Optimal power peak shaving using hydropower to complement wind and solar power uncertainty

Energy Conversion and Management 209 (2020) 112628

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Optimal power peak shaving using hydropower to complement wind and solar power uncertainty

T

Benxi Liua,b, , Jay R. Lundc, Shengli Liaoa,b, Xiaoyu Jina,b, Lingjun Liua,b, Chuntian Chenga,b ⁎

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 University of Technology, Dalian 116024, China c Department of Civil and Environmental Engineering, University of California Davis, Davis 95616, USA b

ARTICLE INFO

ABSTRACT

Keywords: Hydro-wind-solar Generation scheduling Peak shaving Linear programming

Booming renewable energy development, such as wind and solar power, with their intermittency and uncertainty characteristics, pose challenges for power grid dispatching, especially for power grid peak shaving. In this paper, coordinated operation of hydropower and renewable energy in a provincial power grid is explored to alleviate fluctuation and aid peak shaving. Considering their aggregate effect, this study aggregates wind power plants and solar power plants into a virtual wind power plant and a virtual solar power plant, respectively, and the forecasted error distribution of wind and solar power is analyzed with kernel density estimation. Then, based on the principles of using hydropower to compensate for fluctuating wind and solar power, a day ahead peak shaving model with the objective of minimizing residual load peak-valley difference is built, which introduces chance constraints for forecast errors and coordinate hydropower operation with wind and solar power. To simplify the solution, the proposed model is recast as a successive linear programming problem. Day-ahead scheduling case studies of a provincial power grid system indicate that the proposed model can conduct peak shaving effectively, hydropower can compensate wind and solar power fluctuation, improve the stability of combined output, and make better use of renewable energy. Therefore, this study provides an alternative approach for peak shaving operation of power system with hydropower and increasing integration of wind and solar power in China and other places worldwide.

1. Introduction Renewable energy sources, mainly wind and solar power are growing rapidly globally in recent decades to decarbonize the economy while meeting rapidly expanding demand for electricity [1]. Wind power capacity has surged from 9936 MW in 1998 to 564,347 MW in 2018, and the solar power from 305 MW to 487,829 MW in the same period, with annual growth rates of 22.4% and 44.6% respectively in the last 20 years [2]. As the world’s largest carbon footprint since 2004, China has faced widespread criticism. Currently, China has pledged to reduce carbon intensity by 40%-50% in 2020 and reduce by 60%-65% in 2030 compared to 2005 [3]. To support these goals, since electricity generation is the largest carbon emission sector dominated by coal fired thermal power, China has expanded renewable energies on a large scale and been the largest contributor to renewable energy growth in recent years. By the end of 2018, the installed capacity of hydropower, wind power and solar power capacity has surged to 352,260 MW,

184,260 MW and 174,630 MW respectively [4]. For wind and solar power, as shown in Fig. 1, their capacity share has reached 10% and 9% in 2018 from about 2% and nearly 0% in 2009, with annual growth rates of 30% and 167% respectively in China over this period, and the corresponding share of thermal power has decreased from 74% to 60%. Integration of renewables can reduce carbon emission and help alleviate climate change [5]. However, these weather driven power sources are characterized by uncontrollable, intermittent [6], and uncertainty [7], so large-scale integration of wind and solar power is an operational challenge [8]. Complementary operation of indeterminate power sources with traditional hydro/thermal power plants or energy storages like pumped hydropower [10] and compressed air energy storage [11] can help power systems accommodate the fluctuations of non-dispatchable generation and accept larger amounts of wind and solar power. In this, hydropower has the advantages of providing fast response to load variability, robustness to weather fluctuations and energy storage

Corresponding author. E-mail addresses: [email protected] (B. Liu), [email protected] (J.R. Lund), [email protected] (S. Liao), [email protected] (X. Jin), [email protected] (L. Liu), [email protected] (C. Cheng). ⁎

https://doi.org/10.1016/j.enconman.2020.112628 Received 15 November 2019; Received in revised form 27 January 2020; Accepted 19 February 2020 0196-8904/ © 2020 Elsevier Ltd. All rights reserved.

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Nomenclature A. Acronyms VWP VSP CHS PDF CDF

aggregated virtual wind power plant aggregated virtual solar power plant cascaded hydropower system probability density function cumulative density function

Pt P

L

_ l L

Pl

m, l

Rm, t q

upper total discharge limit of plant m at period t lower turbine discharge limits of plant m at period t

_ m, t

of of of of of

qm, t PmH Vm, beg Vm, end Uc Lu Ld

time periods wind power plant solar power plant hydropower plant piecewise segment

D. Parameters and variables

xi h n Ct

upper storage limit of reservoir m at period t lower total discharge limit of plant m at period t

_ m, t

_ m, t

index index index index index

PiW ,t P jS, t PtAW PtAS PmH, t PtAW , a PtAS, a PtAW , f PtAS, f AW t AS t

Vm, t R

j, m

number time periods number of hydropower plants number of transmission lines number of wind power plants number of solar power plants number of piecewise segments of hydropower plant performance curve

C. Indices t i j m k

Hm, t V

t Vm, t Qm, t Rm, t qm, t sm, t Um

B. Constants T M L NW NS K

available maximum output of plant m at period t available minimum output of plant m at period t positive complementary confidence level negative complementary confidence level time period duration storage of reservoir m at periodt natural incremental inflow of reservoir m at period t total discharge of reservoir m at period t turbine discharge of hydropower plant m at period t spill flow of reservoir m at period t direct upstream hydropower plant set of reservoir m water transportation time periods from reservoir j to m water head of hydropower plant m at period t lower storage limit of reservoir m at period t

PmH,,tmax PmH,,tmin

m

m , j, t

output of wind power plant i at period t output of solar power plant j at period t output of VWP at period t output of VSP at period t output of hydropower plant m at period t actual output of VWP at periodt actual output of VSP at periodt forecasted output of VWP at period t forecasted output of VSP at period t forecasted error of VWP at period t forecasted error of VSP at period t value of sample i for kernel density estimation the band width for kernel density estimation number of samples for kernel density estimation load demand of power grid at period t system load reserve rate total output of the hybrid system of hydro, solar and wind power at period t minimum transmission power of line l

PH qk qk ak Wm

upper turbine discharge limits of plant m at period t maximum power ramping capacity of hydropower plant m initial storage of reservoir m expected final storage of reservoir m plant set of a cascaded hydropower system residual load upper boundary residual load lower boundary weight of hydropower generation of plant m auxiliary non-negative variable to ensure output of plant m and j from period t to t + 1 change in the same direction hydropower plant output turbine flow of piecewise segment k upper turbine flow of piecewise segment k slope of linear approximation of piecewise segment k total water discharge of hydropower plant m

E. Functions

F ft ( ) W

ft ( ) S

ft ( )

K( ) Pr{ } FtW 1 ( ) FtS 1 ( )

maximum transmission power of line l weight of power output of plant m to transmission line l

through reservoirs [12]. It is preferred as a power source to compensate for the variability of wind and solar power [13]. Therefore, it is essential to study the compliment operation of hybrid hydro-wind-solar power system to realize large scale renewable energy integration and power system economic and safe operation. At present, many research papers have been published on traditional power sources coordinated operation with renewable power sources. Gupta et al. [14] provided an optimized scheduling strategy for hydropower considering increasing solar and wind generation, and using non-linear optimization and logical optimization comparation to reveals the rule of thumb can perform well. Gebretsadik et al. [15] proposed a method to assess reliability of combined hydro and wind

objective function estimated probability density function of period t

estimated probability density function of wind power at period t estimated probability density function of solar power at period t Gaussian kernel function probability function of chance constraint inverse function of wind power forecast error cumulative probability function inverse function of solar power forecast error cumulative probability function

power operation based on assumed perfect foresight wind generation pattern. Silva et al. [16] analyzed the complementarity of hydropower and offshore wind power from seasonal scale. Wang et al. [17] proposed a model to improve wind power generation and reduce thermal power generation and CO2 emissions by the combined operation of hydro-thermal-wind system. Yuan et al. [18] established a hydrothermal-wind scheduling model with multi objectives of minimize economic cost and carbon emission. Wang et al. [19] proposed a multiobjective model to maximize power generation and minimize output fluctuations for a coordinated hydro-wind-solar power system. Ming et al. [20] proposed a coordinated hydro-solar model that can improve day ahead hydro-solar generation scheduling by increasing total energy 2

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Fig. 1. Trends of China’s wind and solar power development. Data Source: [4,9].

all plants. Moreover, since wind power plants [26] and solar power plants [27] have typical aggregate effects, the power sources in the same area are generally integrated into the same backbone power network. So, it is possible to aggregate the wind power plants as a single virtual plant, and solar power plants as a single virtual plant. Describing the uncertainty of wind and solar power generation is central in modeling this system. Many studies consider wind and solar power output or forecast error as a given probability distribution. For instance, Wang et al. [28] developed a hydro-wind-thermal coordination model describing wind forecast error as a normal distribution. Yang et al. [29] use a model to calculate related solar power plant output and explore long-term operating rules for hydro-solar integrated system. Some studies describe the uncertainty by constructing many scenarios. Baringo et al. [30] use a set of scenarios to model wind-power production, market prices and demand bids to derive an optimal offering strategy of wind power producer in electricity market. Xu et al. [31] use scenarios to represent wind and solar power uncertainty in day-ahead unit commitment modeling, however, building many scenarios greatly increases the difficulty of modeling large-scale hybrid systems. Different from the above studies, the present study focuses on the peak shaving in a regional power system with substantial hydropower resources integrated with large-scale wind and solar power plants. Considering hydropower to compensate for intermittent and uncertainty of wind and solar power, a hybrid hydro-wind-solar coordinated day-ahead scheduling method is proposed. First, this paper aggregates all wind power plants as a virtual wind power plant and all solar power plants as a virtual solar power plant. Based on wind and solar power forecast error, non-parametric kernel density estimation is used to describe the uncertain generation of these power sources. With the objective of smoothing the residual load as much as possible, a peak shaving model considering the forecasted generation of wind and solar power is built. Based on the forecast error distribution, chance constraints are used to avoid power shortages from insufficient wind and solar power generation, and negative reserve chance constraints are used to absorb wind and solar power over-generation. To reduce solution difficulty, the method recasts the min-max objective into a linear expression, and converts the chance constraints and non-linear constraints to deterministic and linear forms, using successive linear

production and decreasing hydropower units’ total online time. However, these studies mainly focused on hydro/thermal-wind-solar coordinate operation strategy, which is not suitable for power system dayahead peak shaving. With higher integration of wind and solar power, the anti-peak regulation characteristic requires more robustness scheduling of power system, especially for peak shaving. There are many studies have been published on hydropower or multipower system peak operation. In [21], maximum power generation and daily peak shaving demands was considered in a multi-objective operation model for the monthly operation of an interprovincial hydropower system. Notton et al. [22] developed a simulation tool using pumped storage hydropower plant to compensate wind and solar power plants satisfying peak shaving requirement in an island. Feng et al. [23] developed an optimal dayahead operation model for hybrid hydro-thermal-nuclear system considering peak shaving requirement of several power grids. Wang et al. [24] developed a hydro-thermal-wind-solar coordinated model considering objectives of maximize renewable energy generation and smooth residual load, aimed to reveal the relationship between hydropower outflow and coordinated effectiveness under different water availability scenarios. Cheng et al. [25] proposed a three-step hybrid algorithm to optimize pumped storage hydropower plants that serving peak shaving of multiple power grids in day-ahead scheduling. However, these studies mainly focus on peak shaving operation with hydropower or pumped hydropower plants coordinate operation with a single or small number of wind or solar power plants, or just consider wind and solar power as renewable that needs to generate with priority but ignore their uncertainty. Considering uncertainty of wind and solar power uncertainty is very essential for peak shaving. For a regional power system integrated with a large number of wind and solar power plants, it is hard to consider the uncertainty of each plant. That is, how to consider a large number of wind and solar power plants and their uncertainty during scheduling is a critical issue. Typically, for a regional power system, there is a wind power and solar power generation forecast system that can analyze prediction models and calibrate parameters for each power plant according to its characteristics and geographical meteorological conditions. Therefore, this generally has higher accuracy than using a single model to forecast the generation of 3

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programming to solve the problem. For a provincial power grid in southwestern China, simulation results show that the proposed model could reduce the peak-valley difference effectively and facilitate use of wind and solar power. The contributions of this paper are:

t

AS

=

PtAS, f

PtAS, a Pt

AS, f

× 100%

(4)

Forecast error for wind power and solar power is usually assumed as a normal distribution [28]. However, studies show that neither normal distribution [34] nor simple statistical analysis [35] fit the forecast error well. Kernel density estimation can address unknow probability distributions with only discrete samples [36]. This paper employs nonparametric kernel density estimation in Eq. (5) to estimate the forecast error distribution.

(1) aggregation of wind power plants and solar plants as a virtual plant respectively and use non-parametric kernel density estimation to describe the forecast error; (2) proposed a short-term day-ahead hybrid hydro-wind-solar coordinated peak operation model minimizes residual load peakvalley difference and introduces constraints to compensate for wind and solar power forecast errors; (3) the model is cast as a successive linear program, and case studies illustrate its effectiveness and applicability.

f (x ) =

1 nh

n

ke i=1

x

xi h

(5)

where n is the number of samples, and x i is the value of sample i. In this study, samples are the forecast errors of wind and solar power computed from historical data in each period. Moreover, a Gaussian kernel and a rule-of-thumb bandwidth estimator recommended by Scott [37] is used in this paper. For simplicity, this paper considers that the forecast errors of different energy sources at each period are independent. Fig. 2 shows the PDF curve and CDF curve of wind power forecast error at one period. Fig. 2(a) shows that the forecast error is non-normal and skewed, and the forecast error mainly concentrates in the interval of [−35%, 55%]. Fig. 2(b) shows the cumulative probability curve and the 5% and 95% confidence limits around the mean. With this method, this paper estimates the distribution of the actual generation of each period with the forecasted value.

The rest of this paper is organized as follows. Section 2 describes the method to analyze the uncertain distribution of wind and solar power generation. Section 3 gives the formulation of hybrid hydro-wind-solar peak shaving model, and the recast objective function, non-linear constraints as a successive linear programming problem. Numerical simulation and the discussion and example are presented in Section 4. Finally, Section 5 presents conclusions and possible further studies. 2. Analysis of wind and solar power forecast errors Wind and solar power generation vary greatly with weather conditions, and it is hard to forecast accurately. For regional power grid dispatch, a generation schedule usually considers all types of load and power sources in the region. With the aggregation effect, forecast error decreases with the aggregation of diverse geographical wind [32] and solar power plants [33]. Therefore, this paper aggregates all wind power plants as a virtual wind power plant (VWP) in this study. And the same for aggregating all solar power plants as a virtual solar power plant (VSP).

3. Hybrid hydro-wind-solar peak shaving model This section describes the objective function of the hydro-wind-solar peak shaving model in detail first, followed by the constraints of the proposed model in Section 3.2. Finally, Section 3.3 explains how to solve the model.

NW

PtAW =

PiW ,t

3.1. Objective

(1)

i=1

Generally, to stimulate the development of renewable generation, promote the consumption of renewable energy, and reduce carbon dioxide emission, many countries give priority to renewable generation in the power grid and market [38], with priorities or a preferential order of power types [39]. The day-ahead peak shaving problem seeks an optimal smooth residual load, as frequent adjustment of thermal power output reduces energy efficiency and increases carbon dioxide emissions. Therefore, an objective function considers that gives renewable energy priority to obtain a smooth residual load is as following:

NS

PtAS =

P jS, t

(2)

j=1

Analysis of the deviation between predicted output and actual output of renewable energies is helpful for improving scheduling for other power sources. Generally, the prediction error of the same period in the same season is similar. Therefore, the forecast error of aggregated wind and solar power can be calculated as following: t

AW

=

PtAW , a Pt

PtAW , f AW , f

× 100%

F = min{ max (Ct

(3)

1 t T

Pt )}

Fig. 2. Wind generation forecast error PDF curve and CDF curve of one period. 4

(6)

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B. Liu, et al. M

PmH, t + PtAW , f + PtAS , f ; t

Pt =

[1, T ]

where Pr{ } is the probability of the expression in the braces. PmH,,tmax is the available max output of hydropower plant m at period t . is the M Ct ) is the maximum output of hyconfidence level. ( m = 1 PmH,,tmax dropower plants considering system reserve. This constraint means that when the actual generation of wind and solar power is less than the expected value, hydropower plants can compensate the deviation with enough capacity with confidence level of to avoid power shortage.

(7)

m=1

where F is the objective. Ct is the load demand of power grid at period t . Pt is total generation of the hybrid system of hydro, solar and wind power. T and M are the number of time periods and hydropower plants respectively. PmH, t is the power output of hydropower plant m at period t , which can be a function of net head and water flow as following:

PmH, t = g (Hm, t , qm, t ); m

[1, M ], t

(2) Negative compensate constraint, for hydropower to accommodate increased wind and solar power generation:

(8)

[1, T ]

where Hm, t and qm, t are water head and turbine discharge of hydropower plant m at period t respectively. Obviously, Ct Pt is the residual load of period t to be satisfied by other power sources like thermal power. This objective function means that the difference of maximum value and minimum value of residual load should be as small as possible. Ideally, the residual load can keep a same value, in which case, thermal power can get the best performance which saves fuels and reduces carbon dioxide emission.

M

(1) System reserve:

× Ct ; t

[1, T ]

M

[1, T ]

Vm, t + 1 = Vm, t + Qm, t +

(10)

m=1

_ l

m, l m=1

where P

_ l

L

× P mH, t

L

Pl ; t

[1, T ], l

[1, L]

L

V

and Pl are the minimum and maximum transmission power

_ m, t

Ct + PtAW ,a + PtAS, a

Pt

;m

[1, M ], t

Vm, t

Vm, t ; m

where V

_ m, t

[1, M ], t

[1, T ]

(15)

and Vm, t are the lower and upper storage bounds of hy-

dropower plant m at period t . (3) Total discharge constraints:

R

_ m, t

Rm, t

Rm, t ; m

where R

_ m, t

[1, M ], t

[1, T ]

(16)

and Rm, t are the lower and upper bounds of total dis-

charge of hydropower plant m at period t . (4) Turbine discharge constraints:

q _ m, t

(1) Positive compensate constraint, for hydropower to accommodate reduced wind and solar power generation: PmH,,tmax

[1, T ]

(2) Reservoir storage constraints:

(11)

3.2.2. Compensate constraints for wind and solar power Wind and solar power generation are intermittent and cannot be forecast accurately. When actual renewable generation deviates from the expected value, actual peak shaving output will deviate from the expect generation and require other power sources to compensate for the deviation, which may reduce overall efficiency. This paper introduces constraints to use hydropower to compensate for the intermittence and forecast errors to ensure the residual load can remain smooth.

M

[1, M ], t

where Rm, t = qm, t + sm, t . Vm, t is the storage of plant m at period t . Qm, t , Rm, t and sm, t are the natural incremental inflow, total discharge and spill flow of plant m at period t respectively. Um is the direct upstream hydropower plant set of plant m . j, m is water transportation time periods from hydropower plant j to m. t is the time period duration.

of line l . m, l is the weight of power output of plant m to transmission line l . This constraint is adopted to represent the power transmission limits of some key tie-lines. For a regional or provincial power grid considered in this paper, only certain large hydropower plants connection to the tie-lines are considered.

Pr

Rm, t × t; m

j, m

(14)

(2) Transmission limit: M

Rj, t j Um

These constraints represent system positive and negative reserve requirement respectively, usually considered as a certain proportion ( in this paper) of the load.

L

(13)

(1) Mass balance constraints:

(9)

m=1

P

[1, M ], t

3.2.3. Hydropower constraints

M

× Ct ; t

;m

[1, T ]

3.2.1. System constraints

PmH, t

Pt

where PmH,,tmin is the available minimum output of hydropower m at M period t . is the confidence level. ( m = 1 PmH,,tmin + Ct ) is the minimum output of hydropower plants considering system reserve. This constraint means that when actual generation of wind power and solar power exceeds the forecasted value, hydropower plants can reduce output to free load for renewables to ensure absorption of un-adjustable renewable generation priory. These two constraints assure the renewable energies can be consumed and that the hybrid system provides expected generation within confidence levels. Increasing the reliability of the hybrid system helps improve the safety and reliability operation of the overall power system.

This model consider system constraints, compensate constraints for wind and solar power forecast error and physical limitations of hydropower plants.

PmH, t )

Ct + PtAW , a + PtAS, a

m=1

3.2. Constraints

(P mH, ,tmax

PmH,,tmin +

Pr

qm, t

where q

_ m, t

qm, t ; m

[1, M ], t

[1, T ]

of hydropower plant m at period t .

[1, T ]

(5) Power output ramping constraints:

m =1

(12) 5

(17)

and qm, t are the lower and upper bounds of turbine discharge

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|PmH, t

PmH ; m

P m, t 1|

[1, M ], t

[1, T ]

where Lu and Ld are the auxiliary variables; constraints (23) and (24) are the introduced to represent the possible upper and lower boundaries of residual load. With these function and constraints, the objective function is recast to a linear function. As shown in Fig. 3, when the collected generation of hybrid hydrowind-solar system is small, then the theoretical optimal residual load will consist of a curve at the bottom of the valley and a straight line at the peak show as “residual load 1”. When the collected generation is large enough, then the residual load can be a perfect straight line that the difference between maximum and minimum residual load is equal to 0, that means the objective value can reach its theoretical optimal value equivalent to 0. However, when the collected generation exceeds the electricity requirement between the peak and valley, objective function Eq. (22) can only ensure that the residual load curve would be a straight line, and there may be countless optimal solutions. For example, as shown in Fig. 3, both “residual load 2” and “residual load 3” can reach the same optimal value that is 0. However, it is obvious that “residual load 3” is better than “residual load 2” with the same input condition, that “residual load 2” may result in a lot of curtailment. Therefore, to avoid this situation, increase power generation of the hybrid hydro-wind-solar system, and reduce residual load left for thermal power, this paper introduces an auxiliary maximum hydropower generation part in the objective function as following:

(18)

where PmH is the maximum power ramping capacity of hydropower plant m . (6) Initial storage constraints:

Vm,0 = Vm, beg ; m

(19)

[1, M ]

where Vm, beg is the initial storage volume of hydropower plant m . (7) Expected final storage constraints:

Vm, T

Vm, end; m

(20)

[1, M ]

where Vm, end is the expected final storage volume of hydropower plant m. (8) Simultaneous regulation constraints:

(PmH, t + 1

PmH, t )(P jH, t + 1

P jH, t )

0; m , j

U c, t

[1, T

1]

(21)

where is the set of a cascaded hydropower system. This constraint means the output of all plants of a cascaded hydropower system should change with the same trends in each period. For cascaded hydropower plants with complicated hydraulic and electrical connections, this constraint is helpful for avoiding frequent output fluctuation during power peak shaving [24]. In this paper, this simultaneous regulation constraint is adopted to keep the output of each hydropower plant of the same cascaded hydropower system change in the same direction.

Uc

M

F = min (Lu

M

(22)

Ct

Pt ; t

[1, T]

(23)

Ld

Ct

Pt ; t

[1, T]

(24)

t

(25)

t=1

3.3.2. Deterministic equivalent of stochastic constraints In the proposed model, the chance constraints (12) and (13) makes it difficult to solve directly. From the analysis in Section 2, these constraints can be converted to equivalent linear form by introducing the cumulative distribution curves of wind power forecast error and solar power forecast error. According to Eqs. (3) and (4), AW , f PtAW , a = Pt (1 + tAW ) , PtAS, a = PtAS , f (1 + tAS ) , then constraint (12) can be convert to:

3.3.1. Objective function recast The min-max objective function of Eq. (6) can be formulated more linearly using the following equivalent form by introducing auxiliary variables and constraints:

Lu

PmH, t

where m is the weight of hydro generation of plant m , which can be a very small value since the objective of this model is peaking operation.

This model includes min-max objective function, stochastic constraints and non-linear hydropower constraints, it’s not easy to solve directly. This section describes, in detail, recast the stochastic constraints into deterministic form and the non-linear objective function and constraints into linear solvable problem.

Ld}

m m =1

3.3. Solution method

F = min{Lu

T

Ld )

Pr

PmH,,tmax

Ct + PtAW , f (1 +

t

AW

) + PtAS, f (1 +

t

AS

)

Pt

m =1

(26) as Pt =

M m=1

PmH, t

+ Pt

AW , f

+ Pt

AS , f

Fig. 3. The demonstration of the peak operation of hybrid hydro-wind-solar system. 6

, this can be converted to:

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Pr PtAW , f

AW t

+ PtAS, f

t

M

AS

M

PmH, t

m=1

PmH,,tmax

where ak is the slope of the linear approximation of segment k. qk and qk are the turbine flow and its up limit of segment k. Since the dischargepower curve is concave, that means ak > ak + 1 for k [1, K 1], from the objective Eq. (25), as maximum generation is a part of the objective, with the mass balance constraints, the model will give priority to use the segment with the highest feasible slope, that means ifqk + 1 > 0 , then qs = qs qs 1 for s [1, k ]. So, this piecewise linearization method does not need auxiliary binary variables and remain as a maximization LP problem. For head sensitive hydropower plants, the power performance can be expressed as a family of fix-head curves, with each curve representing a specific head, and successive linear programming can be used to solve this problem. The main procedures are given as follows with details from [40]. Step 1: Estimate total water discharge of each plant by Eq. (35), assumed that discharge is a constant value at each period, then the initial water head of each period can be estimated.

Ct

m=1

(27) As: t

t

AW

AS

= FtAW

1 (1

(28)

)

= FtAS 1 (1

(29)

)

) are the inverse functions of the CDFs of where ) and FtAS ( ) respectively. Then constraint wind and (12) can be converted to the equivalent deterministic form as following: ) and FtAS 1 ( solar power FtAW (

FtAW 1 (

M

M

PmH, t m=1

Ct + PtAW , f FtAW

PmH,,tmax

1 (1

) + PtAS, f FtAS 1 (1

)

m=1

(30)

) + PtAS, f FtAS 1 (1 ) is the positive comwhere PtAW , f FtAW 1 (1 pensation capacity. Similarly, for constraint (13), with the PDFs and CDFs estimated by Eq. (5), tAW = FtAW 1 ( ) , tAS = FtAS 1 ( ) , then it can be converted to the following form: M

M

PmH, t m=1

PmH,,tmin +

Ct + PtAW , f FtAW

1(

) + PtAS, f FtAS 1 ( )

m=1

T

Wm = Vm, beg

AS , f

FtAW 1 (

PmH, t ) =

0; m , j

m , j, t

m, j, t

× (P jH, t + 1

(31)

FtAS 1 (

U c, t

[1, T

P jH, t ); m , j

1]

U c, t

[1, T

1]

(32)

In this section, the overview of a provincial power grid is briefly described. Then, three case studies include dry season, flood season and long run in different seasons were adopted to show the effectiveness of the proposed model. Finally, the required hydropower compensation capacity in different quarter under different confidence level is discussed in Section 4.3.

3.3.4. Linearization of hydropower performance Hydropower generation performance is a nonlinear function because of storage-forebay water level curve, discharge-tail water level curve, turbine release-performance curve. It is not easy to solve largescale nonlinear optimization problems, so studies often simplify hydropower generation performance functions as fix-head and head sensitive cases. For fixed-head hydropower plants, many studies use a single linear function [40] or piecewise linearization with binary variables [41] to approximate turbine discharge-performance curve. However, this is not appropriate for a day-ahead hydropower peak shaving scheduling as discharge may change frequently and a single linear function deviate from realistic discharge-performance function a lot, and introducing binary variables will transform the problem from LP to MILP which is more difficult to solve. As shown in Fig. 4, a hydropower plant with a given head usually has a non-linear concave turbine discharge-generation curve. Therefore, a concave piecewise linearization is introduced to approximate the release-production curve [42]. Hydropower power output with fixed head or upper storage can be expressed as:

0

qk

qk

K k=1

ak qk

qk

1k

q0 = 0

(35)

4. Case study

(33)

These equations introduce auxiliary variables m, j, t , and change the original nonlinear equation into a linear equation and a non-negative variable.

PH =

Wj j Um

Step 2: Set the performance parameters of the given head with the piecewise linearization values. Then use the LP solver to find the solution for the current subproblem. Step 3: When head gap of the current iteration and last iteration satisfy the terminal condition or the number of iterations exceeds the maximum iterations, then stop. Otherwise, update the head and the release-power output curve, and recalculate with LP. By this time, the model is recast as a deterministic equivalent, and a practical successive linear programming peak shaving model is formulated considering the compensating operation of hydropower with wind and solar power.

3.3.3. Linearization of simultaneous regulation constraints The simultaneous regulation constraint (21) can be recast into the following equivalent form:

(PmH, t + 1

Qm, t t + t=1

) + Pt ) is the negative compensation where Pt capacity. Through these transformations, the chance constraints can be converted to deterministic equivalent form. AW , f

Vm, end +

[1, K ] Fig. 4. Piecewise linearization turbine discharge-performance curve.

(34) 7

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4.1. Overview of a provincial power grid in southwest China

large, and with the optimal operation of the proposed model, residual load for thermo-electric generation is a straight line during the planning period. The peak-valley difference decreases from 17,324 MW to 0 MW, and the load rate increases from 76.4% to 100%, shows that the proposed model can use hydropower to conduct peak shaving and compensate for wind and solar power variations effectively. Since the output of hybrid hydro-wind-solar system is large enough, it can not only shave the peaks, but also take some baseload. Compare to the observed schedule data, although the observed schedule has a similar curve with the proposed model optimal result, there is a peak-valley difference of 553 MW left. Moreover, the load rate and standard deviation also demonstrate that the proposed model provides better results. Fig. 8 shows the forecast and actual output of wind and solar power and the upper and lower bounds respectively. Since the hybrid hydrowind-solar peak shaving model has considered forecast error of wind and solar power, hydropower has enough compensation capacity to offset their uncertainty and the system can fully absorb their generation as long as the actual output is between the upper and lower bounds. Wind power is important in shaving the second peak around 18:00, that can offer expected 4000 MW output, and at least 2000 MW with the confidence of 95%. Moreover, Fig. 9 shows the optimal scheduling of each hydropower plant. It shows that all plants generate relatively smoothly, and all plants participate in peak shaving by utilizing their regulation capacity.

The proposed method was tested for real-world day-ahead peak shaving scheduling to get quarter-hourly generation schedules of each hydropower plant in a provincial power grid in southwest China. This province has developed extensive hydropower resources concentrated in two cascaded hydropower systems (CHSs) shown in Table 1. Other power sources mainly include coal fired thermal, wind and solar power. The province mainly relies on these two CHSs for peak shaving. To meet increasing energy demand and implement low carbon development, as shown in Fig. 5, wind and solar power have developed rapidly in this province. Installed wind power capacity has surged from 670 MW in 2011 to 8480 MW in 2018 and solar power surged from 20 MW in 2011 to 3100 MW in 2018. The fast development of these sources challenge power dispatching, especially the intermittence of wind and solar power for peak shaving. With the forecasted output data from wind and solar power generation forecast system and actual output data collected by the provincial dispatching department. Fig. 6 shows the quarter-hourly lower and upper bounds of forecast error of wind and solar power when = 0.95, = 0.95 (estimated by Eqs. (3)–(5)), in addition with the mean capacity factor forecast output, in each quarter. As shown in Fig. 6(a), there are some clear insights: 1) wind power generation has higher output during the first quarter, with capacity factor about 0.5, and this value is very low in the third quarter, only about 0.16. The other two quarters are about 0.3–0.35. 2) Wind power has high generation at night and low in the daytime in every quarter, with the lowest generation around 10 a.m. 3) the forecast error bound is about [−50%, 50%], however, this bound is larger in the third quarter when wind generation is low, which means it is more difficult to forecast wind generation during the wind valley season. Fig. 6(b) has some clear insights as well: 1) solar power generation peaks at noon in every quarter which is contrary to wind power. 2) The mean capacity factor in the third quarter is a little less than the other 3 quarters, perhaps because the third quarter is rainy season with rain and clouds affecting solar generation. 3) The forecast error bound is about [−50%, 30%]. These figures show these two power sources are somewhat complementary for daily scheduling.

4.2.2. Case study 2: flood season performance A typical flood season load curve is used to test the proposed model for this case study with the same and of case 1. Like the previous case, Table 3 shows that both Model 1 and Model 2 have better results in terms of peak-valley difference, load rate and standard deviation than observed schedule data, and Model 2 has better results in these terms than Model 1. Fig. 10 shows that hydropower output of both Model 1 and Model 2 are limited by upper bounds during the peak periods around 11 AM, as Model 2 does not consider the uncertainty of wind and solar power, it has higher scheduling output during the peak periods and results in a smaller remaining peak-valley difference. However, Fig. 11 shows that the actual output of wind is much lower than the forecast value in some periods. For Model 1, since the actual generation of both wind and solar power are between their upper and lower bounds, and it has considered their uncertainty, the reserved hydropower compensation capacity can offset differences between forecast and actual output and offer enough generation to avoid power shortage. However, for Model 2 and observed scheduling, they may face power shortages during these periods if there are not enough other power sources for reserve. Therefore, the proposed model (Model 1) which considers wind and solar power uncertainty is more practicable for real operations. As shown in Fig. 12, all plants in basin R2 are scheduled with full

4.2. Case studies Since this province has a distinct dry season (first quarter) and flood season (third quarter), scenarios in different seasons are chosen to test the model’s performance in day-ahead peaking operation with a quarter-hour time step. To show the effectiveness of the proposed model, the proposed model was named as Model 1, and a second model (Model 2) which is the same as model 1 except without considering forecast error of wind and solar power is adopted. Moreover, the observed schedule data from power dispatching department is used to verify the effectiveness of the proposed model.

Table 1 Basic information of the hydropower plants.

4.2.1. Case study 1: dry season performance The first quarter is in the dry season, with less rain and natural inflow to the reservoir and greater wind generation. Set = 0.95, = 0.95. Table 2 shows detailed results. Fig. 7 shows the quarter-hourly generation peak shaving optimal results of the hybrid hydro-wind-solar system obtained by the proposed model for a typical day in dry season, which includes the original load, residual load, output of each power source, and hydropower upper bounds and lower bounds, it shows that Model 1 and Model 2 have different hydropower bounds. Both Fig. 7 and Table 2 show that these two models have same results. For Model 1, hydropower has enough regulation capacity for peak shaving and compensating for wind and solar power uncertainty. For Model 2, however, may not prepare enough compensation capacity in actual operation since it does not consider wind and solar power uncertainty. Table 2 shows that the original peak-valley load difference is very

Basin

Plant

Installed capacity (MW)

Regulation capability

Total storage (108m3)

R1

A B C D E F G H I J K L

900 4200 1670 1350 5850 1750 2400 2000 2400 1800 2160 3000

Daily Multiyear Seasonal Seasonal Multiyear Seasonal Daily Daily Daily Daily Daily Weekly

3.16 149.14 9.2 9.4 237.03 11.4 7.72 8.82 9.13 5.07 17.18 20.72

R2

8

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full capacity during other periods. To the contrary, the power plants of R1 CHS have larger reservoirs and better regulation capacity, the output of R1 CHS decreases to very small in some periods to reduce the peak-valley difference. Furthermore, compare to case 1, although the peak-valley difference of the original load is about 10% lower, peakshaving is more challenging during the flood season as some hydropower plants have very limited regulation capacity. 4.2.3. Case study 3: long run performance 12 scenarios in 4 quarters are adopted to further test the effectiveness of the proposed model (Model 1). Table 4 shows results of each scenario and Fig. 13 shows the original, residual load curves, the output of each energy sources and the lower and upper bounds of hydropower. There are some clear insights as following.

Fig. 5. Evolution of the installed capacity of wind and solar power in the study area from 2011 to 2018.

(1) The proposed method gets good results in every scenario, and most scenarios can get perfect straight residual load lines. (2) The greater the forecasted output of wind and solar power, the higher the required compensation capacity with constraints (30) and (31), which may limit the peak regulation capacity of hydropower and result in remaining peak-valley difference (Scenarios 3 and 5). Take Scenario 5 as an example, its peak-valley difference of original load is not the largest compare to Scenario 4 and Scenario

capacity during high load periods, and the peak shaving capacity, system reserve capacity, and compensation capacity mainly relies on plants of basin R1 CHS. This is because all reservoirs of R2 CHS have little regulation capacity as shown in Table 1. During the flood season, when inflow exceeds the regulation ability of these reservoirs, they try to use the limited storage during the valley periods and generate with

Fig. 6. The lower and upper bounds when

= 0.95,

= 0.95 and the mean capacity factor of wind and solar power in each quarter. 9

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Table 2 Peak shaving results of hybrid hydro-wind-solar system of case 1. Model Original load Model 1 Model 2 Observed schedule

Item

Avg (MW)

Max (MW)

Min (MW)

Peak-valley difference (MW)

Load rate (%)

Std. (MW)

Residual load Improvement (%) Residual load Improvement (%) Residual load Improvement (%)

26,390 10,080 61.8 10,080 61.8 10,086 61.8

34,545 10,080 70.8 10,080 70.8 10,339 70.1

17,221 10,080 41.5 10,080 41.5 9786 43.2

17,324 0 100 0 100 553 96.8

76.4 100 23.6 100 23.6 97.6 21.2

6560 0 100 0 100 205 96.9

Note: Peak-valley difference = Maximum – Minimum, Load rate = Average/Maximum * 100%, larger load rate means smaller variation of the load curve.

6, but it has higher forecasted wind output, and require corresponding higher negative compensation capacity (shown as hydropower lower bounds in Scenario 5 in Fig. 13). During 1:45–6:15, although hydropower is scheduled at lower bounds, there is still a residual load peak-valley difference during these periods. (3) With large installed hydropower capacity, the peak-valley load difference does not directly affect the peak shaving effect, but it may affect the result due to insufficient reserve capacity or compensation capacity. It can be seen that the four scenarios (Scenarios 4, 5, 6 and 10) with largest peak-valley load differences of more than 20,000 MW all achieved good results. Three of them (Scenarios 4, 6 and 10) achieved perfect straight residual load lines, the left one (Scenario 5) has some residual peak-valley difference left because of negative compensation capacity limit. (4) The original load rate in the third quarter is obviously higher than in other quarters. This is mainly because of the large inflows concentrated in this quarter, hydropower has a higher generation, and some hydropower generation needs to be exported to other regions. During this quarter, some hydropower plants are running at the baseload to avoid spill ad cannot participate in peak regulation (detailed in case 2), the peak shaving capacity is limited by the system reserve constraint and compensate constraints for wind and solar power, there will be some residual peak-valley differences left. So, higher load rates may mean limited regulation capacity and result in some residual load peak-valley difference. Nevertheless, the residual load curves optimized by the proposed model are quite smooth and easy for other power source scheduling. (5) As shown in Table 4 and Fig. 13, there are 5 scenarios where the residual load peak-valley difference is larger than zero (Scenarios 3, 5, 7, 8, 9) and can be divided into 3 types. The first type (Scenarios 3 and 5) is mainly due to lower bounds of negative system reserve constraint and negative compensate constraint. In contrast, the

second type (Scenarios 7 and 9) is mainly due to upper bounds of positive system reserve constraint and positive compensate constraints. The third type (Scenario 8) is mainly because some hydropower plants need to generate with full capacity to avoid spill and can offer limited peak regulation capacity during the flood season. As shown in Fig. 14, the plants of R2 CHS try to use the limited storage during the valley periods and generate with full capacity during other periods, the plants of R1 CHS try to shave the peak as much as possible. Although hydropower output does not reach upper or lower bounds, there’s still residual load peak-valley difference. (6) As mentioned in Fig. 6 and shown in Fig. 13, although wind and solar power have different generation characteristics in each quarter, hydropower can compensate for wind and solar power uncertainty with a 95% confidence level in each scenario and ensure that the hybrid hydro-wind-solar system can generate electricity according to the plan in most scenarios. 4.3. Discussion Table 5 shows the required hydropower compensation capacity during each quarter for different confidence levels. Some insights include: 1) Wind power generation forecast error is larger in the third quarter. However, as mean generation is much lower than the value of first quarter, so, this may require less compensation capacity. 2) Although the required hydropower compensation capacity is smaller in the third quarter, as mentioned in case 2, the third quarter is the flood season, some hydropower plants with small reservoirs and limited regulation capacity can offer less compensation capacity during this quarter, and complementary scheduling relies on hydro plants with large reservoirs. So, if installed capacity of wind and solar power continue growth quickly, this province may face more frequent shortage of

Fig. 7. Optimal peak shaving result of hydro-wind-solar system in case 1. (Note: The number 1 and 2 of labels in the legend means results of Model 1 and Model 2). 10

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Fig. 8. The forecast and actual output of wind and solar power and their lower and upper bounds in case 1.

Fig. 9. The output of each plant of these two CHSs in case 1. Table 3 Peak shaving results of hybrid hydro-wind-solar system in case 2. Model Original load Model 1 Model 2 Observed schedule

Item

Avg (MW)

Max (MW)

Min (MW)

Peak-valley difference (MW)

Load rate (%)

Std. (MW)

Residual load Improvement (%) Residual load Improvement (%) Residual load Improvement (%)

39,922 17,276 56.7 17,280 56.7 17,285 56.7

46,496 19,272 58.6 18,042 61.2 17,801 61.7

30,606 17,124 44.1 17,250 43.6 15,418 49.6

15,890 2,148 86.5 793 95.0 2,383 85.0

85.9 89.6 3.8 95.8 9.9 97.1 11.2

5,237 461 91.2 144 97.3 747 85.7

peak shaving resources in each quarter especially in the third quarter. 3) Higher confidence level requires higher hydropower compensation capacity, since wind and solar generation predication accuracy is low, appropriate curtailment of their generation may greatly reduce needed compensation capacity and add power system stability. As shown in Table 5, the required positive compensation capacity can drop from 4551 MW to 3562 MW in the first quarter when confidence level decreases from 0.95 to 0.9; and reduce much more when the confidence level continues decrease. 4) Positive compensation capacity is always larger than the negative values with the same confidence value of and , meaning that renewable energies may generate less than the forecast value. For this situation, different positive and negative confidence levels may be a good choice. For instance, as shown in Table 5, when set = 0.8 and = 0.95, the positive and negative compensation requirements will be 2433 MW and 2771 MW respectively in the first quarter. Improving wind and solar power forecast accuracy may help alleviate peak operation. If the forecast error is reduced by 20%, the required compensation capacities calculate from Eqs. (30), (31) and shown in Table 5 can decrease 20%, which means more hydropower

can participate in peak-shaving. Take scenarios in case 3 that has remaining peak-valley difference as examples. Assume that forecast error of wind and solar power reduced by 20%, re-simulate these scenarios, and the results are shown in Table 6. Compare to the results in Table 4, the residual peak-valley difference of Scenarios 3, 5, 7 and 9 reduces significantly. As the remaining peak-valley difference of these scenarios is mainly affected by the upper and lower limits as mentioned before, improving wind and solar power forecast accuracy will be very helpful. However, as restricted by limited hydropower regulation capacity and avoid hydropower spill as mentioned before, the remaining peak-valley difference of Scenario 8 does not change. 5. Conclusions Rapidly growing wind and solar generation capacities bring challenges to the operation of power systems, especially for peak shaving. Coordinating these intermittent power sources with traditional power generation, especially hydropower, can improve efficiency and economy. For a provincial or regional power system with installed 11

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Fig. 10. Optimal peak shaving result of hydro-wind-solar system in case 2 (Note: The number 1 and 2 of labels in the legend means results of Model 1 and Model 2).

Fig. 11. The forecast and actual output of wind and solar power and their lower and upper bounds in case 2.

Fig. 12. The output of each plant of these two CHSs in case 2.

hydropower plants and large-scale wind and solar power integration, this study proposes a method to coordinate operation of a hybrid hydrowind-solar system for power system peak shaving. This paper aggregates all wind and solar power into a virtual wind power plant and a virtual solar power plant respectively. Based on the forecasted and actual generation differences, non-parametric kernel density estimation is used to describe the forecast error of renewable energies. A hybrid hydro-wind-solar complementary peak shaving operation model is proposed with constraints considering hydropower compensation for

wind and solar power forecast errors. This model with a non-linear objective and non-linear constraints is recast as a successive linear programming problem. Case studies of a provincial power grid in southwestern China show that: 1) the proposed model can shave peaks effectively by taking advantage of the flexible regulation ability of hydropower plants; 2) the hybrid system can use hydropower to compensate for wind and solar power, absorb renewable generation and avoid power shortage, and help smooth residual loads for other power sources within confidence levels in different seasons; 3) Different 12

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Table 4 Comparison of the original load and the residual load of the proposed model. Quarter

Scenario

Item

Average (MW)

Peak (MW)

Valley (MW)

Peak-valley Difference (MW)

Load rate (%)

Std. (MW)

First quarter

Scenario 1

Original Optimal result Original Optimal result Original Optimal result Original Optimal result Original Optimal result Original Optimal result Original Optimal result Original Optimal result Original Optimal result Original Optimal result Original Optimal result Original Optimal result

24,875 10,964 27,173 8615 23,029 9,688 29,387 12,844 29,522 12,035 28,863 14,313 40,048 17,762 36,861 16,516 40,944 18,476 30,296 11,418 24,938 6,549 32,106 13,072

32,988 10,964 34,551 8615 30,268 9,700 38,074 12,844 38,259 12,514 38,225 14,313 46,278 18,920 42,224 16,719 47,228 20,186 38,545 11,418 31,880 6,549 39,734 13,072

16,509 10,964 17,861 8615 15,953 9,527 17,660 12,844 17,734 10,396 16,361 14,313 31,883 15,984 29,308 16,024 34,099 17,146 18,487 11,418 17,287 6,549 23,802 13,072

16,479 0 16,690 0 14,315 172 20,414 0 20,525 2,118 21,864 0 14,395 2,935 12,916 695 13,129 3,041 20,058 0 14,593 0 15,932 0

75.4 100.0 78.6 100.0 76.1 99.9 77.2 100.0 77.2 96.2 75.5 100.0 86.5 93.9 87.3 98.8 86.7 91.5 78.6 100.0 78.2 100.0 80.8 100.0

6072 0 6376 0 4849 38 7550 0 7525 858 8443 0 4602 1275 4771 311 4353 1324 7594 0 5352 0 5923 0

Scenario 2 Scenario 3 Second quarter

Scenario 4 Scenario 5 Scenario 6

Third quarter

Scenario 7 Scenario 8 Scenario 9

Fourth quarter

Scenario 10 Scenario 11 Scenario 12

Fig. 13. The original load and optimized results of the proposed model in 12 scenarios in different quarters.

confidence levels can be adopted according to available hydropower regulation capacity, and improved renewable energy forecast accuracy can help reduce needed hydropower compensation capacity.

This model can be used in the coordinated operation of hydropower and new energy in day-ahead peak-shaving scheduling, to improve the efficiency of other thermopower sources, alleviate renewable energy 13

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Fig. 14. The output of each station of these two CHSs in Scenario 8. Table 5 Average hydropower compensation capacity estimation under different confidence level in each quarter. Quarter

Confidence levels

First quarter

Second quarter

Third quarter

Fourth quarter

0.95 0.9 0.8 0.7 0.95 0.9 0.8 0.7 0.95 0.9 0.8 0.7 0.95 0.9 0.8 0.7

0.95 0.9 0.8 0.7 0.95 0.9 0.8 0.7 0.95 0.9 0.8 0.7 0.95 0.9 0.8 0.7

Max wind error (capacity factor)

Max solar error (capacity factor)

Compensation capacity (MW)

+, a

−, b

+, c

−, d

+, e

−, f

0.24 0.17 0.10 0.06 0.22 0.16 0.10 0.06 0.16 0.11 0.05 0.03 0.21 0.18 0.16 0.11

−0.39 −0.31 −0.21 −0.14 −0.26 −0.21 −0.15 −0.11 −0.15 −0.13 −0.10 −0.07 −0.27 −0.21 −0.15 −0.10

0.25 0.12 0.09 0.09 0.08 0.06 0.04 0.03 0.20 0.13 0.06 0.03 0.14 0.10 0.06 0.03

−0.39 −0.30 −0.21 −0.15 −0.35 −0.31 −0.25 −0.21 −0.35 −0.31 −0.26 −0.22 −0.35 −0.30 −0.23 −0.18

4551 3562 2433 1628 3308 2755 2089 1579 2326 2020 1603 1308 3343 2747 1948 1388

2771 1824 1150 780 2121 1518 963 617 1976 1329 652 336 2191 2070 1570 1006

Note: ‘+’ means positive, ‘−’ means negative, e = −1*(b * wind capacity + d * solar capacity), f = (a*wind capacity + c * solar capacity), k and l is similar with e and f, wind capacity = 8480 WM, solar capacity = 3100 MW, which are the corresponding capacity of the study area by the end of 2018. Table 6 Re-simulation results of selected scenarios and comparison to Table 4 assuming wind and solar power forecast reduce by 20%. Scenario

Scenario Scenario Scenario Scenario Scenario

Peak-valley difference of residual load (MW)

3 5 7 8 9

Load rate of residual load (%)

Re-simulation result

Compare to case 3

Improvement (%)

Re-simulation result

Improvement (%)

0 1798 2763 695 2919

−172 −320 −172 −0 −122

100 15.1 5.9 0 4

100 96.6 94.7 98.8 92.1

0.1 0.4 0.8 0 0.6

curtailment and avoid power shortages. It can provide practicable references for provincial or regional power grid day-ahead scheduling. Since renewable energy sources such as wind and solar power will continue to grow with climate change and growing energy use, the proposed model may be adopted in other places. Further research can study effective coordination to operate renewable energy with other power sources and with energy storage facilities such as pumped hydropower or batteries.

curation, Validation. Xiaoyu Jin: Visualization. Lingjun Liu: Investigation. Chuntian Cheng: Supervision, Project administration.

CRediT authorship contribution statement

Acknowledgements

Benxi Liu: Conceptualization, Methodology, Writing - original draft. Jay R. Lund: Writing - review & editing. Shengli Liao: Data

This study is supported by the National Natural Science Foundation of China (Grant No. 51709035, 51979023), the China Scholarship

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Council (Grant No. 201806065023), and the Fundamental Research Funds for the Central Universities (Grant No. DUT19TD32). The authors would like to thank the comments of the editors and anonymous reviewers which significantly improved the quality of this manuscript.

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