probability. The long series of daily-scale historical inflow data is sorted according to the hydrological frequency method and then substituted into the DWLC model study. The smaller the frequency of inflow conditions, the greater the nonspillage probability of DWLCR. As a result, a certain kind of DWLCR can be expressed as a function of the non-spillage duration d and the non-spillage probability p.
According to these two conditions, with a set D of non-spillage duration and a set P of nonspillage probability, the resulting set A of feasible solutions for DWLCR can be expressed as: Several DWLCRs are recorded in the feasible solution set A. Taking a single reservoir as the object of study, for a certain period t, given any set of d i (i ∈ {1, 2, ..., m}) and p j (j ∈ {1, 2, ..., n}) , the reservoir is filled after a non-spillage duration. Then the initial reservoir capacity of the time period V t k can be derived. The relevant equations are as follows.
where h = the time sequence; Q t+h k (p j ) = the inflow of hydropower plant k at period t + h corresponding to the non-spillage probability p j .
The reservoir usually obtains higher economic benefits when it maintains at higher heads as much as possible. Assuming no spillage occurs for consecutive periods d i , and the hydropower plant generates power at maximum output to get more power generation benefits, the reservoir gets full at the end of periods. The reservoir capacity V t k at the beginning of the periods is derived through Eq. (12). Calculate time by time, and a DWLCR can be derived.

DWLCR Selection
Each combination of conditions in solution set A corresponds to a DWLCR, and these DWLCRs show different adaptability to different operation targets. It is a typical multiobjective optimization problem considering the effects of power generation, peak shaving, and spillage on operation benefits. To simplify this problem, the weight factors of different objectives are substituted, and the importance degree of each objective is determined according to the weight magnitude. The fuzzy preference method (Guo et al. 2008) is used to screen DWLCR that best meets the objective, and its objective function can be expressed as follows.
where w i = the weight of objective i, I ∑ i=1 w i = 1 , I = the total number of operation targets; r i,j = the relative superiority of the feasible solution j objective i, when the larger the value of the objective function the better, , when the smaller the value of the objective function is better, For each DWLCR in the solution set, simulations are performed to evaluate the power generation, spillage, and variance of remaining load expectation values conditional on historical inflow data in recent years, as shown in Eqs. (1)-(3). The operation target weights w 1 , w 2 , w 3 are set, and the local optimal solution is selected from the feasible solutions by the transformed Eq. (13). The obtained DWLCR can be used to guide the operation process.

Objective Functions
When the grid determines the total power of the hydropower system, a power distribution strategy should be applicated to meet grid demands. This work is carried out based on operation results guided by DWLCR. In other words, the operation results of DWLC are the initial solution of the power distribution.
Lund (2000) summarized the marginal hydropower value of present release from either reservoir in a parallel reservoir system based on Sheer's study as: where z T = the marginal benefit of economic benefit is with the release volume. The first term is the economic benefit for the current period, related to the electricity price P 0 and head for the current period H S 0 , and is a constant term. The second term is the economic benefit from the current period to the reservoir storage full. It is related to the change of electricity price P r , generation volume Q and average head of that period concerning the release volume H(S f ) T , as a function of the release volume. The third term is the generation loss generated after the reservoir is full, which is related to the electricity price of the period P f , the head of water after the reservoir is full H(K) , and the proportionality factor , and is also a constant term. The order of storage and release is determined according to this rule, and the larger z T is, the higher the marginal benefit of release. For Eq. (14), the first term can be neglected in this study because it is a constant if the power generation or output is taken as the decision variable. The second term can be understood as the loss of power generation electricity caused by the reduced head when a certain amount of water is released for the remaining period of the dispatch cycle, which is called the non-storage loss for that period. The consideration of spillage in the third term is characterized by a proportionality factor. In this study, it is necessary to explore the spillage loss term further in depth.
For areas with excessive inflow, the risk of spillage cannot be ignored. The risk of spillage is characterized by abandoned power in this study, which represents the amount of power that can be generated by spillage. When the system load increases, hydropower plants with high abandoned power will increase their output. The power distribution strategy takes the minimum abandoned power as the main objective and the minimum non-storage loss as the secondary objective. In extreme rainfall, the method will consider the current water level and inflow of each reservoir in the parallel system to determine the abandoned power. According to this objective function, the reservoir with more spillage will be discharged in advance for power generation to make more use of inflow for power generation and increase the power generation of the whole parallel system. The objective function can be expressed in the following form. where Z t c +1 k , Z � t c +1 k = the water levels of reservoir k before and after increasing power generation at time t c + 1 , respectively; T c = length of control period for increasing power generation at calculation period t c affecting subsequent time periods, generally until the end of reservoir refilling.

Constraints
The relevant constraints are the same as for DWLC model, and are shown in section 3.2.

Discriminant Calculation
The abandoned power is obtained by the following method for any hydropower plant. From the calculation period, the maximum reservoir storage capacity V t c in,k during the nonspillage duration d i is: When the frequency of inflow is p j , the amount of inflow from calculation period V t c in,k (d i , p j ) in time d i is: Comparing the inflow with the maximum available storage, the spillage and abandoned power can be calculated from following equations.
where VS t c k (d i , p j ) = the spillage of hydropower plant k from calculation period t c corresponding to the non-spillage duration d i and the non-spillage probability p j ; ES t c k (d i ) = the abandoned power of hydropower plant k from calculation period t c corresponding to the non-spillage duration d i .
The non-storage loss can be calculated for any hydropower plant from the following equations.

Power Distribution
The output of each hydropower plant is adjusted according to the objective function. As the output of the hydropower plant changes, the abandoned power and non-storage loss will change, so multiple output distribution is required. The specific steps are as follows.
1. Obtain the water level and output conditions of each hydropower plant and the total system load at the beginning of the period, and determine the power difference value p target . Set the non-spillage duration d i , which is recommended to be selected from 1 to 7. 2. From the historical daily scale incoming data, calculate the abandoned power of each hydropower plant according to Eqs. (17)-(21). Set Δp t = 1 , and calculate the non-storage loss according to Eqs. (22)-(25). 3. Order the release of the current period by the minimum abandoned power and the minimum loss of non-storage, giving priority to the hydropower plant with the minimum abandoned power. If the same abandoned power is used, priority will be given to the one with the lower non-storage losses. 4. Divide p target evenly into parts, and if the output needs to be increased, increase the output p target ∕ to the first hydropower plant in the sequence according to the sorting result, and if the output needs to be reduced, reduce the output p target ∕ to the first hydropower plant in the reverse order according to the sorting result. The larger is, the more accurate the power distribution result is and the longer the time required for calculation. 5. Apply the new output conditions, go back to step 2, and repeat times, the system load is fully distributed, and the power distribution is completed for that period.

Engineering Background
Chongqing is located in southwestern China, which has spill water resources. There are 19 different rivers providing hydroelectric energy in Chongqing, which is located in southwestern China. And most of the rivers have only one hydropower plant with high regulating ability. In this study, 10 representative hydropower plants were selected as the subjects, all of which have seasonal regulating abilities and above, and relatively large reservoir capacity. Other hydropower plants have mostly very limited regulating abilities, so they are not analyzed in this study. In the description of runoff uncertainty, there are quite mature research results (Ramaswamy and Saleh 2020). However, due to the limited runoff data, this study only ranks according to the historical runoff data, and the runoff uncertainty is represented by different frequencies of inflow (Fig. 2). For any hydropower plant, the non-spillage duration D = (1, 2, 3, 4, 5, 6, 7) and the nonspillage probability P = (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0) are set. The equal weights consider the power generation, spillage, and peak shaving demand during the operation of the hydropower plant. The characteristic parameters of each reservoir are shown in Table 1.

Results and Analysis
Based on the non-spillage duration and non-spillage probability, 70 DWLCRs selected for each reservoir are obtained from DWLC model. All the to-be-selected DWLCRs are separately guided for the operation of multi-scenario inflow, and the average power generation and average spillage for each DWLCR operating under multi-scenario inflow conditions  Figure 4 shows the relationship between generation and spillage with the nonspillage duration for the non-spillage probability of 0.7 and 0.9. The positive direction of the vertical axis in Fig. 3 corresponds to the increasing water level of DWLCR. The scatter diagram shows that for different DWLCRs, the spillage basically increases with the increase of power generation, which can infer that at higher water levels, although the risk of spillage increases accordingly, more power generation benefits can be obtained from higher heads. In the upper part of the scatter diagram, there is a large amount of spillage and a smaller amount of power generation, which indicates that too high water levels are not conducive to increasing the power generation benefit due to the loss of water for power generation.
The optimal DWLCR of each hydropower plant can be obtained from simulation results in the feasible solution set A. The results are shown in Fig. 4. Compared to the regular water level control method that the water level varies evenly, emptying the reservoir at the beginning of May and filling it up at the end of September, the DWLC method has better performance in the degree of membership function (Eq. 13), and the results  are presented in Table 2. Under the DWLC rule, the 6-year simulated operation results in an average generation of 55.2 billion kWh, a 6.1% improvement compared to 52.0 billion kWh of the regular rule. Using the resulting DWLCR as a guide for operation, the results obtained may differ from the generation orders issued by the grid and require corresponding output corrections. The power distribution strategy is used to allocate a load of each hydropower plant according to the system load for the next 5 days for a certain period. To analyze the impact of the proposed method on spillage, the power distribution approach, which only considers the non-storage loss, is set as the regular distribution approach, and the power distribution approach, which considers both the abandoned power and the nonstorage loss, is set as the optimal distribution approach. After the power distribution, both approaches achieve power balance in each period. Table 3 shows the results of the first release ranking (1 is the highest priority). Figure 5 shows the process of power distribution for both approaches over a 5-day period. Table 4 shows the spillage for each period.
As seen from Tables 3, 4, and Fig. 5, the optimal approach more significantly releases water from reservoirs with higher abandoned power in the first few periods compared to the regular approach, reducing the risk of spillage in the subsequent periods. Although there is no reduction in spillage in the first two time periods, from the third period onwards, it can be observed that the optimal approach significantly reduces spillage.

Conclusion
This paper proposes an optimal operation method combining DWLC and power distribution for parallel hydropower systems. The DWLC can generate an optimal DWLCR according to different operation targets, which provides favourable benefits if there is no power grid demand. The power distribution strategy is determined to achieve real-time power balancing when faced with load changes in a short period. The primary conclusions can be summarized as follows.
1. This method applies to the optimal operation of parallel hydropower systems with small reservoir capacity in flood season. Although operation according to a suitable DWLCR does not necessarily lead to optimal operation results in a particular case, this method can obtain better expectations for different inflow scenarios. 2. This method can generate different power generation schedules that meet different operation requirements while capturing intra-month spillage phenomena that are difficult to observe with regular operation methods. 3. The setting conditions of the model are ideal, and other uses, such as navigation, ecology, and irrigation, are not considered. The effect may not be obvious under some extreme inflow conditions. In order to ensure the maximum comprehensive benefits, reservoirs operate at a high-water level most of the time, and their safety needs to be further verified.
The DWLC proposed in this paper is simple and practical. This method is conducive to coordinating the normal operation of the parallel systems and giving full play to the compensatory regulating ability of each reservoir.
Author Contribution All authors contributed to the study conception and design. Xinyu Wu and Chuntian Cheng provided the idea of the model construction. Xinyu Wu and Yuan Lei collected study data and analyzed model performance results. The first draft of the manuscript was written by Yuan Lei and revised by Qilin Ying. All authors read and approved the final manuscript.
Funding The research work described in this paper is supported by the National Nature Science Foundation of China (52179005 and 91647113).
Data Availability Some data, models, or code generated or used during the study are available from the corresponding author by request.

Declarations
Ethical Approval Not applicable.