Multiobjective optimal operations for an interprovincial hydropower system considering peak-shaving demands

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Multiobjective optimal operations for an interprovincial hydropower system considering peak-shaving demands Jianjian Shen a, b, *, Chuntian Cheng a, b, Sen Wang c, Xiaoye Yuan a, Lifei Sun a, Jun Zhang d a

Institute of Hydropower and Hydroinformatics, Dalian University of Technology, Dalian, 116024, China Key Laboratory of Ocean Energy Utilization and Energy Conservation of Ministry of Education, Dalian, 116024, China c Key Laboratory of the Pearl River Estuarine Dynamics and Associated Process Regulation, Ministry of Water Resources, Guangzhou, 510611, China d State Grid Zhejiang Electric Power Company, Hangzhou, 310000, China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Multiobjective Optimal operation Interprovincial hydropower system (IHS) Peak shaving Security constraints

Large-scale power transmission in China poses a challenge for the operation of an interprovincial hydropower system (IHS) connected by ultrahigh-voltage direct current (UHVDC) lines. This study develops a multiobjective optimization model for the monthly operation of an IHS considering daily peak-shaving demands. This model is first transformed into a sequence of single-objective subproblems with different peak-to-valley ratio constraints. A multiphase multigroup coordination strategy is proposed in which the plant group for each month is deter­ mined based on the hydrological seasons of the different rivers. A multidimensional search method is then utilized to optimize the monthly generation schedules, where the energy target for a typical day is proportional to the daily load share in each month. Finally, a load-reconstruction-based strategy is used to determine feasible hourly generation schedules and obtain the peak-to-valley ratios to evaluate the monthly operation schemes. The above model and method are applied to a real-world IHS related to the Xiluodu-Zhejiang UHVDC power transmission project. The obtained Pareto solutions reveal the interaction between the energy production and peak-shaving objectives and show a significant improvement in peak shaving with a small decrease in energy production. Moreover, in typical inflow scenarios, all operation schemes enhance the total energy generation from local hydropower plants while leaving that from Xiluodu almost unchanged compared with the conven­ tional operation results.

1. Introduction The mismatch in the distributions of energy resources and electricity consumption in China motivates large-scale interprovincial power transmission [1,2]. The national project titled “West-East Electricity Transfer” was proposed to address this situation. One of its major goals is to transmit hydropower generated in the southwest to eastern regions, which consume over 40% of the country’s total electricity but lack en­ ergy resources. Almost all large hydropower plants on the Jinsha River, Lancang River, Yalong River, Dadu River, and others that have been newly commissioned over the past two decades serve well-developed provinces such as Shanghai, Zhejiang, Jiangsu, and Guangdong via a high-/ultrahigh-voltage (UHV) power network. Figure 1 shows the UHV direct current (UHVDC) power network for hydropower transmission in China. The maximum capacity for interprovincial hydropower trans­ mission exceeded 80 GW by the end of 2017. Such large-scale

hydropower indeed alleviates power shortage problems in the recip­ ient regions but poses new challenges regarding the operation and management of hydropower systems and power grids [3]. At present, it is difficult to efficiently collect hydropower from the southwest and deploy it in response to peak loads. This is because the transmission schedules are usually established based on the individual dispatch of large hydropower plants or heavily depend on surplus electricity from the sending power grids, with inadequate consideration of the needs of the receiving power grids [4,5]. As a result, these power grids often receive a large amount of power during valley times, which makes it difficult to schedule local plants. The pressure of balancing peak de­ mands is thus aggravated. There is an urgent need for the coordination of operations policies between the UHV hydropower source regions and the local energy in recipient regions. In particular, some essential operational needs, especially the peak-shaving demands of the power grids, should be considered when coordinating operations. This study focuses on the operations of an interprovincial

* Corresponding author. Institute of Hydropower and Hydroinformatics, Dalian University of Technology, Dalian, 116024, China. E-mail addresses: [email protected] (J. Shen), [email protected] (C. Cheng). https://doi.org/10.1016/j.rser.2019.109617 Received 21 April 2018; Received in revised form 26 July 2019; Accepted 19 November 2019 1364-0321/© 2019 Published by Elsevier Ltd.

Please cite this article as: Jianjian Shen, Renewable and Sustainable Energy Reviews, https://doi.org/10.1016/j.rser.2019.109617

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Nomenclature

pm;t pdm;i;t

A. Acronyms IHS interprovincial hydropower system UHV ultrahigh voltage UHVDC ultrahigh-voltage direct current POA progressive optimality algorithm DDDP discrete differential dynamic programming ZJPG Zhejiang power grid

pm;t pm;t

average generation of plant m in period t average generation of plant m in period i of a typical day of month t upper and lower bounds on the power generation of plant m in period t

pdm;i;t pdm;i;t upper and lower bounds on the power generation of plant m in period i of a typical day of month t pli;t pli;t

Qlm,t Qm,t qm,t qm;t Qnm,t QTm k;t

B. Indices i time period index for a typical day k upstream plant index m hydropower plant index t time period index C. Constants I total number of time periods in one day Km total number of upstream plants for hydropower plant m M total number of hydropower plants, M ¼ M1þM2 M1 total number of local hydropower plants M2 total number of outside hydropower plants T total number of time periods over the whole optimization horizon Δt time period duration Ω set of months in the dry season δ number of seconds in an hour, which is used to ensure the same units on both sides of the equation; here, δ is equal to 3600

Sm,t Sm;t ; Sm;t Vm,t vm v

0

Zm;T Zm;T 0

Zm,t Zm;t Zm;t

D. Variables Ci;t original load in period i on a typical day of month t Ct monthly average load demand in period t 0 Em;t Em;t calculated total energy production of plant m in period t and the corresponding control target

Δpdm 0 Δpt

hydropower system (IHS) consisting of sending hydropower plants and local hydropower plants in a receiving power grid. The optimization problem exhibits great complexity. First, the hydropower plants located at the starting and ending points of UHV lines usually have unique characteristics in terms of geography, weather, hydrology, etc., which can easily lead to large differences in runoff [6,7]. Therefore, difficulty arises in determining how to use these runoff characteristics. Second, coordinated operations impose more complex operational needs and restrictions on large hydropower plants. In particular, diverse objectives such as energy production maximization, peak response for power grids, and other comprehensive requirements may conflict and compete [8–10]. These requirements will give rise to additional complexity in hydropower system operations. Third, the hydropower generated in the southwest is transmitted by UHVDC lines operating at 500 or 800 kV. Thus, power transmission safety [11] is a top priority for the IHS. Optimal operations must comply with certain network safety con­ straints, including transmission capacity limits, a maximum ramping capacity, and requirements regarding transmission flow stability. These constraints make the optimal coordination of the hydropower plants more complex and difficult. As complex and challenging tasks, hydropower scheduling and op­ erations have received extensive attention in the literature [12–15]. Many studies focus on the optimization of the hydropower system at the generation side and therefore usually neglect important power network safety constraints [16–18]. Consequently, the optimized hydropower generation schedules might require further improvement to satisfy the requirements of power system operations. Thus, the application of the corresponding optimization models and methods to an IHS may be

upper and lower bounds on the UHVDC transmission power in period i of a typical day of month t spill flow of reservoir m in period t inflow into reservoir m in period t turbine discharge of hydropower plant m in period t upper bound on the turbine discharge of plant m in period t local natural inflow into reservoir m in period t delayed flow from upstream reservoir k into reservoir m in period t with a time delay total discharge of reservoir m in period t, Sm;t ¼ qm;t þ Qlm;t minimum and maximum limits on the total discharge of reservoir m in period t storage volume of reservoir m in period t minimum number of time periods in which the power generation of plant m is at a local extremum minimum number of time periods in which the transmission power is at a local extremum final forebay elevation of reservoir m at the end of period t and the corresponding specified target forebay elevation of reservoir m in period t upper and lower bounds on the forebay elevation of reservoir m in period t maximum ramping capacity of plant m in period t maximum variation of transmission power via UHVDC lines

difficult. Certainly, some studies have placed emphasis on safety-constrained hydropower system operations. Norouzi et al. [19] considered the DC power flow equation and transmission flow limits in hydro/thermal unit commitment. They modeled the network safety constraints as linear constraints. Johannesen et al. [20] presented optimal short-term hydro scheduling including safety constraints. In that study, the original problem was decomposed into a hydro sub­ problem and an electrical subproblem, and the stability constraints were addressed using a special linear programming technique. Similarly, power balance and transmission flow limits have also been considered and effectively handled in other related studies [21,22]. However, these previous works have not accounted for the new requirements regarding power stability that must be considered for UHV transmission lines to avoid frequent power flow fluctuations between multiple successive operating periods. These are highly coupled constraints and thus intro­ duce particularly high complexity into the modeling of the IHS. In addition to these constraints, complex operational tasks are required in present hydropower systems. All hydropower plants are required to provide as much energy as possible while also being able to quickly respond to the peak loads of power grids. The operational targets are complicated because of the great difficulty of coping with short-term peak-shaving demands in long-term optimal operations and the lack of any existing method to address this problem. This is a new multi­ objective optimization problem with complex constraints, which is distinct from conventional single-objective optimization problems and other multiobjective optimization problems [23,24]. With existing optimization models, it is difficult to accurately capture the optimization problem of interest. Thus, a new, suitable model is needed. On the other 2

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separately if their hydrological seasons are not consistent. A multidi­ mensional search method is utilized to optimize the monthly operations of the hydropower plants. This method combines a group strategy, the POA, and discrete differential dynamic programming (DDDP) to alle­ viate the dimensionality challenges of hydropower optimization. The main advantages of the POA and DDDP are that they reduce the numbers of stage variables and state variables, respectively, in each iteration. The group strategy is mainly used to reduce the number of hydropower plants that are optimized at one time. Its basic principle is to divide all plants into several groups in accordance with their hydraulic connec­ tions and relative spatial positions and then optimize only one group of plants at a time, with the results for the other groups of plants being fixed. During optimization, a load-reconstruction-based strategy is used to handle the time-coupled network safety constraints such that feasible hourly generation schedules for peak shaving can be easily obtained. Finally, the developed optimization model and method are validated based on a real-world IHS. To provide a clear understanding of this study, its main contributions are briefly summarized here. The novel aspects of this study are the development of a multiobjective optimization model for the monthly operations of an IHS considering daily peak-shaving demands and the proposal of an effective solution methodology. Here, two coupled tem­ poral scales are considered. Monthly operations and hourly generation scheduling are integrated to obtain improved operation schemes for the considered hydropower system. This contribution is distinct from the outcomes of previous studies.

hand, many studies have been conducted in relation to the design and operation of multipurpose reservoir systems or hydropower systems. Yazdi and Moridi [25] focused on the determination of the design pa­ rameters for multipurpose cascade hydropower reservoir systems using optimization objectives formulated to minimize the squared deviation of release from demand, maximize the total amount of produced energy, and maximize the system reliability. Moridi and Yazdi [26] utilized flood damage at downstream sites and the loss of hydropower genera­ tion as optimization objectives to determine the optimal allocation of flood control capacity. Karamouz et al. [27] examined the dimensions of a dam diversion system. They developed an optimization model for minimizing construction costs and the expected value of damage caused by floods. Liu et al. [28] studied multiobjective operations with flood control and hydropower generation, and the progressive optimality al­ gorithm (POA) and a smooth support vector machine were adopted to solve the multiobjective optimization problem. Other requirements such as water quality protection of reservoirs and water sharing in trans­ boundary rivers were respectively studied by Yazdi and Moridi [29] and Avarideh et al. [30]. These previous works are different from the gen­ eration operations considered in this study, but they provide good insight for dealing with the multiobjective operations of multipurpose reservoir or hydropower systems. Mathematically, the considered problem is a large-scale, highdimensional, nonlinear, multiobjective optimization problem. Various optimization methods have also been suggested to address such prob­ lems. Yeh [12], Simonovic [13], Momoh et al. [14], and Labadie [15] successively presented detailed discussions of the advantages and dis­ advantages of these methods. The reader is referred to their articles for further information about the related work that has been presented in the literature. It is generally agreed that the choice of methods depends on the characteristics of the considered system, such as the system scale, operational tasks, objectives and constraints. Due to the high complexity of the problem, it would be difficult to completely solve the optimization problem considered here with any existing optimization technique. Therefore, the present study aims to develop an efficient and practical methodology that can be applied in real-world engineering. In partic­ ular, many of the specific complex conditions and constraints, such as coupled operational needs at multiple time scales, also require addi­ tional suitable solution strategies or techniques. As mentioned above, the considered problem exhibits high complexity. Three major aspects of this complexity can be summarized as follows: 1) the problem involves two coupled time scales; 2) multi­ objective requirements are considered; and 3) there are many complex power plant operation constraints and network safety constraints, some of which are spatiotemporally coupled. These sources of complexity result in great difficulty in addressing this problem. In this paper, an optimization model with two objectives, namely, maximizing energy production and responding to peak loads in the receiving power grid, is developed. The first objective is widely understood, while the second is formulated as the minimization of the ratio of the peak-to-valley load difference to the load peak of the residual load curve in the dry season. The main goal is to shave the peak loads so as to flatten out the residual load variations that must be addressed by the low-efficiency coal-fired plants in the receiving power grid in China. In this way, the numbers of times that coal-fired units need to be shut down and put back into operation can be reduced, and their operational efficiency can be enhanced accordingly. The model is treated with the constrained method and solved using a new methodology. To establish this meth­ odology, a multiphase multigroup coordination strategy is first proposed by considering the differences in hydrology and operational tasks be­ tween hydropower plants on different rivers. In this strategy, the entire time horizon is split into several phases that reflect different combina­ tions of hydrological seasons of the multiple rivers located at the starting and ending points of the UHVDC lines. In each phase, all hydropower plants will be optimized simultaneously if their hydrological seasons are consistent, or they will be divided into several groups and optimized

2. Case study: an IHS connected by UHVDC lines This study is based on a real-world IHS. The hydropower system consists of the Xiluodu plant on the Jinsha River in Yunnan Province and hydropower plants on the Qiantang River and Ou River in Zhejiang Province, with Xiluodu serving Zhejiang via a UHVDC line of 1680 km in length. Figure 2 shows the schematic layout of the IHS. The Xiluodu hydropower plant is installed with 18 identical hydropower units with capacities of 700 MW. The total installed capacity is therefore 12,600 MW; this is the second highest capacity of any plant in China and the third highest in the world. As one of the main power suppliers for the West-East Electricity Transfer project, this plant provides more than 40% of its total energy production to Zhejiang Province every year. The maximum power fed into the Zhejiang power grid (ZJPG) reaches 8000 MW, which accounts for approximately 20% of the average load demand of the ZJPG. Thus, the Xiluodu plant plays a significant role in the operation and management of the ZJPG. The other 9 hydropower plants included in the IHS are local to the recipient province. Because of the centralized dispatch scheme currently used in China, all hydropower plants considered in this study are directly operated by the ZJPG and its higher authorities. Their operators are responsible for designing the generation schedules for these plants to meet the power supply and peak demands of the ZJPG. In this situation, determining how to obtain rational and efficient operation schemes by coordinating the operations policies of these hydropower plants is crucial. However, the generation schedules of Xiluodu and the other hydropower plants are currently defined separately, lacking effective coordination and a proper optimi­ zation model. Thus, the coordinated management of generation capacity and storage considering the different hydrological characteristics of these regions offers great potential for improving the generation schedules of all hydropower plants. This paper emphasizes the overall operations of the IHS, such that all plants are considered and scheduled together. Based on the realistic demands of the ZJPG, the developed methodology is utilized to generate monthly generation schedules for a year while considering hourly peakshaving demands. The data on the operation conditions and constraints come from real monthly and hourly generation schedules for the ZJPG. Note that the hourly load demands for a typical day of each month have been determined based on the historical load data from the ZJPG over 3

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the last two years. Table 1 lists the main characteristics of the Xiluodu plant and the local plants. Here, the situation of the Tankeng hydro­ power plant is unique. The flood water level of this plant is equal to the normal water level during the Meiyu period. This period is the main flood season (usually from June to July). To analyze the effect of IHS operations on the reservoir levels in the flood season, the normal water level is used in this study as the upper boundary within the time horizon. In addition, it should be mentioned that all hydropower plants consid­ ered in this study are mainly designed for generating electricity. Other demands are not considered in the current research.

K X

Qm;t ¼ Qnm;t þ

■ Turbine discharge limits

■ Minimum and maximum discharges

8 F1 ¼ MaxðE1 þ E2 Þ > > M1 X T > X > > < E1 ¼ pm;t � Δt

■ Minimum and maximum forebay elevations (8)

Z m;t � Zm;t � Z m;t ■ Minimum and maximum power generation

(9)

pm;t � pm;t � pm;t 3.2.2. Operation constraints on a typical day ■ Total energy balance

(10)

0

Em;t ¼ Em;t Here, Em;t is determined as follows: 0

0

Em;t ¼ pm;t � Δt �

I X Ci;t Ci;t ; Ci;t ¼ Ct I i¼1

(11)

■ Minimum and maximum power generation pdm;i;t � pdm;i;t � pdm;i;t

(12)

■ Maximum ramping capacity � � � d � �pm;i;t pdm;i 1;t � � Δpdm

(13)

■ Stability requirements for power generation over multiple successive operating periods � �� � pdm;i αþ1;t pdm;i α;t pdm;i;t pdm;i 1;t � 0; ​ α ¼ 1; 2; ⋯vm (14)

0 m ¼1 t¼1

Objective 2: responding to the peak loads of the power grid. 0

3.2.3. Power network safety constraints

MaxCi;t Min Ci;t 8 1�i�I 1�i�I > > F2 ¼ Min 0 > > t2Ω MaxCi;t <

■ Transmission power limits

(2)

1�i�I

0

(7)

Sm;t � Sm;t � Sm;t

(1)

m¼1 t¼1

> > M2 X T X > > > : E2 ¼ pm0 ;t � Δt

Ci;t ¼ Ci;t

(6)

0 � qm;t � qm;t

In general, for the long-term optimal operations of hydropower systems in China, the aim is to determine monthly generation schedules and reservoir levels over one year. A number of objective functions have been applied to address this problem. Based on the operational tasks of the considered hydropower system, two objectives are chosen here. One is to maximize the total energy production so that clean renewable en­ ergy can be employed to the greatest possible extent. This objective is commonly used for hydropower optimization [31,32]. The objective function value should be equal to the total accumulated energy pro­ duced by the hydropower plants in all periods. The corresponding for­ mula is given in Eq. (1). The other objective is to respond to the peak loads of the power grid. It is well known that the significant peak-to-valley differences in load demand in China impose a heavy electricity balancing burden [33]. Due to the lack of flexible generation capacity in most provincial power grids, especially coal-dominated power grids, the flexible regulation of hydropower plants is mainly required to respond to peak demands. Therefore, this objective can be formulated as minimizing the ratio of the peak-to-valley load difference to the load peak of the residual load curve in the dry season (referred to as the peak-to-valley ratio for short). The purpose is to flatten the re­ sidual load curve obtained by subtracting the hydropower generation from the original load, thus allowing the large number of low-flexibility coal-fired units to be easily and efficiently scheduled. The objective function for peak shaving is formulated as shown in Eq. (2). Objective 1: maximizing the energy generated by the hydropower system.

M X

(5)

0

Zm;T ¼ Zm;T

3.1. Objective function

> > > > :

(4)

■ Reservoir level target

3. Problem formulation

0

QT mk;t

k¼1

pdm;i;t

pli;t �

m¼1

M2 X

pdm0 ;i;t � pli;t

(15)

0

m ¼1

■ Maximum variation of transmission power � � M M2 �X � X 0 � 2 d d � pm;i;t 1 pm;i;t � � Δpt � � � m¼1 m¼1

3.2. Constraints 3.2.1. Long-term operation constraints

(16)

■ Continuity equation Vm;tþ1 ¼ Vm;t þ δ � Qm;t

qm;t

� Qlm;t Δt

■ Stability requirements for power transmission over multiple succes­ sive operating periods

(3)

4

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8 M2 X > > > pd > < m¼1 m;i M2 > X > > > pdm;i :

Renewable and Sustainable Energy Reviews xxx (xxxx) xxx M2 X

βþ1;t

M2 X β;t � 0;

if

pdm;i;t

M2 X

m¼1

m¼1

m¼1

M2 X

M2 X

M2 X

βþ1;t

m¼1

pdm;i pdm;i

β;t � 0;

m¼1

if

pdm;i;t

m¼1

m¼1

pdm;i

; ​ β¼1;2;⋯v pdm;i

season. It is generally agreed that the flood-season plants should usually be responsible for satisfying the base load to ensure that the clean hy­ dropower energy capacity is fully employed and that spillage is avoided, while the dry-season plants will often be required to provide flexible peak power during times of high load. In Figure 4, panels (a) and (b) show two typical generation profiles, one for a base plant and one for a peak plant. In the flood season, most hydropower plants, especially lowflexibility plants, are usually operated as base plants for power supply. In the dry season, they may serve as peak plants. The different roles of hydropower plants in balancing electricity demands are closely related to hydrological periods. Based on the complex characteristics and various operational tasks of the IHS, a multiphase multigroup coordination strategy is developed. In this strategy, the entire time horizon is split into several phases that represent different combinations of hydrological periods of the multiple rivers in the source and recipient regions of the UHV transmission project. In each phase, if all hydropower plants are in the same hydro­ logical period, i.e., the flood season or the dry season, they will be optimized simultaneously using the same objective. If the hydrological periods of these plants are not consistent, they will be divided into two groups, one corresponding to the flood season and the other corre­ sponding to the dry season. In this case, the operations of the group of flood-season plants will be optimized to maximize the total generated energy, while the operations of the other group will be optimized based on peak-shaving demands. This process consists of two steps. In the first step, the flood-season plant operations are optimized to determine the monthly generation schedules. Thus, their total hourly generation pro­ file on a typical day can be calculated and subtracted from the original load to obtain the residual hourly load profile, as shown in the following equation:

1;t > 0 0

1;t < 0

(17)

4. Solution methodology 4.1. Solution framework The above multiobjective optimization problem is difficult to directly solve because of the high complexity of the objective functions and tightly coupled constraints. In the usual way, the original problem is transformed into a set of single-objective optimization models with different levels of peak-shaving demands [34,35]. A multiphase multi­ group coordination strategy is then developed to solve each single-objective optimization model. In this strategy, the time horizon is divided into multiple phases with different combinations of the hydro­ logical seasons of the rivers. During optimization, the hydrological seasons in each phase are first identified to determine the plant groups. A load-reconstruction-based strategy is then used to determine a feasible hourly generation schedule for peak shaving with the given daily energy target. In this strategy, the daily energy demand is first calculated using Eq. (11). All outside hydropower plants are then aggregated into a single virtual plant, and the hourly generation of this virtual plant and that of each local plant are gradually determined using load reconstruction and a load shedding method. Thus, the peak-to-valley ratio can be calculated and used in the multidimensional search method that is applied to optimize the monthly generation schedules. Pareto-optimal solutions are generated by setting different bounds on the peak-to-valley ratio. Figure 3 depicts the whole solution methodology for determining the multiobjective optimal operations of the IHS. This methodology includes three main components, which are described in the following subsections.

Cr1 i;t ¼ Ci;t

0

(19)

The residual load demand obtained above is the input to the second step, in which the monthly schedules of the dry-season plants are opti­ mized by using a model with a peak-to-valley ratio constraint (Eq. (20)). The detailed order of the calculations for this coordination strategy is shown in Figure 5. Correspondingly, the plant groups in each phase are also determined. � � 8 maxCr2 min Cr2 > 1�i�I i;t 1�i�I i;t > > > < maxCr2 i;t 1�i�I (20) > > > M Mf > X > : Cr2 ¼ Cr1 pdm;i;t i;t i;t

The constrained method is used to treat the peak-shaving objective as a constraint, as shown in Eq. (18). Thus, the multiobjective problem can be transformed into a single-objective problem. For this transformed single-objective problem, a set of values for the upper bound in Eq. (18) are considered to analyze the interaction of the two proposed objectives. That is, a varying constraint is considered. For each boundary value R of the constraint, an optimal solution for the single-objective problem is generated.

m¼1

4.4. Determining the hourly generation profiles on a typical day

0

MaxCi;t MinCi;t
pdm;i;t

m¼1

4.2. Addressing multiobjective optimization

rt ¼

M1 X

(18)

The hourly generation profiles of the hydropower plants on a typical day are required to calculate the value of the peak-to-valley ratio for use in the optimization model. It is evident that the hourly generation scheduling process involves many complex operation constraints. In addition to Eqs. (10)–(14), which apply to all hydropower plants, the safety constraints for UHV power transmission given in Eqs. (15)–(17) must also be imposed on the operation of the outside hydropower plants. These constraints exhibit highly temporally coupled characteristics, thus giving rise to considerable complexity and difficulty in determining the hourly power generation schedules of the hydropower plants, particu­ larly outside ones. How these constraints should be handled is the key question. The aforementioned load-reconstruction-based strategy can be used to effectively cope with these constraints and produce the hourly generation schedules. This strategy consists of four main steps.

where rt is the peak-to-valley ratio for a typical day in period t and R is the upper bound, which is initially obtained from the optimized result of the energy production maximization model. To generate Pareto-optimal solutions for the peak-shaving objective and the energy production maximization objective, R will be gradually decreased during the opti­ mization process. 4.3. Coordinating outside plants and local plants in the receiving power grid The rivers at the starting and ending points of UHV power trans­ mission lines are usually separated by thousands of kilometers and exhibit great hydrological differences. For example, in the same month, some plants in such a hydropower system may be in regions that are in the flood season, while others may be in regions that are in the dry 5

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(1) The first step is to determine the daily energy demand in accor­ dance with the monthly power generation. For this, Eq. (11) is used. (2) In the second step, all outside hydropower plants are aggregated into a single virtual plant. Correspondingly, the daily energy demands and operation constraints are converted into equivalent restrictions on this virtual plant, as represented by the following formulations in Eqs. (21)–(24).

successive approximations that converge to the optimal result. Suppose that the number of discrete states (such as forebay levels) is S; then, there will be SM different combinations of the states of M plants. Each state combination represents a potential solution to the problem. Since S is usually set to at least 20 for determining the optimal operations, considerable computational effort will be required to calculate all combinations. Therefore, DDDP is used to reduce the number of discrete states to enhance the computational efficiency and alleviate the memory requirements imposed on the computing hardware [39,40]. In this method, a small range of increments is applied to both sides of the initial state sequence to form a corridor that allows for several discrete values, where each discrete value represents one specified storage increment. For example, if the number of discrete values were to be set to 3, the number of state combinations to be calculated would decrease to 3M . Moreover, based on the conditions of interest, the increment can be gradually changed to a smaller value or can be biased to only one side of the initial decision sequence. In addition, for the case in which the number of plants is so large that the optimization problem still cannot be solved within a reasonable amount of time, the group strategy is utilized to divide all plants into groups based on their hydraulic connections and relative spatial posi­ tions [41]. The purpose is to reduce the size of the problem that must be solved at any one time and thus alleviate the dimensionality limitations. This strategy consists of three steps. In the first step, all hydropower plants located on hydraulically unrelated rivers are divided into different groups. If the number of plants in the same river basin is still large, these plants are further divided in the second step into two or more smaller groups depending on whether they are located on tribu­ taries or the main branch of the river. In other words, all hydropower plants in each group should belong to either the same tributary or the main branch of the river after this step is implemented. If the number of plants in one group is still larger than the specified value, the third step is applied to further separate these plants in order from upstream to downstream. Note that the hydraulic connections between cascaded hydropower plants must still be considered to satisfy the continuity equation, although these plants may belong to different groups. Fig. 7 illustrates the core principle of the multidimensional search method to provide a clear understanding of this method.

The daily energy demand of the virtual plant is formulated as M2 X

Et ¼

(21)

Em0 ;t 0

m ¼1

The minimum and maximum amounts of power generated by the virtual plant are expressed as ! ! M2 M2 X X d l l;f d l Max (22) p m0 ;i;t ; pi;t � pi;t � Min pm0 ;i;t ; pi;t 0

0

m ¼1

m ¼1

The maximum variation of the power generation of the virtual plant is described as ! M2 X � l;f � �p (23) pl;f � � Min Δpd 0 ; Δpl i;t

i;t 1

m

t

0

m ¼1

The stability requirements for power transmission during multiple successive operating periods are expressed as � d;f � 00 pi;t pd;f pd;f pd;f (24) i γ;t i γþ1;t i 1;t � 0; ​ γ ¼ 1; 2; ⋯; v where v00 is the minimum number of time periods in which the power generation of the virtual plant is at a local extremum, v00 ¼ Maxðv1 ;v2 ;:::; 0 vM2 ; v Þ. (3) The third step is to generate the hourly generation schedule of the virtual plant. The original load curve is first reconstructed considering constraints (21)–(24), as shown in Figure 6. The purpose is to flatten the frequent fluctuations in the load curve so that the load shedding method that is commonly used in China for peak operations can be utilized to quickly determine a suitable working state of the current plant and feasible operation sched­ ules for peak shaving. The main principle is to move the shedding position in Figure 6 upwards or downwards until the calculated energy production is equal to the specified energy demand. In this case, the shadowed area will represent the hourly generation profile. For details on this strategy, the reader is referred to our previous work [36]. (4) The task in the fourth step is to determine the hourly generation of each hydropower plant. The generation profile of the virtual plant obtained in the previous step is taken as the load curve that will be served by all outside plants while satisfying constraints (11)–(14). This problem is similar to the hourly generation scheduling of the virtual plant because it is subject to the same types of constraints and conditions. Therefore, the hourly gen­ eration schedule of each plant can be determined using the same procedure applied in the third step.

5. Results and discussion When the long-term operations of the hydropower plants are considered in monthly increments and the operations on a typical day are considered in hourly increments, the long-term optimization model includes 360 decision variables and 1090 constraints, whereas the shortterm optimization model includes 2880 decision variables and 48,960 constraints. Thus, the total numbers of variables and constraints for the current optimization problem are 3240 and 50,050, respectively. This large-scale problem was solved on an HP PC running Windows 7 with an Intel(R) Core(TM) i3-3110 2.4 GHz CPU and 4 GB of RAM. The computation time was approximately 98 s, which is acceptable for the real-world generation operations of power grids. An analysis of the differences in hydrological characteristics between the different rivers and the optimization results are described and discussed in the following three subsections. 5.1. Analysis of differences in hydrological characteristics between rivers

4.5. Optimizing hydropower operations via the multidimensional search method

The different hydrological characteristics of different rivers serve as an important basis for the coordinated operation of hydropower plants and provide opportunities to improve the operations policies and in­ crease the energy production or dispatchable power of the entire system. Therefore, a hydrological characteristic analysis of the IHS is necessary before optimization. In this case study, the historical monthly inflows in the target regions

The abovementioned multidimensional search procedure, which combines the group strategy, the POA, and DDDP, is mainly used to optimize the hydropower system operations. The POA decomposes the considered problem into a series of two-stage multidimensional sub­ problems [37,38]. A general two-stage solution process is iteratively applied to successive overlapping stages of the problem, yielding 6

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over several decades are employed. Figure 8 illustrates the multi-year average monthly inflow profiles of Xiluodu and other plants. The inflow at Xiluodu is obviously far larger than those at the local hydro­ power plants in the recipient province. To clearly show the differences between two hydropower plants in these different regions, a directional statistics method based on the annual maximum is used to identify the flood seasons. In this method, the flood samples from each year are taken as time vectors, each of which points in the direction of the time point at which the largest flood occurred in that period of time. Based on the directional similarity of the sample vectors, reasonable estimates of the positions of the segmentation points for the flood season can be deter­ mined synthetically. For more details, the reader is referred to Reference [42]. The obtained flood seasons of two rivers located in two sides of UHV power transmission line are shown in Figure 9. It is obvious that the hydropower plants in the two regions show greatly different main flood seasons. At Xiluodu, the flood period begins in June and ends in late October; throughout this time period, the monthly inflow is larger than the average. The main flood season corresponds to August and September. In contrast, the flood season for local plants such as Tankeng is from April to early October, with the main flood season spanning from early June to early August. Therefore, great hydrological differences but good complementarity exist between the hydropower plants located on the different rivers. This situation offers great potential for the coordi­ nation of the operations policies based on the full use of the large available reservoir storage. Thus, the monthly generation scheme throughout a year and the hourly generation schedule for peak shaving during a typical day may be more effectively managed by considering these hydrological characteristics.

[200~]. Correspondingly, the solutions are categorized into the first, second, and third groups, respectively. The first group of solutions favors energy production maximization, while the third group of solutions fa­ vors the peak-shaving objective. Intermediate compromise solutions can be found in the second group. Compared with the first group of solu­ tions, the intermediate compromise solutions decrease the total energy production of the hydropower system and increase the peaking power available in response to load variations. Similarly, the intermediate compromise solutions increase the energy production and decrease the peaking power by comparison with the third group of solutions. As specific examples, three representative solutions (r ¼ 0.276, 0.260, and 0.248) are selected from category III for comparison with the extreme solutions in Table 2. It can be inferred that the energy production during the dry season, especially from January to March, increases with a decrease in r. This means that hydropower generation is shifted from the flood season to the dry season to enhance the peak-shaving ability of the power grid. Moreover, compared to the category I solution, the solution with r ¼ 0.276 shows a decrease of 2.2% in the peak-to-valley ratio but a decrease of only 0.02% in energy production (approximately 7 million kWh). Compared to the category II solution, the solution with r ¼ 0.248 shows a decrease of 2.1% in the peak-to-valley ratio and a decrease of 0.8% in energy production (approximately 300 million kWh). In fact, the two objectives considered represent different roles of a hydropower plant in balancing power demands, namely, the peak position and the base position. A preference for one particular role will have a significant impact on the chosen operations policy. In real-world engineering, the system operators should select a suitable operation scheme depending on the specific demands of the situation. In general, Figure 11 shows that introducing the peak-to-valley ratio objective into the conventional operation optimization problem will accelerate the reduction of hy­ drological energy utilization efficiency when the peak-to-valley ratio is less than 2.68, especially at a value of 0.258. The main reason is that a reduction in the reservoir water levels leads to a significant reduction in power generation efficiency, which in turn causes a reduction in total electricity production. These findings imply that a reasonable bound on the peak-to-valley ratio should be carefully selected to establish the trade-off between these two important operation objectives.

5.2. Pareto-optimal solutions for multiobjective operations This subsection is devoted to the analysis of Pareto-optimal solutions for multiobjective operations. The Pareto solutions are generated by considering different upper bounds on the peak-to-valley ratio in the optimization model; all generated solutions are feasible. Figure 10 shows the nondominated solution set, which illustrates the trade-off between energy production maximization and peak-to-valley ratio minimization. To evaluate the optimization result, the Pareto front is classified into three categories, namely, the extreme solution that maximizes F1, the extreme solution that minimizes F2, and other solu­ tions. In other words, category I represents the single-objective optimi­ zation model considering energy production only. This solution reflects the maximum generation ability of all hydropower plants, with a maximum energy production of 36.01 billion kWh. This solution makes full use of the available hydrological energy but neglects the demands from the power grids; thus, its practicability may be severely under­ mined. Category II corresponds to the single-objective optimization model considering peak-to-valley ratio minimization only. The resulting peak-to-valley ratio of 0.243 reflects the case in which the generation of the hydropower plants most closely follows the power demands. The operation scheme in this category is beneficial for response to peak demands because it effectively utilizes the operational flexibility of the hydropower plants, especially large ones, in southwestern China. However, the total electricity production may be greatly decreased compared to that in category I, revealing the interaction between the quantity of electricity and power. In category III, the more electricity is generated by the hydropower plants over the time horizon, the less they can provide flexibly dispatchable power for meeting the peak loads of the power grid on a typical day. To identify the specific trade-off between the two objectives and help operators easily choose a suitable solution, a decision preference coef­ ficient dp is introduced to further divide the solutions in the third category into three groups. This coefficient represents the rate of decrease of energy production with an increase in the peak-to-valley ratio and is formulated as shown in Eq. (25). Specifically, the values of this coefficient are separated into ranges of [0,35), [35,200), and

dp ¼

ΔE* Δr*

(25)

where ΔE* ¼ difference in energy production between two neighboring solutions and Δr* ¼ difference in peak-to-valley ratio between two neighboring solutions. 5.3. Operation schemes in different inflow scenarios This subsection presents an analysis of the sensitivity of the proposed model and method to reservoir inflows. Three typical inflow scenarios corresponding to dry, normal, and wet years are used to simulate the operations of the considered IHS. Figure 11 and 12 present the reservoir level profiles of Xiluodu and Tankeng, respectively, under these different inflow conditions. Figure 13 shows a comparison among the total power generation profiles of all hydropower plants in the different inflow scenarios. It should be noted that these results are obtained from the equilibrium solutions in category III to present an effective trade-off between the two optimization objectives considered in this study. First, all operation conditions and constraints are checked, including the target reservoir level at the end of the time horizon, the upper and lower limits on the reservoir level, the power generation limits, and the peak-to-valley ratio limits. The optimized operation results for every hydropower plant in the different inflow scenarios effectively satisfy these constraints, demonstrating that the optimization results are cor­ rect. Second, the differences in compensation dispatch for the IHS under the three inflow conditions are analyzed. When the reservoir inflow gradually changes from low magnitude in dry year to high level in wet 7

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high-load hours. The corresponding operation scheme is found via the abovementioned category II solution. For a more extensive analysis, a compromise solution with r ¼ 0.268, the category I solution and the category II solution are selected to analyze the operations policies of the Xiluodu plant and the local hy­ dropower plants. Here, we choose the Tankeng plant as a typical representative of the local hydropower plants to avoid showing a large amount of data. Figure 14 depicts the monthly generation profiles of Xiluodu and Tankeng. Figure 15 shows the reservoir levels at Tankeng over the whole year. In the first two solutions, Xiluodu works at higher reservoir levels and follows generation schedules similar to that in the conventional scheme. This result is mainly related to the peak-shaving requirements of the power grid. Both solutions consider relatively loose bounds on the peak-to-valley ratio, so they can be easily satisfied through occasional adjustment of the monthly generation profile. In contrast, there is a large decrease in generation during the dry season in the category II solution to meet the peak demands of the ZJPG. The reason is that the requirement for the peak-to-valley ratio is set at a high level, requiring a significant improvement relative to the conventional operations policy. The key is the competitive relationship among the multiple purposes of the IHS. In this case, there are also obvious changes in the operation schedules of the local hydropower plants, such as Tankeng, in terms of reservoir level and power generation. Coordinated operation allows the Tankeng plant to generate less energy during the dry season to keep the reservoir level at a higher value. The reservoir level at the end of April is higher by almost 10 m compared to that in the conventional operation scheme. This is because Xiluodu provides more energy to guarantee the required dispatchable capacity for peak shaving over this period. As a result, the total energy generated by the Tankeng plant shows a maximum increase of approximately 6.5% among the several considered solutions. In summary, the coordination of operations between outside plants and local plants in the receiving power grid is able to effectively enhance the total energy generated by local hydropower plants while leaving the total energy production from the Xiluodu plant almost unchanged. Moreover, the more flexible capacity available from large outside hy­ dropower plants can help improve the peak-shaving ability of coaldominated power grids.

year, there is a slightly slow decline of the pre-flood water levels of Xiluodu but the process of water storage from flood season to the end of the year are basically same. This is dependent on the total inflow volume into Xiluodu, which exceeds the regulatory storage except in dry year. By contrast, the water levels at local power plants such as Tankeng also decrease from March to June, but the overall water level remains rela­ tively high. These findings imply that the interprovincial coordination of operations supports the ability of local hydropower plants to maintain high power generation efficiency before the main flood season in Xiluodu, even with large variations in the inflow conditions. The annual distribution of electricity production shows that all local hydropower plants contribute an average 61% of the total annual energy generation in January–May, November and December. In this period, the minimum monthly power generation of the hydropower system increases as the inflow increases. The two indicators obtained with the proposed method are significantly higher than those obtained with the conventional method based only on the energy production maximization objective; this difference can be attributed to the impact of the peak-to-valley ratio constraint. In this case, increasing the dry-season power production of hydropower plants that account for a small proportion of the ZJPG de­ mand can effectively improve the ability to rapidly respond to load changes. Moreover, it should be noted that improving the system’s peakshaving ability will inevitably reduce the total power generation of the hydropower system. According to the above analysis, it can be inferred that the compensatory dispatching of interconnected hydropower plants tends to appropriately reduce the preflood water levels at hydropower plants and increase the power generation production when the inflow magnitude increases. However, the water levels at local power plants remain relatively high over the whole time horizon, implying that implementing complementary operations between sending hydropower plants and local hydropower plants in receiving power grid can effec­ tively reduce the sensitivity caused by inflow. In this way, high power generation efficiency can be guaranteed for these local hydropower plants over the whole time horizon. 5.4. Comparison with the conventional method In real-world operations, the local hydropower plants in Zhejiang Province and the Xiluodu plant are usually dispatched individually. The former are optimized with the objective of maximizing firm generation, and the latter is optimized with the objective of maximizing energy production. This method, referred to as the conventional method in the present study, often fails to produce satisfactory operation schedules for the receiving power grid due to the lack of coordination between the plants. Therefore, a comparison between the present optimization re­ sults and those of the conventional method is presented to demonstrate the merits of the proposed method. The optimized solutions with a peak-to-valley ratio of 0.266 or larger contribute more electricity production than the 35.96 billion kWh ob­ tained in the conventional operation scheme, whose peak-to-valley ratio is 0.278. It can be inferred that there exists a feasible solution set in which the values of both optimization objectives are better than the actual operation results. This improvement is mainly attributable to joint operations and effective cooperation. Considering the hydrological differences between Xiluodu and the local hydropower plants in Zhe­ jiang enables better spatiotemporal generation allocation to enhance the total electricity production. Meanwhile, the production of more elec­ tricity in the dry season allows these plants, especially the local plants, to offer more dispatchable generation for the power grid. If only the best peak-shaving behavior for the ZJPG were to be sought, all hydropower plants would decrease their energy production by approximately 1.9%, but a significant decrease of as much as 12.6% in the peak-to-valley ratio would be obtained. This finding implies that the power grid can achieve a considerable savings in peaking generation capacity with the loss of only a small amount of electricity. This approach could be quite useful for alleviating peak-shaving pressure and ensuring the power balance in

6. Conclusions Hydropower systems in China are facing new and tremendous operational challenges due to the large-scale development of longdistance power transmission via a UHV power network in the past two decades. This paper develops a multiobjective optimization model for IHS operations with peak-shaving demands subject to operation con­ straints at two coupled temporal scales as well as network safety con­ straints. Another contribution of the present work is to propose a complete solution methodology that integrates a multiphase multigroup coordination strategy, a load-reconstruction-based strategy, and a multidimensional search technique. This methodology can be used to solve the optimization model and obtain effective and efficient Paretooptimal solutions. The proposed model and methodology are applied to a real-world IHS that consists of the Xiluodu plant on the Jinsha River and local hydropower plants on the Ou River and the Qiantang River. A case study is presented involving the design of monthly generation schedules considering the peak-shaving demands throughout a typical day. The results show that (a) different nondominated solutions from the Pareto front present different energy production and peak-shaving results; (b) a significant improvement in peak-shaving ability can be achieved with only a small decrease in total energy production; (c) there is a set of feasible solutions that have advantages over the conventional operation scheme in terms of both objectives; and (d) coordinated operation be­ tween outside hydropower plants and local ones can allow the reservoir levels of local plants to be raised during the dry season to enhance their 8

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generation capabilities while allowing outside plants to bear more re­ sponsibility for peak-shaving tasks in the local power grid. It should be emphasized that the differences in hydrology, generation capacity, reservoir storage, etc., between the source and recipient regions offer great potential to implement coordinated operations, and such coordi­ nation should receive more attention to enable better scheduling and management of such complex hydropower systems in China. In future work, potential improvements in the coordination among UHVDC hydropower and various types of power plants, such as hydro, thermal, wind, and solar plants, can be investigated to generate more efficient operation schemes and transmission schedules. The present model was developed with one receiving power grid. Extensions of the proposed model and method with multiple receiving power grids are recommended as a subject for future work. In addition, this study con­ siders several inflow scenarios to calculate the deterministic operations under real-world dispatching demands. The inflow stochasticity could also be considered in the problem description to enable further

optimization of the operation schedules as well as risk analysis. Data availability The data used in this study are mainly from the cooperation projects with power grid corporation, which are confidential and may only be provided with restrictions. Acknowledgments The National Natural Science Foundation of China (No. 51579029, 91547201), the open research fund of the Key Laboratory of Ocean Energy Utilization and Energy Conservation of the Ministryof Education (LOEC-201806) and the Fundamental Research Funds for the Central Universities (DUT19JC43). The authors are very grateful to the anony­ mous reviewers and editors for constructive comments and suggestions.

Appendix. Figures

Fig. 1. UHVDC power network for hydropower transmission in China.

Fig. 2. Distribution map of interprovincial hydropower system.

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Fig. 3. Methodology for multi-objective optimal operations of interprovincial hydropower system.

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Fig. 4. Roles of hydropower plants in different hydrology seasons.

Fig. 5. Calculation orders of hydropower plants in different hydrology seasons.

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Fig. 6. Principle of load reconstruction-based strategy for determining power generation.

Fig. 7. Principle of multidimensional search method.

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Fig. 8. Monthly average inflows over several decades.

Fig. 9. Flood seasons of two rivers located in two sides of UHV power transmission line.

Fig. 10. Perato optimal solutions of multi-objective operation.

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Fig. 11. Reservoir level profiles of Xiluodu with different inflow scenarios.

Fig. 12. Reservoir level profiles of Tankeng with different inflow scenarios.

Fig. 13. Monthly generation schedules of all plants with different inflow scenarios.

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Fig. 14. Monthly generation schedules from different solutions.

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Fig. 15. Reservoir levels of Tankeng plant over a year.

Appendix. Tables

Table 1 Characteristics of hydropower plants. Hydropower plant

River

Regulating ability

Installed capacity (MW)

Normal level (m)

Dead Level (m)

Xiluodu Shanxi Tankeng Jinshuitan Shitang Sanxikou Hunanzhen Huangtankou Xinanjiang Fuchunjiang

Jinsha River Ou River Ou River Ou River Ou River Ou River Qiantang River Qiantang River Qiantang River Qiantang River

Seasonally Multi-yearly Yearly Yearly Daily Daily Yearly Daily Multi-yearly Daily

12,600 200 604 305 85.8 99.9 320 88 850 354

600.00 142.00 160.00 184.00 102.50 18.00 230.00 115.00 108.00 23.00

540.00 117.00 120.00 164.00 101.10 17.75 190.00 114.00 86.00 21.50

Table 2 Optimized results from different solutions. Solution

Energy production (F1)/billion kWh

Peak-valley ratio (F2)

Category I Category II Category III

36.013 35.260 36.007 35.904 35.561

0.282 0.243 0.276 0.26 0.248

The first group The second group The third group

References

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