A refined load distribution method for large hydropower stations with multiple units considering constraints of substations

Large hydropower stations often undertake peak regulation tasks during the non-abandoning water season. This requires a reasonable arrangement of unit commitment and load distribution (UCLD). In this study, a load distribution model considering constraints of substations was established, and a novel refined and practical method (RPM) was proposed by considering a whole plant-substation module, substation-unit module, and adjustment module along with practical strategies. The Three Gorges hydropower station in China was selected to demonstrate the effects of the RPM. The results showed that the RPM with high calculation timeliness can obtain UCLD schemes with high rationality and practicability under the constraints of the whole plant, substations, and units. The relative mean absolute error of the simulation outflow could be controlled within 1%, thereby providing a valuable reference for forecasting the outflow process which involving the unit level.


I. INTRODUCTION
In the context of carbon emissions peaks [1] and carbon neutrality [2], the energy structures and industrial layouts of various countries have changed to different degrees. In recent years, the installed capacities of new energy sources such as wind and solar energy have been growing rapidly in China [3]. However, the uncontrollability and randomness caused by their grid-connected operations also pose a significant threat to the security and stability of the grid [4]. Hydroelectric energy is a type of clean and conventional energy with a flexible peak regulation capacity, and is generally a superior choice for addressing with these difficult problems [5]. Therefore, studying the operation management of hydropower stations is of great significance for optimizing the national energy structure, smoothly promoting the connection of new energy sources with the grid, and gradually realizing the goal of carbon neutrality.
Large hydropower stations often undertake peak regulation tasks from the grid, and are expected to carry out the load instructions strictly, arrange unit commitment and load distribution (UCLD) reasonably, and adapt to load fluctuations quickly [6], [7]. UCLD is a typical nonlinear large-scale optimization problem with complex constraints [8], [9]. The computation burden of UCLD grows exponentially with an increase in hydro units. Cheng et al. [10] pointed out that when the number of hydro units is greater than 10, a dimensionality disaster occurs. In the past few decades, a large number of studies have attempted to solve the UCLD problem; these studies can be broadly divided into three categories.
The first category is the traditional methods, mainly including lagrangian relaxation (LR) [11], [12], nonlinear programming (NLP) [13], mixed integer linear programming (MILP) [14], [15] and dynamic programming (DP) [10], [16]. These methods have gained elegant achievements for specific scenarios with simple constraints and small scales. However, their respective flaws are the main obstacles for the extensive application. LR can achieve a quick solution, but it is generally difficult to find suitable lagrange multipliers [17]. Nonlinearity is a prominent feature of the UCLD problem, and is mainly caused by the dynamic operation characteristics of hydro units. Raouia Taktak et al. [18] summarized that NLP is a very efficient method to model and solve the UCLD model. However, it is not easy to manage the nonlinearity of NLP in general, as approaches tend to be mainly limited by the scale of the problem and solving tools. While MILP can be aided by normalized and efficient commercial software solvers, such as LINGO and CPLEX. More importantly, there is no limit to the size of the problem. Xiang Li et al. [19] utilized MILP to optimize the hydro unit commitment. However, linearization has an extreme influence on the solution feasibility. Especially for large hydropower stations with numerous units and complex constraints, the poor quality of linearization can easily lead to unsatisfactory dispatching results [18]. Along with MILP, DP has been one of the most classical algorithms for solving the UCLD model. More importantly, it is skilled in addressing nonlinear optimization problems. But with an increase in units, the calculation efficiency of DP decreases sharply [10].
The second category includes heuristic methods, such as the particle swarm optimization [10], [20], ant colony algorithm [21]- [23], genetic algorithm [24], [25], evolutionary programming [26], [27] and differential evolution [28], [29]. These algorithms usually determine a suboptimal solution by simulating a specific natural phenomenon. In general, the modeling steps can be divided into three steps. First, the sequence of the unit on-off states during the dispatching period is abstracted into a single particle, and an initial population is generated. Second, the iteration and update rules of the population are defined according to different natural phenomena, and constrainthandling strategies are formulated. Finally, the rules and strategies are executed until the termination conditions are met. Compared with traditional methods, heuristic methods have powerful searching abilities and high calculation timeliness [30]. However, the dispatching results are often not convergent, leading to deep confusion regarding the actual dispatching decisions of hydropower stations; this has become the main obstacle to their application in production practice [31].
The third category includes methods with practical strategies. Based on a detailed analysis of the operation characteristics of the target hydropower station, the concerns affecting the dispatching results are analyzed, and practical methods are proposed. For example, Erlon Cristian Finardi et al. [32] proposed a two-phase decomposition strategy to yield accurate and practical results for the UCLD in the Brazilian regulatory framework. Thomas K. Siu et al. [33] designed the hierarchical approach to determine the optimal hydroelectric unit generation schedules. Li Linfeng et al. [34] used Xiluodu as the research object, and proposed a practical method for increasing and decreasing the type and number of hydro units under varying load conditions. Zhou Jianzhong et al. [35] proposed a load-adjustment method with regime-changing conditions for the Yuanshui cascade hydropower stations in China. Methods with practical strategies such as these can be regarded as summaries of practical dispatching problems; in general, they greatly improve the solution efficiency for the UCLD problem, and have strong engineering application value. In contrast to the first two categories, proposals of this type of method require a full analysis and understanding of the case, and the effects and performances also need to be repeatedly verified [36].
In recent years, a large number of large hydropower stations have been successively put into operation in China, such as Wudongde and Baihetan [37], [38]. Large hydropower stations with multiple hydro units are often divided into several substations according to the relative positions of the units. Moreover, different substations are usually connected to different outgoing transmission lines, and the constraints for the substations from the grid and those from the substations themselves are not identical. Previous studies on the UCLD problem have considered many constraints; these all can be divided into a constraint set for the hydropower station level (CS-HSL) and a constraint set for the unit level (CS-UL). However, so far there hasn't been a method to deal with the UCLD problem considering the constraint set of the substation level (CS-SL) (hereafter the " UCLD-SC"). Hence, this has led to new and urgent requirements for power generation enterprises to strengthen the refined dispatching and management levels of hydropower stations, and to consider the substations constraints when modeling the UCLD problem; this is the main motivation of this study. In this study, the UCLD-SC model is established and a novel refined and practical method (RPM) is proposed for the first time. The Three Gorges hydropower station (TGHS) is selected as a case study for verifying the high applicability and timeliness of the RPM.
The remainder of this paper is organized as follows. Section II introduces the study case. Section III describes the UCLD-SC model, including the objective function, constraint sets, and other modeling principles. Section IV introduces the RPM in detail. Section V presents the test results and discussion, and Section VI summarizes the conclusions of this study.

II. STUDY AREA AND HYDRO UNIT LAYOUT
The Three Georges Water Conservancy Project (TGWCP) is located on the main stream of the Yangtze River in China [39]. Gezhouba [40], with daily regulation capacity, is located 38 km downstream from the dam site of the TGWCP, and has a remarkable jacking effect for the TGHS. The locations of the TGWCP and Gezhouba are shown in Figure  1, and the engineering layout of the hydro units in the TGHS is presented in Figure 2. As shown in Figure 2, the TGHS is equipped with 34 units; these belong to nine different types (marked with different colors). According to the layout of the units and connection mode of the outgoing transmission lines, the units are divided into four substations: the power substation, left bank substation, right bank substation, and land substation. Moreover, these four substations are physically independent from each other. In particular, there are two bus connection switches distributed in the left and right substations, and the on-off state of each switch is restricted by the grid. The main reason for choosing the This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. TGHS as the study case is that it has the largest number of substations and the most complex constraints. Meanwhile, it also has the unique problem of the bus connection switch.
The other large hydropower stations mentioned in the introduction section are simplified cases of the TGHS.

Ⅲ. PROBLEM FORMULATION
Based on the investigations from Three Gorges Cascade Dispatch & Communication Center (TGCDCC) [41], which governs the TGHS, the optimal operation efficiency of the hydro units is often not pursued deliberately in the actual dispatching, that is, the hydropower stations do not always maintain the minimum water consumption conditions for a given load. There are two main reasons for this phenomenon. First, the operations of the hydro units are restricted by the grid, and cannot maintain the optimal operation conditions at all times. Second, from the perspective of safety, for both the grid and hydropower station, it is desired that the outputs of units are relatively stable. Hence, the goal of the UCLD-SC is to determine a suboptimal dispatching scheme in which UCLD are reasonably arranged. The objective function [10] is defined as follows. (The modeling variables in this paper can be viewed in Appendix.)

A. OBJECTIVE FUNCTION
Notably, the time scope of this study does not include the abandoning water season.

B. CONSTRAINTS
After discussions with the site dispatchers in the TGCDCC, all of the concerned constraints are considered and grouped into the triple constraint sets. Different constraints have different degrees of importance. The important constraints are treated as rigid constraints, i.e., the dispatching results should obey them. The soft constraints are marked directly when introduced. Soft constraints are of less importance to the solution, and may have a significant influence on the feasibility of the solution. When a feasible solution cannot be found, the soft constraints should not be considered.

B-I. CONSTRAINT SET-HYDROPOWER STATION LEVEL (CS-HSL)
(1) Water balance constraint (2) Load-balance constraint , , , Here, we number the same units from the perspectives of the whole plant and substation to make the subsequent statements clearer.
This constraint is the first soft constraint. The operation efficiency is relatively high when units operate around the rated output. Therefore, the operating number of units should be limited.

B-II. CONSTRAINT SET-SUBSTATION LEVEL (CS-SL)
( This constraint is another soft constraint. It can avoid obtaining a solution in which the output of the entire plant is excessively concentrated in certain substations while the output of the other substations is almost zero. (3) Stable operation zone (SOZ) constraint of substations The operation area of a hydro unit can be divided into a SOZ and forbidden operation zone (FOZ). The SOZs of units in TGHS are all continuous. For example, Figure 3 shows the distribution of the SOZ and FOZ for Unit 3 in the left bank substation. If a gross water head is not exactly equal to the ordinate values in Figure 3, gross water head and bounds of SOZ will be calculated by linear interpolation. In view of this, the SOZs of substations can be expressed as follows: Notably, the empirical formula for t Q and the water difference is often utilized to determine tail t Z in the TGCDCC at present; the definition is shown in Equation (17).

B-III. CONSTRAINT SET-UNIT LEVEL (CS-UL)
Except for nonlinearity, the combinatorial aspect and discontinuous characteristics are the other two major characteristics of the CS-UL. The combinatorial aspect is caused by discrete factors such as the on-off states of the units, and the discontinuity is limited by the FOZs of the units [42]- [44].
This is the third soft constraint, and is used to avoid frequent unit starts and stops to a certain extent.   According to the site investigation, except for the triple constraint sets in Section III-B, the modeling of the UCLD-SC also needs to meet certain load distribution and unit operation principles, as follows. Principle-I: The units of the TGHS should operate to exceed 70% of the expected output (i.e., 490 MW), as guided by the manufacturer's instructions. Principle-II: the units should not frequently cross between the SOZ and FOZ. Principle-III: Load fluctuations with a small amplitude should be handled by the units currently in operation as much as possible.

IV. METHODOLOGY
When the UCLD-SC contains triple constraint sets, there are three levels of output from the hydropower stations, i.e., the outputs of the entire plant, substations, and units. Therefore, we divide the solving method into two modules: the whole plant-substation module (WPSM) and substation-unit module (SUM), in which the WPSM determines the initial output of each substation, and SUM determines UCLD. Notably, the initial output of substations as determined by the WPSM may not satisfy the SOZ constraint of substations, which is a focus in the RPM. Therefore, an adjustment module with practical strategies (AM-PS) is proposed for adjusting the output of substations. The details of these modules and the overall flowchart are described in the following sections.

A. WHOLE PLANT-SUBSTATION MODULE (WPSM)
The central distribution principles of WPSM are that the condition of UCLD should keep as stable as possible between periods, namely the Principle-III in Section III-C. When the load fluctuation is large, it should be jointly undertaken by substations. In this way, output of all the substations can vary relatively smoothly, which is beneficial to safe operation. The specific steps of the WPSM at period t are as follows. ( Here, the parameter of unit commitment is ,1 kt− s . If the unit commitment at period t-1 satisfy output requirements at period t, the unit commitment will remain original. This obeys the Principle-III in Section III-C.    (33) and Equation (34).
The process of the one-step output adjustment is terminated until t N  is equal to zero or the adjustable output spaces have been run out. Then, Thus far, the WPSM processes have been introduced. The installed capacities of substations are different, so the one-step output adjustment in Step (5) and (6)  For substations in FSS, calculate adjustable output spaces.
Discretize by , and conduct the one-step output adjustment in Step (5) repeatedly .

B. SUBSTATION-UNIT MODULE (SUM)
There are three categories of methods can be used to determine the UCLD. Here, the UCLD should be determined with stability and high timeliness. Simultaneously, all units need to operate in high efficiency zones. Heuristic methods are excluded because results not convergent. While, the third category methods with practical strategies are proposed with difficulty to determine UCLD for so many constraints. They are more skilled in designing solution framework, namely the RPM in this paper. In the traditional methods, except the high timeliness, DP perfectly satisfies the solution requirements of UCLD-SC with 34 units. Jia Benjun et al. [45] worked out the optimal load allocation table (OLAT) for the whole plant (OLAT-WP) with the aim of minimizing the streamflow consumption by using DP (Here, the Principle-I in Section III-C can be satisfied easily.), and quickly calculated the output and outflow by consulting the OLAT-WP. More fortunately, Zhang Yongchuan [36] has already proved a generalization theorem regarding the optimization principle, and showed that a lookup based on the OLAT-WP could also achieve the optimal allocation under any unit commitment.
Here, the specific meaning of any unit commitment reflects the maintenance of units, and some units cannot participate in the load distribution. This discovery leads to an important conclusion that solving the UCLD-SC requires only one OLAT that contains all of the units, namely, the OLAT-WP. When the output and unit commitment of a substation are known, the load distribution of units can be determined by N is equal to 0 P , the following steps illustrate how to allocate it to the units in substation k, excluding units l and m.
(1) Select row P at the OLAT-WP.
(2) Excluding unit l and unit m, calculate the total output at row P , and mark it as 1 P .  N . Hence, the load distribution of substation k is determined. 1 ,, ,

C. ADJUSTMENT MODULE WITH PRACTICAL STRATEGIES (AM-PS)
After the adjustment of AM-PS, the outputs of substations will meet the SOZ constraint, and the unit commitment is also determined, so that the SUM can be directly invoked to determine output of units. The steps of AM-PS are as follows. ( , , , , ,  Figure 6. , , , gross gross gross gross , , ,  Sub are all equal to zero, the AM-PS is terminated. Otherwise, the process returns to Step (1), and the switching strategy is updated to expand the range of the SOZ at substation k. Specifically, when executing the startup operation, the unit with the lowest lower bound of the SOZ has the highest priority for startup, which can lower the lower bound of SOZ. Conversely, when executing the shutdown operation, the unit with the lowest upper bound of the SOZ has the highest priority for shutdown, which can raise the upper bound of SOZ.
The flowchart of AM-PS is as follows:

D. OVERALL FLOW
Based on the three modules, the overall flow of the UCLD-SC is as follows; the flow chart is shown in Figure 8.
(1) Read the items of the calculation conditions, including the triple constraint sets, inflow and supply streamflow, load instructions, FWL of Gezhouba within the dispatching period, and the output and online/offline times of the units at the initial time of dispatching.

E. COMPARATIVE EXPERIMENTS AND EVALUATION INDEXES
To comprehensively demonstrate the effects of the RPM, comparisons and analysis will be carried out from two aspects. The first aspect is to test computational efficiencies and outflow accuracies of RPM, which can be assisted by comparative experiments. Hence, the variable comprehensive efficiency coefficient method (VKM) [45] and the water consumption rate method (RM) [46] for their high precision and timeliness are selected to calculate the outflow as contrast experiments. The FWL process of Gezhouba was set as the actual process in three methods. The steps of contrast experiments were as follows.
(1) The items for the calculation conditions were read.
(2) According to the initial FWL, the constraints of FWL and outflow, determined the feasible range of the FWL at the end of period t, as follows: In this study, the load instructions were set for the actual process. The TGHS has been running steadily for many years and has accumulated significant dispatching experience, in which the hydro units often operate under suboptimal conditions. Hence, the high accuracy of outflow means that the calculated streamflow is close to the actual value. The pivotal evaluation indexes [47] are listed as follows: This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
The second aspect is to analyze the rationality and practicability of UCLD results from RPM. However, there are currently no relevant methods for solving the UCLD problem considering constrains of substations. Hence, some insight discussions, such as the operation efficiency of units, the on-off situations of units, the situations of crossing between SOZ and FOZ, and load transfer situations of substations [47], are made based on the UCLD results from RPM.
In addition, the cases were selected from 2021, as only the actual operation data, including the repair plans for the units and on-off states of the bus connection switches from January 1 to April 7 in 2021 were available. Utilizing the actual data during the non-abandoning water season from 2018 to 2020, the mean of water consumption rate was calculated as 11.

B. Results and Discussion
In 2021, the bus connection switch of the left bank was separated from January 1 to January 20, and was closed from January 21 to April 7. However, the right one was always closed. Therefore, a typical scenario was selected as the simulation case in each period, namely January 12 and April 1, respectively. A bimodal operation condition with a large peak modulation amplitude was selected as the typical scenario. This condition exhibited a multi-stage load fluctuation, and the aftereffects of the tail water level were stronger, thereby providing a good test of the effects of the RPM. In addition, the main function of the power substation is to supply power to the TGHS. In daily dispatching, the power substation seldom participated in optimizing the dispatching for the UCLD-SC. Therefore, we set the condition of the power substation as the actual condition. Under the parameters mentioned in Section V-A, the three methods were utilized to calculate the outflow process. While the RPM are utilized to solve the UCLD-SC with triple constraint sets. The key information for the two cases is presented in Table II. In the above, Number is the total number of substations in the corresponding scenario. L-off indicates that the bus connection switch of the left bank is separate. The value before / is the amplitude of the peak regulation, whereas the latter is the rate between the peak regulation amplitude and installed capacity of the TGHS. Repairing lists the units in the repair status. The unit of peaking amplitude is MW, while the units of other parameters are listed in APPENDIX.
First, evaluation indexes of the two scenarios are listed in TABLE III. Overall, the effects of VKM and RM are almost the same. For the perspective of timeliness, the cases are conducted on a personal computer with a 1.6 GHz processor and 8 GB of RAM. For each case, identical calculations are performed ten times and the average time consumption of the three methods is less than 1 s, indicating that looking up the OLAT-WP to determine the load distribution is efficient to solve the UCLD problem with 34 units. Comparing with VKM and RM which not involving units, the calculation efficiency of RPM not decreases significantly.
For the perspective of accuracy, even though the MAE of RPM is slightly larger than that of the VKM and RM, its MAE can be limited within 100 m 3 /s, and its RMAE can still be limited within 1%. The differences between the simulated and actual flows are shown in Figure 12 and 13. Seen from the figures, the simulation outflow tracks the trend of the actual process well. Even in periods with large fluctuations in output and outflow, the outflow error does not increase significantly, reflecting the high robustness of RPM. High accuracy in simulating outflow makes the FWL deviation controlled within 0.1 m throughout the entire period. In addition, it should not be ignored that most of the streamflow as simulated by the VKM and RM is less than the actual one. The actual streamflow through the units usually falls within an excellent allocation scheme. Hence, the reliability of VKM and RM is questionable. While, the RPM can consider the unit commitment and provide more specific dispatching advice for hydropower stations.  Next, analysis comes to rationality and practicability. For the operation efficiency, TABLEs IV and V show the output processes of units. Except for the units in the power substation, no operating unit undertakes an output of less than 490 MW. (The installed capacity of units in the power substation is only 50 MW.) By statistics, the indicators Nummax on January 12 and April 1 are equal to 24 and 18, respectively. Though the Nummax on April 1 is slightly larger than the given one in TABLE II, the outputs of units are determined by consulting the OLAT-WP, the load distribution within each substation is optimal. Hence, the operational efficiency of units can be guaranteed, not to mention this constraint (i.e., Formula (12)) is a soft constraint.   For the on-off situations, Figures 14 and 15 show the scheme for starting up and shutting down units. During the entire period, there is no frequent switching for the units, and the on-line/off-line time is larger than 4 h. The conversions of units in the on-off state are based on the unit combination in the previous period, and make full use of the adjustable output spaces (utilized in WPSM and AM-PS) to cope with the load fluctuations. Hence, the soft constraint limiting the switching of units in Formula (19) can be satisfied in general. These practical strategies reduce the unnecessary crossings between SOZ and FOZ, improving the safety of the power grid and hydro units. For the load transfer situations, Figure 16 and 17 show the output trend of substations following load instructions. (Here, the output of the power substation is located at the bottom, and it is too small to be visible in the figures.) The one-step output adjustments in WPSM and AM-PS all take the installed capacity difference of substations into account. Hence, there is no load transfer with a large amplitude between substations, and all substations can cooperate with each other to share the fluctuations of the load instructions. All substations ensure that at least one unit is in operation, thereby preventing some substations from undertaking an empty load. When solved the UCLD problem, the three categories of methods mentioned in Introduction not consider the SOZ constrains of substations. Hence, it is quite possible that in some peak regulation conditions, the UCLD results from them do not meet the actual needs of power generation enterprises. While, the RPM successfully solves the UCLD-SC problem, and combine calculation efficiencies and accuracies. This is the outstanding advantages of RPM.

VI. CONCLUSION
In this study, substation constraints are considered in the unit commitment and load distribution of hydropower stations for the first time. The complex constraints in the UCLD-SC are classified into three constraint sets, namely, CS-HSL, CS-SL, and CS-UL. A refined and practical method for solving the UCLD-SC is proposed, and contains three submodules. Among them, the WPSM and AM-PS jointly guarantees the output of substations following load fluctuations, and determine the reasonable unit commitment. While, the SUM guarantees the calculation timeliness, load distribution scheme and efficiencies of units. Owing to its representativeness and complexity, the TGHS is selected to verify that the RPM is a highly time-efficient and adaptable method for the UCLD-SC, and is of vital importance for large hydropower stations to execute critical dispatching, such as arranging unit commitments and forecasting the FWL process of the next day. The main research contributions of this study are as follows.
(1) A unit commitment and load distribution model considering the constraints of substations is abstracted and established. Subsequently, a refined method with practical VOLUME XX, 2021 14 strategies (RPM) for solving the UCLD-SC is proposed for the first time.
(2) The RPM can guarantee the operation efficiency of units, avoid units frequently crossing between SOZ and FOZ, and reasonably arrange substation output to jointly undertake load fluctuations, namely the UCLD scheme by RPM is with high rationality and practicability.
(3) The calculation timeliness and accuracy of RPM are almost identical to that of VKM and RM. The relative MAE of the simulation outflow can be controlled within 1%. The RPM involving the unit level can be treated as a trustworthy method for forecasting the FWL and outflow processes.
The RPM has good timeliness and rationality, but there remain some drawbacks. In particular, the SUM can only guarantee the optimal allocation of the streamflow within each substation, and the AM-PS does not consider the efficiency of the units from the whole plant, which resulting in a violation of the first soft constraint on April 1. Thus, improving the strategies in the AM-PS requires further study.