Combined Heat and Power dispatch considering Advanced Adiabatic Compressed Air Energy Storage for wind power accommodation

Energy Conversion and Management 200 (2019) 112091

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Combined Heat and Power dispatch considering Advanced Adiabatic Compressed Air Energy Storage for wind power accommodation

T

Yaowang Lia, Shihong Miaoa, , Binxin Yina, Ji Hana, Shixu Zhanga, Jihong Wanga,b, Xing Luob ⁎

a

State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Hubei Electric Power Security and High Efficiency Key Laboratory, School of Electrical & Electronic Engineering, Huazhong University of Science & Technology(HUST), China b School of Engineering, University of Warwick, UK

ARTICLE INFO

ABSTRACT

Keywords: Advanced Adiabatic Compressed Air Energy Storage (AA-CAES) Combined Heat and Power (CHP) dispatch Wind power accommodation Off-design performance Mixed Integer Linear Programming (MILP)

As an attractive large-scale clean energy storage technique, Advanced Adiabatic Compressed Air Energy Storage (AA-CAES) can store and generate both electricity and heat, which has great application potentials in Integrated Electricity and Heating Systems (IEHSs). However, few studies have been reported on Combined Heat and Power (CHP) dispatch by implementing AA-CAES in IEHS. This paper presents a CHP dispatch model considering the participation of AA-CAES for IEHS and mitigating wind curtailment. The mathematical model of AA-CAES for CHP dispatch is developed based on Mixed Integer Linear Programming (MILP). The off-design characteristic of AA-CAES is considered in the modelling process. The benefits of AA-CAES for mitigating wind curtailment and decreasing operation costs in IEHS has been studied. The scheduling analysis based on the developed model and the existing model without considering off-design characteristics has been carried out. The study shows that the participation of AA-CAES in CHP dispatch can increase the flexibility of IEHS, and it can lead to an apparent reduction of wind curtailment and operation costs. Also, AA-CAES’s off-design conditions need to be considered in the dispatch; otherwise it can cause the misjudgment of AA-CAES’s state of charge, which in turn results in costly operation.

1. Introduction Wind power generation has been progressed rapidly in recent years in response to the challenges of worldwide fossil energy crisis and environmental pollution issue. In China, the total installed capacity of wind power reached 188 GW in 2017, contributing to 35% of the global wind power capacity [1]. However, due to the intermittent and unpredictable features of wind power, its large-scale integration brings great challenges to power system reliability. In China, about 41.9 billion kWh wind power was curtailed in 2017, which takes about 13.7% of total wind power [2]. The wind curtailment problem is more severe in the regions with large proportion of Combined Heat and Power (CHP) units, because the electric generation of CHP units is restricted by their heat demand and it results in the lack of system power regulation flexibility. For example, over 70% of the heat loads has been supplied by centralized CHP units in Jilin Province and its curtailment rate reached 21% in 2017, resulting in the economic loss of $180.8 million [2,3]. Therefore, how to mitigate wind curtailment efficiently, especially for the Integrated Electricity and Heating System (IEHS) with large proportion of CHP unit, has aroused much attention.

⁎

Among various solutions for mitigating wind curtailment, Advanced Adiabatic Compressed Air Energy Storage (AA-CAES) recently attracts great interest due to its merits of long lifetime, low cost, large scale and the ability of multi-carrier energy storage and generation [4,5]. AACAES is a new technology development direction of Conventional Compressed Air Energy Storage (C-CAES) [5]. Unlike C-CAES, AA-CAES uses Thermal Energy Storage (TES) to store and reuse compression heat, which makes it needs no fuel for electric generation and has higher efficiency than C-CAES [6]. AA-CAES can also easily integrate with External Heat Sources (EHSs), such as solar collectors and electrical heaters, and uses the heat produced by EHSs for heat supply and electric generation [7,8]. Due to the possession of TES, AA-CAES can not only store and generate electricity, but also store and supply heat. This unique characteristic makes AA-CAES has great application potential in IEHS for wind power accommodation [4]. Extensive work has been done in optimal schedule strategy of CCAES in Electric Power System (EPS). The biding and offering strategies of C-CAES were studied in Refs. [9–12]. Shafiee et al. developed bidding and offering curves of a C-CAES facility based on robust optimization [9]. The C-CAES was taken as price maker and participated in

Corresponding author. E-mail address: [email protected] (S. Miao).

https://doi.org/10.1016/j.enconman.2019.112091 Received 14 April 2019; Received in revised form 19 September 2019; Accepted 20 September 2019 Available online 01 October 2019 0196-8904/ © 2019 Elsevier Ltd. All rights reserved.

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day-ahead electricity markets [9]. An information gap decision theory based risk-constrained biding and offering strategies for C-CAES participating day-ahead electricity market is proposed by Shafiee et al. [10]. Nojavan et al. proposed optimal biding and offering strategies for CCAES in deregulated electricity market using Robust optimization approach [11]. A stochastic programming-based optimal bidding strategy for C-CAES participated in energy and reserve markets was developed by Akbari et al. [12]. In Refs. [13,14], C-CAES was taken as price taker, and the optimal self-scheduling strategy for C-CAES was proposed to gain more profits in electricity market. Shafiee et al. developed a CCAES self-scheduling model considering its thermodynamic characteristics [13], and the test results indicated that the neglect of thermodynamic characteristics of C-CAES can result in significant loss of profit in the case of participating in energy and reserve markets. Attarha et al. proposed a self-scheduling model for a combined wind and C-CAES system participating in energy market based on robust optimization [14]. The EPS scheduling strategy considering the participation of CCAES was studied in Refs. [15–21]. In Refs. [15,16], C-CAES was considered in unit commitment problem and the optimal day-ahead scheduling strategy for the EPS containing thermal units and a C-CAES was proposed. In Refs. [17,18], Thermal units, a C-CAES, renewable energy sources and flexible loads were co-optimized to increase district EPS operation economic. Li et al. proposed a real-time scheduling model for EPS considering the dynamic characteristics of C-CAES to decrease EPS operation costs [19]. Aliasghari et al. investigated the benefits and applicability of look-ahead scheduling of EPS containing wind power farms and C-CAESs [20]. Sedighizadeh et al. proposed an optimal joint energy and reserve scheduling model considering frequency dynamics, C-CAES and wind turbines in EPS [21]. In the above literatures, the scheduling models of C-CAES for EPS dispatch in different time-scales are developed. However, the mathematical model of AA-CAES is different from C-CAES due to the integration of TES, and the ability of storing both electric and heat makes AA-CAES has broader application areas than C-CAES, so that the scheduling strategies of AACAES for new application areas are required. Academic researchers have explored the concept of AA-CAES with a range of different research focuses, such as simulation modelling, efficiency analysis and configuration optimization. A simulation software tool for dynamic modelling and transient control of AA-CAES is developed by Luo et al. [22]. He et al. proposed a dynamic model of AACAES discharging process [5]. The thermodynamic analysis of AA-CAES system was carried out in Refs. [23,24], and simulation results showed that the cycle efficiency of AA-CAES is closely related to the structure and parameters of AA-CAES’s main components [23,24]. Zhang et al. studied the effects of TES on AA-CAES system cycle efficiency [25]. Szablowski et al. presented an energy and exergy analysis of an AACAES system, and the results indicated that the system cycle efficiency can be improved by increasing stage number of compressors, turbines and HEXs [26]. Luo et al. presented a thermodynamic model of AACAES for simulation study. Various system configurations aiming for system efficiency improvement were designed and discussed, and recommendations were made in Ref. [27]. A similar mathematical model of AA-CAES was studied in Ref. [28], and a simulation study on cycle efficiency and energy storage density was carried out [28]. Hartmann et al. conducted a simulation study of cycle efficiency on AA-CAES systems, which shows the efficiency of an ideal isentropic configuration is about 70% [29]. Guo et al. studied the off-design performance of AACAES systems [30]. Based on the research, the efficiency distribution among all stages are obtained [30]. Wolf et al. proposed an AA-CAES design method with cycle efficiency in the range of 52%–60% [31]. The above studies make great efforts to achieve higher cycle efficiency of AA-CAES system, and these studies lay a solid foundation for constructing AA-CAES demonstration plants. At present, several AA-CAES demonstration plants, from the scale of 500 kW–100 MW, are under construction or in the early stage of experimental around the world [6,32–36]. These demonstration projects

greatly promote the application of AA-CAES in EPSs and integrated energy systems. Therefore, developing optimal scheduling methods for EPSs and integrated energy systems considering AA-CAES becomes particularly important. However, there are very limited studies that focus on the scheduling strategy of AA-CAES in EPSs and integrated energy system. In [37,38], AA-CAES scheduling models were developed to estimate the value of AA-CAES in monopoly power markets, energy market and reserve market. Li et al. proposed a joint energy and reserve optimization scheduling model for the EPS with AA-CAES for decreasing EPS operation cost [6]. The constraints of the regulation reserve and contingency reserve provided by AA-CAES were modelled in detail in Ref. [6]. Li et al. proposed an optimal dispatch model for the zero-carbon-emission micro energy internet integrated with AA-CAES [39], and the AA-CAES was taken as an energy hub in the energy internet for supplying both electricity and heat [39]. From the above literature review, it is noticed that large-scale AACAES participating in CHP dispatch for mitigating wind curtailment have not been reported. AA-CAES has the ability of storing and generating both electricity and heat. This merit makes AA-CAES has great application potential in IEHS. Thus, how to make the large-scale AACAES well coordinate with CHP units, wind farms and other scheduling resources in IEHS, in order to decrease system operation cost and mitigate wind curtailment, is an important problem. In addition, in previous studies, none of the proposed AA-CAES dispatch model considered AA-CAES’s off-design performance. In order to suit the demands from the coordination with wind farms, a wide regulation range of charging and discharging powers of AA-CAES is needed [40]. When AACAES works at off-design points, some parameters of AA-CAES system, such as the isentropic efficiencies of compressors and turbines, deviate significantly from their rated values [4,41]. Therefore, the off-design performance of AA-CAES should be taken into account in formulating the dispatch model of AA-CAES. On the above premises, the aim of this paper is to propose a CHP dispatch model for the IEHS containing AA-CAES and incorporating the off-design performance of AA-CAES. The main contributions of the paper are: (1) The first Mixed Integer Linear Programming (MILP) based AA-CAES model considering its off-design performance for CHP dispatch is developed. (2) The first CHP dispatch model considering the coordination of AACAES facilities, CHP units, thermal units, wind farms and the EHSs of AA-CAES is proposed based on fuzzy chance-constrained optimization. (3) The benefits of AA-CAES for mitigating wind curtailment in CHP dispatch are explored, and the potential financial losses of ignoring AA-CAES’s off-design performance in CHP dispatch are assessed, based on the developed AA-CAES model and CHP dispatch model. This paper is organized as follows: in Section 2, the mathematical model of AA-CAES for CHP dispatch is proposed; Section 3 presents the CHP dispatch model involving AA-CAES facilities and wind farms; in Section 4, case studies are carried to assess the benefits of AA-CAES in CHP dispatch, and to analyze the impact of the neglect of AA-CAES’s off design performance on dispatch results; Section 5 concludes the paper. 2. Mathematical model of AA-CAES for CHP dispatch In this section, a MILP-based AA-CAES model considering its offdesign performance is established for CHP dispatch. The system description is first introduced. After that, the mathematical modelling to the operation constraints of AA-CAES components is given. Then, the equivalent linear formulation of the proposed model is presented.

2

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Air flow Hot thermal fluid flow

Electricity consumption

Cold thermal fluid flow

AA-CAES

Electricity supply

Multi-stage air expansion unit

Multi-stage air compression unit

Power flow

M EHS

LP C

HPC

HEX1

HPT

Pressure regulator HEX2

LP T

HEX4

HEX3

exhuast

Ambient air Solar collecter or Electrical Electricity heater

Heat produced by EHS

Air reservoir

Thermal storage reservoir

G

TES working medium reservoir

Heat supply

or

Fig. 1. Schematic diagram of an AA-CAES integrated with EHS.

2.2.1. Operation constraints of compressor and turbine Based on the properties of incoming flow and the outlet pressure, the ideal mechanical power consumption can be calculated by the isentropic compression approach. The ideal mechanical power consumption (Wcs, k ) for compressing the fluid can be calculated by [27],

2.1. Description of AA-CAES system Fig. 1 illustrates the schematic diagram of an AA-CAES integrated with EHS. The components of AA-CAES system mainly includes: i) motor and generator; ii) multi-stage air compression unit; iii) multistage air expansion unit; iv) underground cavern(s) or aboveground tank(s) for compressed air storage; v) two groups of Heat Exchangers (HEXs); vi) thermal storage reservoir and TES working medium reservoir. The AA-CAES works in the process as: during the compression process, the ambient air is compressed via a multi-stage air compression unit into the air reservoir, meanwhile, the heat produced in the stage of air compression is extracted by the cold thermal fluid form the TES working medium reservoir in a group of HEXs, and the hot thermal fluid is stored in the thermal storage reservoir; during the expansion process, the stored high-pressure air is released, heated by the stored hot thermal fluid in another group of HEXs, and then, the heated compressed air is used to drive a multi-stage air expansion unit to generate electricity. The AA-CAES can share its thermal storage reservoir with EHSs to increase the flexibility in supplying heat. During the periods of high heat demand, the stored heat, including compression heat and the heat produced by EHS, is released for heat supply.

Wcs, k = mc, t hin

out , k ,

t

(1)

{1,2. ..,T}

where mc, t is the mass flow rate of the compressor at time t; T is the total scheduling time periods; hin out , k is the ideal enthalpy change from the inlet to the outlet of the compressor k in the ideal isentropic change of state. From the ideal gas law and the isentropic compression characteristics, hin out , k can be calculated by [27],

hin

out , k

1

= cp,a Tc,k,in

1 ,

c, k , t

t

{1,2. ..,T}

(2)

where cp,a is the specific heat capacity of air; is the specific heat ratio; Tc,k,in is the inlet temperature of compressor k; c, k, t is the compressing ratio of compressor k at time t; After using the isentropic efficiency to calibrate the ideal mechanical power to be close to the practical working conditions, the actual power consumed by the compressor k is (Wc, k ) [27],

Wc, k =

Wcs, k

,

t

{1,2. ..,T}

(3)

c,k ,t

2.2. Formulation of AA-CAES operation constraints

Thus, the total actual charging power of AA-CAES (PCAESc, t ) can be calculated by,

According to the multiple time-scale coordinated active power control system, which has been put into real practice in several regional electric power control system in China, the day-ahead scheduling module is performed every 24 h, with a time resolution of 1 h [42]. The day-ahead scheduling of AA-CAES is performed to make the next day’s schedules of its charging power, discharging power and the heat releasing power for heat supply. In the scheduling, the operation constraints of all main components of AA-CAES system should be considered, but the dynamic constraints of AA-CAES system, such as the limits of ramp up/down rates, can be ignored, as CAES can normally switch from minimum power output to maximum power output within 1 h [43]. For the convenience of analysis, the following assumptions are made in the modelling process: i) the ideal air is used; ii) pressure loss in pipes and heat exchangers are negligible; iii) the air reservoir is set to be isothermal; iv) assuming the inlet temperature of each compressor and turbine is close to their rated inlet temperature; v) the thermal storage reservoir is set to be adiabatic.

PCAESc, t =

1

nc

Wc, k,

t

{1,2. ..,T}

(4)

m k=1

where m is the efficiency of motor; nc is the stage number of compressors. Similar to the modelling of compressor, the isentropic expansion method is adopted and the isentropic efficiency is also considered. The ideal mechanical power produced by turbine j (Wds, j ), the actual mechanical power produced by turbine j (Wd, j ) and the actual discharging power of AA-CAES (PCAESd, t ) are described below [27], 1

Wds, j = md, t cp,a Td,j,in 1

Wd, j = 3

d,j,t Wds, j,

t

d, j

,

{1,2. ..,T}

t

{1,2. ..,T}

(5) (6)

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air pressure in the air reservoir ( par,t ) is calculated by (15).

nd

PCAESd, t =

Wd, j,

g

t

{1,2. ..,T}

(7)

j=1

in par, t =

where md, t is the mass flow rate of the turbine at time t; Td,j,in is the rated inlet temperature of turbine j; d, j is the expansion ratio of turbine j; d,j,t is the isentropic efficiency of turbine j at time t; nd is the stage of turbines; g is the efficiency of generator. In practical operation, the isentropic efficiencies of compressor and turbine are variables whose values depend on the operation conditions of compressor and turbine. For example, the isentropic efficiencies of compressor and turbine can normally be about 87%–90% when they work at designed point; however, the isentropic efficiencies is decreased to 65%–70% when they work at 50% compression and generation levels (% of designed compressing/generating powers); the isentropic efficiencies can be lower than 50% when they work at 30% compression and generation levels [4]. In addition, the compressing ratios of the compressors rise with the increase of the pressure in air reservoir [13,19]. Thus, the compressor needs more power to compress unit kg of air to the air reservoir when the pressure in air reservoir gets higher. In AA-CAES system, a pressure regulator is normally set before the 1st stage turbine to control the inlet pressure of turbine, so that the air pressure in the air reservoir hardly affects the operation of turbine. However, with the increasing of the generation level, the expansion ratios of turbines also have a tendency of increase [41]. Based on the above discussion, the isentropic efficiency of compressors, compressing ratios and isentropic efficiencies of turbines are variables during the AA-CAES operation process. Therefore, according to (1)–(4), the relations between the mass flow rate of compressors and AA-CAES charging power can be expressed by (8); according to (5)-(7), the relations between the mass flow rate of compressors and AA-CAES charging power can be expressed by (9).

m c, t = k mPc, t × PCAESc, t k mPc, t = K mPc (PCAESc, t , par,t )

md, t = k mPd, t × PCAESd, t k mPd, t = K mPd (PCAESd, t )

,

,

t

t

{1,2. ..,T}

de par, t =

PCAESc, t

vCAESc, t P¯CAESc,

vCAESd, t P CAESd

PCAESd, t

vCAESd, t P¯CAESd,

vCAESc, t + vCAESd, t

1,

t

{1,2. ..,T}

{1,2. ..,T} t

{1,2. ..,T}

Var

t

{1,2. ..,T}

md, t ,

t

{1,2. ..,T}

=1

(13) (14)

t in par,

t =1

de par,

t,

t

{1,2. ..,T}

(15)

where Rg is the universal gas constant; Tar is the temperature in the air reservoir; Var is the volume of the air reservoir; par,0 is the initial air pressure in the air reservoir; t is the time resolution of day-ahead scheduling (1 h). The upper and lower limits of air pressure are described by,

p¯ar

par,t

par ,

t

{1,2. ..,T}

(16)

where p¯ar and par are the lower and upper limits of the pressure in air reservoir. 2.2.3. Operation constraints of thermal energy storage and release process The TES in AA-CAES mainly includes HEXs and the thermal storage reservoir. In the TES system, the pressurized water is usually chosen as the thermal fluid, because of its high specific heat capacity bring the ability of storing more heat energy within a low temperature range [5,27]. In charging process, the heat released by the compressed air is captured by the cold water in HEXs. The high-pressure air flowing out from the kth compressor is cooled down in the kth HEX, and then flows input into the k + 1th compressor for further compression (or flows into the air storage reservoir). The heat storage power of kth HEX (HHEXck,t ) in air compression stage is calculated by (17), and total heat storage power of AA-CAES in charging process (HCAESc,t ) can be expressed by (18).

(9)

t

Rg Tar

m c, t ,

t

HHEXck,t = m c, t c p,a (Tc, k,out,t

where kmPc, t and kmPd, t are the ratios that represent the mass flow rate per MW of charging and discharging power, respectively; par,t is the air pressure in air reservoir at time t; kmPc, t is presented as a function of PCAESc, t and par,t : say K mPc (PCAESc, t , par,t ) . kmPd, t is presented as function of PCAESd, t : say K mPd (PCAESd, t ) . kmPc, t can be calculated by (1)-(4) with given c,k ,t and c, k , t , and k mPd, t can be calculated by (5)-(7) with given d,j,t . The limits of charging and discharging power should be considered in the scheduling. These constraints are described by (10) and (11), respectively. Additionally, the AA-CAES facility normally does not work in charging and discharging modes simultaneously, and the constraint is modelled by (12).

vCAESc, t P CAESc

Var

par,t = par,0 +

(8)

{1,2. ..,T}

Rg Tar

k

{ 1,2. ..,nc 1},

t

{1,2. ..,T}

Tc, k + 1,in ), HHEXck,t = m c, t c p,a (Tc, k,out,t

t

{1,2. ..,T}

Tar,in ), k = nc , (17) nc

HCAESc,t =

HHEXck,t ,

t

{1,2. ..,T}

(18)

k=1

where Tar,in is the inlet temperature of air reservoir; Tc, k,out,t is the air temperature at the compressor k outlet at time t. Similar to the modelling of HEXs in air compression stage, the heat releasing power of jth HEX (HHEXej,t ) in air expansion stage can be expressed by (19), and the total heat releasing power for heating the compressed air in discharging process (HCAESd,t ) can be calculated by (20).

(10) (11) (12)

where vCAESc, t and vCAESd, t are the binary variables that are used to indicate the operation modes of AA-CAES (vCAESc, t = 1 is charging mode, vCAESd, t = 1 is discharging mode); P CAESc and P¯CAESc are the upper and lower limits of charging power, respectively; P CAESd and P¯CAESd are the upper and lower limits of charging power, respectively.

HHEXej,t = md, t cp,a (Td, j,in

Tar ),

HHEXej,t = md, t cp,a (Td, j,in

Td, j

j = 1, 1,out,t ),

j

t

{ 2,. ..,ng },

{1,2. ..,T} t

{1,2. ..,T} (19)

ng

HCAESd,t =

2.2.2. Operation constraints of air reservoir In this paper, the temperature of air reservoir is treated as the ambient temperature to simplify the analysis. Based on the ideal gas law ( pV = mRT ) and considering the controlled volume of boundary in de (V = 0 ), the air pressure increasing and decreasing rates ( par, t and par,t ) in the air reservoir can be expressed by (13) and (14), respectively. The

HHEXej,t , j=1

t

{1,2. ..,T}

(20)

where Td, j,out,t is the air temperature at the turbine j outlet at time t. Applying the isentropic compression and expansion concept, the air temperatures at compressor and turbine outlet can be calculated by (21) and (22) [27]. 4

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Based on Eqs.(1)-(4) and considering the offdesign characteristics

Charging power

Turbines

Air storage reservoir

Compressors Mass flow rate of compressors

Eq.(8) Eq.(8) contains nonlinear term

Eq.(13)

Mass flow rate of turbines

Eq.(14)

Pressure increasing rate

Based on Eqs.(5)-(7) and considering the offdesign characteristics Eq.(9) Eq.(9) contains nonlinear term

Pressure decreasing rate

Limits of discharging power Eq.(11)

Eq.(15)

Limits of charging power Eq.(10)

Discharging power

Pressure in air reservoir Limits of air pressure

Eq.(16) HEXs

AA-CAES operation constraint

Based on Eqs.(17),(18),(21) and considering the offdesign characteristics

Based on Eqs.(19),(20),(22) and considering the offdesign characteristics

Eq.(12)

Eq.(23) Eq.(23) contains nonlinear term

Eq.(24) Eq.(24) contains nonlinear term

Heat storage power of AA-CAES

Heat releasing power of AA-CAES

Thermal storage reservoir

heat releasing power for heat supply

Eq.(25) Stored heat in thermal storage reservoir Limit of stored heat Eq.(26)

Fig. 2. Block diagram of the whole AA-CAES system model. 1

Tc, k,out, t

1

c,k, t

Tc, k,in 1 +

QHS , t ,

t

{1,2. ..,T} ,

k

{1,2. ..,nc }

t

= min QHS,0 +

c, k , t

(21)

t , Q¯ HS ,

1

Td, j,out,t

Td, j,in 1

1

d, j

d, j, t

,

t

{1,2. ..,T} ,

Q HS

Td, j,in

pressure ratios. Therefore, based on the discussion of the off-design performance of compressor and turbine in Section 2.2.1, the relations between mass flow of compressors and heat storage power of AA-CAES can be expressed by (23); the relations between mass flow of turbines and heat releasing power of AA-CAES can be expressed by (24).

HCAESc,t = kQc m, t × mc, t kQc m, t = K Qc m (PCAESc, t , par, t )

HCAESd,t = kQd m, t × md, t kQd m, t = K Qd m (PCAESd,t )

,

,

t

t

{1,2. ..,T}

{1,2. ..,T}

HCAESrl, )

t

{1,2. ..,T}

(25)

QHS, t ,

t

(26)

{1,2. ..,T}

where QHS,0 is the initial thermal energy stored in thermal storage reservoir; Q HS and Q¯ HS are the lower and upper limits of thermal storage reservoir. HCAESst,t is the heat power of storing heat from EHS; HCAESrl,t is the heat releasing power for heat supply. The limits on HCAESst,t depend on the operation characteristics of EHS. For example, when the solar collector is chosen as EHS, HCAESst,t can be calculated by (27) [44]; when the electrical heater is chosen as EHS, HCAESst,t can be calculated by (28). If there is no EHS integrated with AA-CAES, HCAESst,t = 0 .

According to (21) and (22), the ratios of inlet and outlet temperaT T tures ( c, k,out, t and d, j,out,t ) depend on the isentropic efficiencies and Tc, k,in

HCAESd, + HCAESst,

j (22)

{1,2. ..,nd}

(HCAESc, =1

(23)

HCAESst,t = (

SC SSC It )· SCs, t ,

HCAESst,t = (

EH PEH , t )· EHs, t ,

t

{1,2. ..,T}

(27)

t

{1,2. ..,T}

(28)

where SC is the transfer efficiency between solar and heat; SSC is the area of solar multiple; It is the direct normal irradiance at time t; EH is the energy conversion efficiency of electrical heater; PEH , t is the electric power consumed by electrical heater, and it is the decision variable of electrical heater in day-ahead scheduling; SCs, t and EHs, t are the efficiencies of storing heat from solar collectors and electrical heaters. The output limits of HCAESrl,t is described by (29).

(24)

where kQc m, t is the ratio that represents the heat storage power per kg/s of air mass flow rate in charging process; kQd m, t is the ratio of released heat and air mass flow rate in discharging process. kQc m, t is presented as a function of PCAESc, t and par,t : say KQc m (PCAESc, t , par, t ) ; kQd m, t is presented as a function of PCAESd, t : say KQd m (PCAESd,t ) . kQc m, t can be calculated by (17), (18) and (21) with given c,k,t and c, k, t , and kQd m, t can be calculated by (19), (20) and (22) with given d,j,t . The thermal energy stored in the thermal storage reservoir (QHS, t ) can be calculated by (25). This constraint indicates that the compressed air and the heat produced by EHS is wasted after the stored heat in thermal storage reservoir reaches to its upper limit. Constraint (26) enforces that the stored heat in thermal storage reservoir is greater than its lower limit during the scheduling.

0

HCAESrl,t

H¯ CAESrl ,

t

{1,2. ..,T}

(29)

where H¯ CAESrl are the maximum heat storage power. With all constraints of the main components in AA-CAES system presented above, the overall mathematical model of AA-CAES for CHP dispatch is formulated by integration of these constraints. Fig. 2 shows the block diagram of the whole AA-CAES system model with all above equations. 5

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Piecewise linearization on nonlinear function . ,K . . ,K . , KQ m terms K mP Q m mP c

MINLP based model

d

c

d

respectively.

Using big M method to linearize the bilinear terms

BLP based model

z¯ z=1 z¯ z=1 z¯ z=1

MILP based model

Fig. 3. Flowchart of the linear approximation of the mathematical model of AACAES for CHP dispatch.

s¯ s=1 s¯ s=1 s¯ s=1

z, s uarP Ps c, t CAESc z, s uarP pz c, t ar z, s uarP c, t

x¯ x=1 x¯ x=1

The developed mathematical model of AA-CAES for CHP dispatch in Section 2.2 is a mixed-integer non-linear programming (MINLP) based model due to the non-linear relations between decision variables in (8), (9), and (24). The MINLP based model is difficult to be solved with reasonable accuracy and computational time [45]. Therefore, the equivalent linear formulation of the MINLP based model is developed to address this issue. The flowchart of the linear approximation process is shown in Fig. 3. After considering the off-design performance of AA-CAES, the ratios kmPc, t , kmPd, t , kQc m, t and kQd m, t are calculated by the complicated nonlinear functions K mPc , K mPd , KQc m and KQd m . Therefore, a piecewise linearization method is first employed to approximate the nonlinear functionsK mPc , K mPd , KQc m and KQd m . After the piecewise linear approximation [46], the MINLP model is converted into a bilinear programming (BLP) based model. Then, the big M approach [47,48] is used to linearize the bilinear terms. As a result, the MINLP based model is transformed into a MILP based model.

z¯

…

s¯

1,1 kmP , kQ1,1 c cm

1, s kmP , kQ1,csm c

1, s¯ kmP , kQ1,csm c

z¯, s kmP , kQz¯,csm c

= vCAESc, t

x u Pxd, t P CAESd

x¯ x=1

x PCAESd, t

x u Pxd, t P¯CAESd

u Pxd, t = vCAESd, t

,

t

{1,2. ..,T}

(31)

in par, t =

de par, t =

z¯

Rg Tar Var

s¯

z =1 s=1

x¯

Rg Tar

z, s (kmPz,s ·uarP , t ·PCAESc, t ) ,

(kmPx ·u Px ,t ·PCAESd, t) , d

Var

x=1 z¯

s¯

HCAESc,t = x¯

HCAESd,t =

t

d

c

(kmPx kQx m· u Px ,t ·PCAESd, t), d

t

c

c

d

{1,2. ..,T}

(32)

{1,2. ..,T}

z, s (kmPz,s kQz,sm·uarP , t · PCAESc, t ),

d

x=1

t

c

c

t

(33)

{1,2. ..,T}

(34)

{1,2. ..,T}

(35)

and It can be observed that there are bilinear terms u Pxd, t · PCAESd, t in (32), (33), and (34), and these bilinear terms is required to be linearized. z, s uarP ·P c, t CAESc, t

2.3.2. Linearization of bilinear terms The bilinear terms in (32)–(35) can be exactly linearized by the use of big M approach [13,48]. Takes as an example, its equivalent linear form is described as (36). The M is a big enough number. In (36), the M Rg Tar max P is set to be V kmP . The equivalent linear forms of (33)–(35) c CAESc ,max ar can be obtained with the same method, and the M in these equivalent Rg Tar max max max P k P linear forms are set to be V kmP , kmP and c Qc m CAESc,max d CAES d,max ar max max max max kmPd kQd m PCAES d,max , respectively, where kmPc , kmPd , kQmax and kQmax are dm cm the maximum value of kmPc, t , kmPd, t , kQc m, t and kQd m, t . in par ,t

Rg Tar

in par ,t

Rg Tar

Var

z, s z,s kmP P + (uarP c, t c CAESc, t

z, s kmP P + (1 c CAESc, t Var

{1, 2. ..,¯}, s

z

1)M

z, s uarP )M c, t

,

{1, 2. ..,z¯}

t

{1, 2. ..,T },

s (36)

Above all, the MILP-based AA-CAES model for CHP dispatch contains (10)-(12), (15), (16), (25)-(26) and the equivalent linear forms of (32)–(35). 3. Formulation of CHP dispatch model

1

z¯,1 kmP , kQz¯,1 c cm

t

maximum pressure in air reservoir of z segment. After using the piecewise linear approximation and the auxiliary out in binary variables, par, t and par,t can be calculated by (32) and (33), respectively (according to (8), (9) and (13, (14)); HCAESc,t can be calculated by (according to (8) and (23)); HCAESd,t can be calculated by (according to and (24)).

In this section, the CHP dispatch problem involving wind farms and AA-CAES facilities is formulated. Here, the extraction turbine CHP unit, which has been widely used in China, is considered in this paper. In order to mitigate wind curtailment, the wind power is assumed to have the highest scheduling priority, and the penalty of wind curtailment is considered in the scheduling. In order to address the uncertainty issue of wind power output, robust optimization [14,48,49], scenario-based stochastic optimization

Changing power

z ,s kmP , kQzc,sm c

,

s where and P¯CAESc are the minimum and maximum charging x x power of s segment; P CAESd and P¯CAESd are the minimum and maximum discharging power of x segment; parz and p¯arz are the minimum and

Table 1 Values of k mPc, t and kQc m,t after piecewise linear approximation.

z,1 kmP , kQz,1 c cm

z, s uarP p¯ z c, t ar

s P CAESc

2.3.1. Linear approximation of nonlinear function terms The values of kmPc, t and kQc m, t both depend on the charging power and the pressure in air reservoir. Assuming the charging power is divided into s¯ segments and the pressure in air reservoir is divided into z¯ segments; s = 1 and s = s¯ represent the minimum and maximum charging powers of AA-CAES, respectively; z = 1 and z = z¯ represent the lower and upper limits of the pressure in air reservoir, respectively. Table 1 shows the values of kmPc, t and kQc m, t after piecewise linear apz,s proximation. The values of the ratios remain unchanged in a cell. kmP c and kQzc, sm represent the values of kmPc, t and kQc m, t at row z and column s. z,s kmP and kQzc, sm are the scheduling parameters of AA-CAES which can be c obtained before the scheduling. x Similarly, the discharging power is divided into x¯ segments. kmP d and kQxd m represent the values of kmPd, t and kQd m, t at column x. They are also the scheduling parameters of AA-CAES. Note that, the number of segments (s¯ , x¯ and z¯ ) can be determined by the efficiency characteristic of compressor, the efficiency characteristic of turbine and the system compression ratio, with the given accuracy requirement. In order to define which segments that the charging power, discharging power and air pressure are in, two sets of binary variable z, s z, s = 1 represents the charging power is in uarP , u Pxd, t are defined: uarP c, t c, t segment s while the pressure in air reservoir is in segment z at time t; u Pxd, t = 1 represents the discharging power is in segment x at time t. z, s Fig. 4 illustrates how to use uarP to obtain the value of kmPc, t . From c, t z,s z, s Fig. 4, uarPc, t = 1 indicates that the value of kmPc, t is equal to kmP . c z z, s u In the scheduling, uarP and can be obtained by (30) and (31), ar , t c, t

…

z, s ¯ s uarP P c, t CAESc

(30)

z=1 s =1

1

s¯ s=1

z¯ z=1

par , t

s¯ s=1

{1,2. ..,T}

2.3. Equivalent linear formulation

Pressure

z¯ z=1

PCAESc,t

z , s¯ kmP , kQzc,s¯m c z¯, s¯ kmP , kQz¯,cs¯m c

6

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Y. Li, et al.

Fig. 4. Diagram of how to use u Psc, t and uarz ,t to obtain the value of k mPc, t .

[50,51] and chance-constrained optimization [6,52–56] are commonly used in power system scheduling. In robust optimization, a range/band is used to represent the upper and lower bounds of the uncertainty [57]. Compared with the stochastic optimization and chance-constrained optimization, the robust optimization normally has higher solving efficiency, but the solutions obtained by it are usually over conservative owing to the neglect of possibilities of cases [56,57]. In scenario-based stochastic optimization, a large number of scenarios are generated to model the uncertainty [56]. Normally, the quality of solutions increases with a larger number of scenarios. However, the computational requirements also increase with the number of scenarios, and a tradeoff needs to be made between the accuracy and the computational performance [57]. Although the above methods are capable of dealing with the uncertainty of wind power output, they cannot effectively balance the economy and reliability in power system scheduling [56]. In chanceconstrained optimization, the relationships between economy and reliability can be easily controlled by the system operators through adjusting the confidence levels of the chance constraints [56]. Therefore, this method have been widely used in power system scheduling. The uncertainty in chance-constrained optimization can be described by random variables or fuzzy variables. Compared with using random variables to describe the uncertainty, the advantage of using fuzzy variables is that the acquisition of their membership function does not depend on the large amount of historical data [53,56]. Therefore, in the situation where the historical data is difficult to collect or insufficient, using fuzzy variables to describe the uncertainty can be an effective way [53,56]. In this section, the uncertainty of wind power output and load are described by fuzzy variables, and the CHP dispatch model is developed based on fuzzy chance-constrained programming.

T

NTU

min

[(bi PTU, i, t + ci ) + STU,i, t ] +

t=1 i T NCHP

[ t=1 i T NW t=1 i =1

W

p CHP, i ( CHP,i PCHP, i, t

+

h CHP,i HCHP, i, t )

+ SCHP,i, t ] +

PWc, i, t

(37)

where NTU , NCHP and NW are sets of thermal unit, CHP unit and wind farm, respectively; PTU, i, t is the power output of thermal unit i at time t; PCHP, i, t and HCHP, i, t are electricity power output and heat releasing power of CHP unit i at time t; bi and ci are cost coefficients of thermal h p unit i; CHP,i is the cost coefficient of CHP unit i; CHP, i and CHP,i are the fuel consumption per electricity and heat generation of CHP unit i, respectively; W is the unit wind curtailment penalty; PWc,i, t is the wind curtailment of wind farm i at time t; STU,i, t and SCHP,i, t are the start-up costs of thermal unit i and CHP unit i at time t. The mixed-integer linear formulation for the start-up cost of thermal units can be expressed by (38) [59]. Note that, the start-up cost of CHP units can be expressed by the similar formulation of (38).

STU,i, t STU,i, t

CTU, i (v TU,i, t 0

v TU,i, t 1)

,

t

{1,2. ..,T},

i

{1,2. ..,NTU } (38)

where v TU,i, t is the binary variable which is used to indicate the ON/OFF states of thermal units; CTU, i is the cost of a single start-up of thermal unit i. ~ In this paper, the wind power output PW, i, t is expressed by the trit (P Wf, i, t ,PWf, i, t ,P¯Wf, i, t ) , angle fuzzy variable ~ {1,2. ..,T}, i { 1,2. ..,NW } ; where PW, i, t is the wind power output of wind farm i at time t; PWf, i, t is the wind power forecast output of wind farm i at time t; P Wf, i, t and P¯Wf, i, t are minimum and maximum wind power forecast output of wind farm i at time t, respectively. Fig. 5 shows the schematic diagram of wind curtailment in CHP dispatch. When the maximum wind power forecast output (P¯Wf, i, t ) is greater than the maximum acceptable wind power output (PWa, i, t ), there is possibility that wind curtailment occurs. However, the wind

3.1. Objective function The objective function is minimizing the system operation cost which includes operation cost of thermal units, operation cost of CHP units and penalty of wind curtailment. The operation cost of thermal units and CHP units consists of their fuel cost and start-up cost. Since the AA-CAES facility and wind farm need no fuel for generation, the operation costs of AA-CAES and wind farm are neglected. The objective function is formulated as (37), in which the fuel costs of thermal units and CHP units are both defined as linear functions [6,58].

Power Wind curtailment

Maximum wind power forecast output (Pessimistic value of wind power)

Maximum acceptable wind power

Forecast wind power

Time Fig. 5. Schematic diagram of wind curtailment in CHP dispatch. 7

Energy Conversion and Management 200 (2019) 112091

Y. Li, et al.

curtailment evaluated by P¯Wf, t can be too conservative. In this paper, the pessimistic value of wind power (PW, i, t ) is used to evaluate the wind curtailment. The definition of PW, i, t is shown by (39) [60]. When confidence level is set to be 1, PW, i, t reaches its maximum value, which is P¯Wf, i, t ; when confidence level is set to be 0, PW, i, t reaches its minimum value, which is P Wf, i, t .

PW, i, t

~ = inf{r|Cr{PW, i, t

r}

},

t

{1,2. ..,T},

i

PCHP,i

t

{1,2. ..,T},

i

S heat sources, the supply temperature (TN, n, t ) can be calculated by,

t

0

HCHP, i, t

P CHP, i vCHP,i, t t

{1,2. ..,T},

F¯CHP,i vCHP,i, t

H¯ CHP, i vCHP,i, t PCHP, i, t P¯CHP, i vCHP,i, t i

n

S in,m TN, n, t = TN,n, t ,

(42)

{1,2. ..,NDHS}

t

{1,2. ..,NDHS} ,

{1,2. ..,T}, m

(43)

{1,2. ..,NHL,n}

where NHL,n is the set of heat loads of nth DHS. out,m According to the mth heat load, the outlet temperature (TN, n, t ) of mth heat load can be calculated by,

,

(41)

{1,2. ..,NCHP}

{1,2. ..,T},

where NDHS is the set of DHS; HCHP, n, t and HCAESrl,n, t are the heat generation power of CHP and heat releasing power of AA-CAES in nth DHS at time t; cp,w is the specific heat capacity of water; mns is the source S R mass flow rate of nth DHS; TN, n, t and TN,n, t are supply temperature and return temperature of nth DHS at time t. For the DHS whose area is not very large, the thermal loss and water transmission delay can be ignored [62]. The mth heat load inlet temin,m perature (TN, n, t ) is equal to the supply temperature, as shown in (43).

n h CHP,i HCHP, i, t

HCHP, n, t + HCAESrl,n, t R + TN, n, t , cp,w mns

S TN, n, t =

{1,2. ..,NW }

rCHPi HCHP, i, t +

P H CHP,i ,t H CHP,i H CHP, i

Fig. 7. ROR of extraction CHP units.

3.2.1. Operation constraints of district heating system (DHS) A typical DHS mainly consists of heat sources, a heating network and heat loads. The diagram of DHS is shown in Fig. 6. Heat sources, including CHP units and AA-CAES facilities, are used to produce heat. The Feasible Operation Range (FOR) of extraction CHP units, which is shown by Fig. 7, can be modelled by (41) [61,58]. The operation constraints of AA-CAES have been modelled in Section 2. p CHP,i PCHP, i, t

Heat producing power (MW)

A

0

3.2. Constraints

F CHP,i vCHP,i, t

B

PCHP,i

{1,2. ..,NW }

(40)

PCHP, i, t

C

E

where inf{r } is the function used to get lower bound of variable r; Cr{ ·} is the function used to get confidence level; is the confidence 1. level selected by decision makers, 0 Based on the above discussion, the wind curtailment can be calculated by (40).

PWa, i, t , 0),

FOR of extraction CHP

PCHP,i ,t

(39)

PWc, i, t = max(PW, i, t

Electricity output (MW) D

HL, n, m, t , cp,w mnL, m

out,m in,m TN, n, t = TN,n, t

where rCHPi is the power to heat ratio ; vCHP,i is the binary variable used to indicate the ON/OFF states of CHP unit; F CHP,i and F¯CHP,i are the lower and upper bounds on fuel consumption of CHP unit i; H¯ CHP, i is the maximum heat generation power; P CHP, i and P¯CHP, i are the minimum and maximum electric generation of CHP i. In Fig. 7, the slope of line AB is rCHPi ; the slopes of line DC and EA h p P are CHP, i / CHP,i . HCHP, i is the heat generation power that the CHP can operate at its minimum electric generation power (point A in Fig. 7). A heating network consists of supply pipes and return pipes [62]. Supply pipes are used to supply heat by means of hot water, and the return pipes are used to transport return water from heat loads to heat sources after the heat is extracted by heat loads. In this paper, the heating network is assumed to operate with constant flow and variable temperature, which is the most common operation mode of heating network in China due to the lack of control devices at end users [62]. Based on the heat balance constraint, after absorbing heat produced by

n

{1,2. ..,NDHS} ,

m

t

{1,2. ..,T}, (44)

{1,2. ..,NHL,n}

where HL, n, m, t is the mth heat load in nth DHS; is the mass flow rate via mth heat load in nth DHS. The return temperature is determined by the outlet water mixture from all heat loads. It can be calculated by (45).

mnL, m

R TN, n, t

+1

=

1 mns

NHL, n out,m (mnL, m · TN, n, t ),

t

{1,2. ..,T

1},

n

m =1

{1,2. ..,NDHS} ,

m

(45)

{1,2. ..,NHL,n}

In the CHP dispatch, the operation range constraints of supply temperature and return temperature should be considered, the constraints are shown by (46) and (47). S T N, n

S TN, n, t

. .

. Fig. 6. Diagram of a typical DHS. 8

S T¯N, n,

.

t

{1,2. ..,T},

n

{ 1,2. ..,NDHS}

(46)

Energy Conversion and Management 200 (2019) 112091

Y. Li, et al. R T N, n

R T¯N, n,

R TN, n, t

t

{1,2. ..,T},

n

downward spinning reserve provided by CHP units can be modelled by (53) and (54), respectively.

(47)

{1,2. ..,NDHS}

S where and T¯N, n are the lower and upper limits of supply temR R perature; T N,n and T¯N, n are the lower and upper limits of return temperature.

S T N, n

3.2.2. Operation constraints of EPS The operation constraints of EPS mainly include power balance constraint, spinning reserve constraints, thermal unit operation constraints and transmission line capacity constraint. The detail of these constraints is described as follows. Power balance constrain is used to make sure that the total generation and electric demands are balanced at each period. It is described by, NTU

NCHP

PTU, i, t + i

(PCAESd, i, t

i

i

t

{1,2. ..,T}

(48)

where PLf, t is the system forecast load at time t. Spinning reserve is the idling capacity which is usually used to cope with wind power and load forecast errors. The lack of upward spinning reserve may result in power shortage, while the lack of downward spinning reserve may result in wind curtailment. In EPS, thermal units, CHP units and AA-CAES facilities all have the ability of providing spinning reserve. The system upward and downward spinning reserve constraints are expressed in the formulation of fuzzy chance constraint, as shown in (49) and (50), respectively. NTU

Cr

+ (PTU, i, t + RTU, i, t ) +

NCHP

i NCAES i

NTU

t

(PTU, i, t

RTU, i, t ) +

(PCHP, i, t

NW

(PCAESd, i, t

PCAESc, i, t

RCAES, i, t ) +

i

~ PL, t

PWa, i, t

t

(50)

{1,2. ..,T}

where and are the upward spinning reserve provided by thermal unit i, CHP unit i and AA-CAES facility i; RTU, i, t , RCHP, i, t and RCAES, i, t are the downward spinning reserve provided by ~ thermal unit i, CHP unit i and AA -CAES facility i at time t; PL, t is the system load, which is expressed by triangle fuzzy variable. Its ordered triple formulation is (P Lf, t , PLf, t , P¯Lf, t ) , where P Lf, t and P¯Lf, t are minimum and maximum system forecast load at time t. The limits on spinning reserve provided by thermal units mainly depend on their output limits and ramping rates, as shown in (51) and (52), respectively. + RTU, i, t ,

0

+ RTU, i, t

+ RCHP, i, t

{1,2. ..,T}, 0

RTU, i, t

+ RCAES, i, t

min(P¯TU, i v TU, i, t i

i

t (51)

{1,2. ..,NTU }

min(PTU, i, t

{1,2. ..,T},

+ PTU, i, t , rTU, i tR v TU, i, t ),

P TU, i v TU, i, t , rTU, i tR v TU, i, t ),

{1,2. ..,NTU }

p CHP,i

(F CHP,i vCHP,i, t

h CHP,i HCHP, i, t )

,

t (54)

{1,2. ..,NCHP}

+ RCAESc, i, t

0

+ RCAESd, i, t

PCAESc, i, t P CAESc, i vCAESc, i, t P¯CAESd, i vCAESd, i, t PCAESd, i, t

,

max max + max max + + kmP k R k R t (kmP d Qd m CAESd, i, t c Qc m CAESc, i, t ) Rg Tar Var

max + max + RCAESd, i, t + kmP RCAESc, i, t ) t (kmP d c

i

RCAESc, i, t

par (55)

{1,2. ..,NCAES}

Rg Tar Var

PCAESd, i, t

PCAESc, i, t P CAESd, i vCAESd, i, t

max max RCAESd, i, t + kmP RCAESc, i, t ) t (kmP d c

i

{1,2. ..,NCAES}

t

Q HS

,

t

p¯ar (56)

+ + where RCAESc, i, t and R CAESd, i, t are the upward spinning reserve provided by AA-CAES facility i when it operate in charging and discharging modes respectively; RCAESc, i, t and RCAESd, i, t are the downward spinning reserve provided by AA-CAES facility i when it operate in charging and discharging modes respectively. During the stage of formulating dayahead schedules, it is difficult to estimate how much spinning reserve is executed in practical operation (the executed spinning reserve depends on practical wind power output and system load). Therefore, in order to max max ensure the feasibility of the day-ahead schedule, kmP , kmP , kQmax and d c cm max kQd m are used in spinning reserve constraints of AA-CAES. Thermal unit operation constraints consist of the generation output constraints, ramping constraints and minimum ON/OFF duration time constraints. The MILP-based formulation of these constraints can be found in Ref. [59]. Note that, CHP units also have ramping constraints and minimum ON/OFF duration time constraints, and the formulation of these constraints is similar to the relative constraints of thermal units [58]. Transmission line capacity constraint is used to make sure that the transmission flows do not exceed the transmission line capacity. The line flow distribution factors [63] are used to calculate the transmission flow in this model. The formulation of transmission line capacity constraint can be found in Ref. [63]. It has to be pointed that, if there is an electrical heater integrated with AA-CAES, the electrical heater should also be considered as one of scheduling resources in CHP dispatch. The power consumption of electrical heaters should be considered in power balance constraint. The spinning reserve provided by electrical heaters should also be considered in system spinning reserve constraints. The limits on the upward and downward spinning reserve provided by electrical heaters are modelled by (57) and (58). Lower and upper limits of its power consumption is described by (59).

i=1

,

0

{1,2. ..,T}, (49)

RCHP, i, t )

NCAES

PCHP, i, t rCHPi HCHP, i, t rCHP, i tR vCHP, i, t

par , t +

~ PL, t

{1,2. ..,T}

t (53)

RCAES, i, t = RCAESc, i, t + RCAESd, i, t RCAESc, i, t P¯CAESc, i vCAESc, i, t

i

+

RCHP, i, t RCHP, i, t

{1,2. ..,T},

NCHP

i

0 0

0 0 ~ PW, i, t

1

PCHP, i, t

i

,

{1,2. ..,NCHP}

RCHP, i, t

par , t

i=1

, Cr

NW

i

PCHP, i, t

tR vCHP, i, t

0

QHS , t

+ (PCHP, i, t + RCHP, i, t )

+ PCAESc, i, t + RCAES, i, t ) +

(PCAESd, i, t

+ rCHP, i

+ + + RCAES, i, t = RCAESc, i, t + RCAESd, i, t

i

+

0

+ RCHP, i, t

h CHP,i HCHP, i, t p CHP,i

The spinning reserve provided by AA-CAES is limited by its maximum charging/discharging powers, air pressure in air reservoir and stored heat [6,38]. The constraints of upward and downward spinning reserve provided by AA-CAES is described by (55) and (56), respectively.

NW

min(PWf, i, t , PWa, i, t ) = PLf, t ,

P¯CHP, i vCHP,i, t

{1,2. ..,T},

PCAESc, i, t ) +

i=1

+ RCHP, i, t

{1,2. ..,T},

NCAES

PCHP, i, t +

0

t (52)

where P¯TU, i and P TU, i are maximum and minimum output power of + thermal unit i; rTU, i and rTU, i are the ramp up and down rates of thermal unit i; tR is response time of spinning reserve. It is set to be 10 min in China [53]. The power regulation range of extraction CHP units is decided by their heat generation power (see Fig. 7). The limits on upward and 9

Energy Conversion and Management 200 (2019) 112091

Y. Li, et al.

G

1

G

G

G

G

G

G G

G

G G

G

G

G

G

G

G G

G

ES

G

G

W G

G G

G

G

G

ES

G

G

G

DHS

W

G G

Diagram of district heating system HL1

HL2

Nd4

G

Nd1

Thermal unit

ES

W Wind farm

Nd2

G

DHS

Nd5

ES HL3

Heat sources (CHP and AACAES)

G

DHS

ES

G G

DHS

W

ES

G W

DHS

G

W

Nd3

G

G

G

G G

AA-CAES facility

G

ES

CHP unit

G

G G

G

DHS

Fig. 8. Schematic diagram of the revised IEEE 118-bus system.

0

min(P¯EH , i vEH , i, t

+ REH , i, t

{1,2. ..,T}, 0

REH , i, t

i

P EH , i vEH , i, t

i

PEH , i, t

hk+ (x ) =

(57)

P EH , i vEH , i, t , rEH , i tR vEH , i, t ),

t (58)

{1,2. ..,NEH }

P¯EH , i vEH , i, t ,

0, hk (x ) 0 hk (x ), hk (x ) 0 and hk (x ) = ; spehk (x ), hk (x ) 0 0, hk (x ) 0 cially, if hk (x ) = 1, then hk+ (x ) = 1 and hk (x ) = 0 , if hk (x ) = 1, then hk+ (x ) = 0 and hk (x ) = 1. Note that, the triangle fuzzy variable can be considered as the tra~ pezoidal fuzzy variable whose rk2 = rk3 . For example, PW, i, t is the trirk1 = P Wf, i, t , angle fuzzy variable(P Wf, i, t ,PWf, i, t ,P¯Wf, i, t ) , then rk2 = rk3 = PWf, i, t , rk 4 = P¯Wf, i, t . According to the above theorem, the crisp equivalents of Eqs. (39), (49) and (50) are expressed as follows:

t

{1,2. ..,NEH }

min(PEH , i, t

{1,2. ..,T},

+ PEH , i, t , rEH , i tR vEH , i, t ),

t

{1,2. ..,T},

i

{1,2. ..,NEH } (59)

where and REH , i, t are the upward and downward spinning reserve provided by electrical heater; P EH , i and P¯EH , i are the lower and upper + limits of the power consumption of electrical heater; rEH , i and rEH , i are the ramp up and down rates of electrical heater. + REH , i, t

PW, i, t = inf{r|(2

NTU

n

rk2 hk (x )] + (2

k=1

+ h 0 (x ) where

hk+ (x )

[rk 4 hk+ (x )

1)

(PCAESd, i, t

hk (x )

are

the

functions

+ (PCHP, i, t + RCHP, i, t ) +

+ PCAESc, i, t + RCAES, i, t ) +

[(2

2 ) PWf, i, t + (2

1) P W, i, t ]

(2

2 ) PLf, t + (2

1) P¯L, t

i=1

,

rk1 hk (x )]

defined

NCHP

i (61)

NW

(60) and

{1,2. ..,T},

i

k=1

0

t

i NCAES

n

[rk3 hk+ (x )

+ (PTU, i, t + RTU, i, t ) +

i

In the proposed CHP dispatch model, the fuzzy chance constraints ((39), (49) and (50)) can be convert into their crisp equivalents. The transformation process is described bellow: From Refs. [55,64], for the function g (x , ) defined by g (x , ) = h1 (x1) 1 + hn (x n) n + h 0 (x ) , where xk is decision variable; k is trapezoidal fuzzy variable(rk1, rk 2, rk3, rk 4 ) ; hk (xk ) is the function of xk . The fuzzy chance constraint Cr{g (x , ) 0} can be converted into 0.5): the following crisp equivalent (when the confidence level

2 )

r },

{1,2. ..,NW }

3.3. Solving method

(2

1) P¯W, i, t

2 ) PWf, i, t + (2

by 10

t

{1,2. ..,T}

(62)

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Fig. 9. Curves of (a) electric and heat loads, (b) wind power forecast output. Table 2 The costs of each scenario (unit: $). Costs

Scenario 1

Scenario 2

Scenario 3

Scenario 4

Scenario 5

Operation cost of thermal units Operation cost of CHP unit Penalty of wind curtailment System operation cost

260,836 1,316,703 139,931 1,717,470

262,918 1,256,143 22,448 1,541,509

290,900 1,165,163 3698 1,459,761

300,017 1,128,436 3698 1,432,151

328,424 1,085,452 1352 1,415,228

NTU

of the total generation power (8270 MW), which is close to the proportion in Jilin Province. Also, the electric load, heat load and wind power profiles are set based on the real operation data [3]. The curves of total electric loads, total heat loads and wind power output are shown in Fig. 9. The peak electric load of 3697 MW occurs at hour 12. The schedule parameters of thermal units and electricity network parameters come from Ref. [65]. The schedule parameters of CHP units are given in Appendix A [65,66]. Each AA-CAES facility has 50 MW of charging power, 30 MW of discharging power, 30 MW of heat releasing power, and about 5 h of rated discharging capability. The minimum charging and discharging powers of AA-CAES are 40% maximum charging and discharging power. The charging and discharging powers of AA-CAES are both divided into 6 segments. The step is set to be 10% of compression/generation level [13]. The air pressure in air reservoir is divided into 5 segments. The step is set to be 3 bar. The average air pressure of each segment is used to calculate the kmPc, t and kQc m, t . The value of ratios kmPc, t , kmPd, t , kQc m, t and kQd m, t after piecewise approximation is given in Appendix A. The heating network in each DHS is operated with the strategy of fixed flow and variable water temperature. The supply temperature should be between 60 °C and 90 °C, and the return temperature should be between 20 °C and 40 °C [62]. The maximum forecast errors of wind power and electric load are set to be 30% and 10%, respectively. The confidence level of fuzzy chance constraint is set to be 0.95.

NCHP

(PTU, i, t

RTU, i, t ) +

i

(PCHP, i, t

RCHP, i, t ) +

i NCAES

(PCAESd, i, t

PCAESc, i, t

RCAES, i, t ) +

i NW

[(2

1) P¯W, i, t ]

2 ) PWf, i, t + (2

(2

2 ) PLf, t + (2

1) P L, t

i=1

,

t

{1,2. ..,T}

(63)

According to Eq. (61), the pessimistic value of wind power can be calculated by,

PW, i, t = (2

2 ) PWf, i, t + (2

1) P¯W, i, t

(64)

After the fuzzy chance constraints are converted into their crisp equivalents, the proposed CHP dispatch problem is convert into a MILP problem, which can be solved by conventional solvers. 4. Case studies The simulation uses a revised IEEE 118-bus system to demonstrate the effectiveness of the proposed model. The benefits of AA-CAES in CHP dispatch are assessed, and the impact of AA-CAES’s off-design performance on CHP dispatch is analyzed. All simulations are performed in a computer with an Intel Core i5 2.5 GHz CPU and 8 GB memory, and IBM ILOG CPLEX 12.6.3 is employed to solve the optimization problems.

4.2. Simulation results

4.1. System description

4.2.1. Assessing the benefit of AA-CAES for reducing system operation cost In order to assessing the benefits of AA-CAES for reducing system operation cost, five scenarios are set. Scenario 1: there is no AA-CAES facility in the system. Scenario 2: there are 6 AA-CAES facilities in the system, but the AA-CAES facilities can only provide electricity generation. Scenario 3: there are 6 AA-CAES facilities in the system, and the AA-CAES facilities can provide both electricity and heat generation. Scenario 4: there are 6 AA-CAES facilities in the system. Each AA-CAES facility is integrated with the solar collectors of 35,000 m2 solar multiple. The AA-CAES facilities can provided both electricity and heat generation. The detailed schedule parameters and the curve of direct normal irradiance are given in Appendix A. Scenario 5: there are 6 AA-

Fig. 8 illustrates the schematic diagram of the revised IEEE 118-bus system. In the system, there are 48 thermal units, 5 wind farms, 6 extraction CHP units and 6 AA-CAES facilities. There are 6 six-node DHSs in the system. The heat source of each DHS consists of a CHP unit and an AA-CAES facility. These DHSs are connecting to buses 10, 65, 66, 80, 87 and 89, respectively. CHP units are the main heat source in many practical systems in China [3]. According to Ref. [3], the total generation power in Jilin Province (located in Northeast China) is 18GW, where the total capacity of CHP units is 6.6 GW. It takes 36.7% of the total generation capacity. In this case study, the total capacity of CHP units (2990 MW) is 36.2% 11

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Y. Li, et al.

(a)

(b)

(c)

(d )

(e)

(f )

( g)

( h)

Fig. 10. (a) Wind curtailment of scenario 1, (b) wind curtailment of scenario 3, (c) electricity generation schedule of scenario 1, (d) electricity generation schedule of scenario 3, (e) heat generation schedule of scenario 1, (f) heat generation schedule of scenario 3, (g) downward spinning reserve schedule for wind accommodation in scenario 1, (h) downward spinning reserve schedule for wind accommodation in scenario 3

CAES facilities in the system. Each AA-CAES facility is integrated with an electric heater. The AA-CAES facilities can provided both electricity and heat generation. Each electric heater has the maximum electric generation power of 22 MW, and the detailed scheduling parameters is given in Appendix A. The costs of each scenario are shown in Table 2. As shown in this

table, after AA-CAES participates in scheduling (comparing the results of scenario 1 and 2), the penalty of wind curtailment decreases rapidly, which is 16.0% of the wind curtailment penalty of scenario 1. The total operation cost of thermal units and CHP units also has a tendency of decrease, which is $58478 lower than the cost of scenario 1. Comparing the costs of scenario 2 and scenario 3, we can observe 12

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Table 3 Expected and actual costs after the 6 AA-CAES facilities all use simple model for CHP dispatch (unit: $). Costs

Expected

Actual

Operation cost of thermal units Operation cost of CHP unit Penalty of wind curtailment System operation cost

269,574 1,183,505 2896 1,455,975

277,152 1,183,505 2896 1,463,553

cost, including the reduction of wind curtailment penalty and the total operation costs of thermal units and CHP units. In addition, the unique advantages of AA-CAES, such as the ability of flexible integration with various kinds of EHS and the ability of storing and generating both electricity and heat, make it have great performance in CHP dispatch. 4.2.2. Assessing the benefit of AA-CAES for mitigating wind curtailment The scheduling results of scenario 1 and 3 are compared to assessing the benefits of AA-CAES for mitigating wind curtailment. Fig. 10 shows the detailed scheduling results of scenario 1 and 3. The simulation results of wind curtailment of scenario 1 and 3 are shown in Fig. 10(a) and (b). It can be seen from Fig. 10(a) that, the wind curtailment problem is very severe in scenario 1, and the wind curtailment mainly occurs in the periods of low electric demand and high wind power output (1:00–6:00). The most severe wind curtailment occurs at 6:00, in which the wind power output and heat demand is relatively high meanwhile the electricity demand is relatively low. The total wind curtailment of a day is 1396 MW.h. From Fig. 10(b), the wind curtailment is mitigated obviously in scenario 3. The wind curtailment only occurs at 6:00 in scenario 3, and the amount of wind curtailment is decreased by 97.3%. The main reasons that lead to the mitigation of wind curtailment after AA-CAES facilities participate in CHP dispatch are analyzed as follows: The electric generation schedules of scenario 1 and 3 are shown in Fig. 10(c) and (d), respectively. According to Fig. 10(d), the AA-CAES operates in charging mode during the periods of low electric load demand and high wind power output (1:00–6:00). The wind power output can be stored by AA-CAES in terms of high-pressure air and heat, and during the periods of high electric load demand and low wind power output, the AA-CAES operates in discharging mode (11:00–15:00). The compression heat of AA-CAES can not only be reused for its electric generation but also be used for heat supply. Fig. 10(e) and (f) shows the heat generation schedules of scenario 1 and 3, respectively. It can be observe from Fig. 10(e) and (f) that, the heat storage characteristic of the district heating network can increase the heat generation flexibility (the total heat generation may not equal to the heat load demand in each time period). Also, after AA-CAES facilities participate

Fig. 11. (a) AA-CAES schedule obtained from simple model, (b) Actual AACAES schedule following simple model schedule.

that, the heat production ability of AA-CAES lead to $81748 reduction of system operation cost in CHP dispatch. Among them, the reduction of CHP units and thermal units operation costs takes about 77.1% of total cost reduction. The heat production ability of AA-CAES can also lead to an obvious decrease of the wind curtailment penalty. Compared with scenario 2, the wind curtailment penalty in scenario 3 is decreased by 83.5%. After AA-CAES is integrated with EHS and participates in CHP dispatch (see scenario 4 and 5), the system operation costs are further reduced. Comparing the costs of scenario 3 and 4, the system operation cost of scenario 4 is $27,610 lower than the cost of scenario 3, and the cost reduction mainly comes from the reduction of the operation cost CHP unit. It is mainly because the integration of solar collectors makes the AA-CAES have more sufficient stored heat for heat supply to support CHP unit operation. Comparing the costs of scenario 3 and 5, the system operation cost of scenario 5 is $44,533 lower than the cost of scenario 3, and the wind curtailment penalty is also decreased by 63.4%. It indicates that the integration of electric heater can further mitigate wind curtailment. The results in Table 2 indicate that the participation of AA-CAES in CHP dispatch can lead to an apparent reduction of system operation

(a)

(b)

Fig. 12. (a) Expected and actual pressure variation following simple model schedule, (b) expected and actual stored heat variation following simple model schedule. 13

Energy Conversion and Management 200 (2019) 112091

Y. Li, et al.

in CHP dispatch, the total heat generation of CHP units in a day is decreased by 6.5%, due to the support from the heat supply of AA-CAES facilities. According to the FOR of extraction CHP unit (refer to Fig. 7), the CHP unit has larger power regulation range with the decrease of its heat generation power. Therefore, it can be seen in Fig. 10(c) and (d) that, during 1:00–6:00 (the wind curtailment occurs in scenario 1), the electric power output of CHP units can be decreased in scenario 3 to leave more generation capacity for wind power generation. Since the wind power has the inherent characteristic of uncertainty, downward spinning reserve is needed to cope with the situation that practical wind power output is higher than its forecast value. Downward spinning reserve is one of important factors that determines the maximum acceptable wind power output of the system, and more downward spinning reserve preserved for wind accommodation can lead to higher maximum acceptable wind power output. The downward reserve schedules for wind accommodation of scenario 1 and 3 are shown in Fig. 10(g) and (h), respectively. The comparison of these two figures demonstrates that, after AA-CAES facilities participate in CHP dispatch, the downward spinning reserve preserved for wind accommodation is increased obviously. The total downward spinning reserve preserved for wind accommodation of a day in scenario 3 is about 2.3 times of the total downward spinning reserve in scenario 1. In addition, 34.3% of the total downward spinning reserve for wind accommodation is provided by the AA-CAES facilities. It is mainly because that AACAES has great dynamic performance. The results indicate that the advantages of AA-CAES in providing spinning reserve can be one of important factors that help mitigating wind curtailment.

stored heat in 13:00–24:00 is higher than its expected value due to the modification of simple model schedule. The misjudgment of the stored heat can cause the decrease of the superiority of the scheduling strategy. Moreover, if there is EHS, such as solar collectors, integrated with AA-CAES, the misjudgment of the stored heat can result in the waste of stored heat. Table 3 shows the expected and actual costs after the 6 AA-CAES facilities in the system all uses simple model for CHP dispatch. As shown in Table 3, the expected system operation cost of using simple model for CHP dispatch is $3786 lower than the system operation cost obtained from the proposed model (refer to Table 2 scenario 3). The main reason is that the off-design performance of AA-CAES is ignored in simple model and it leads to the overestimation of its state of charge. However, the actual system operation cost following simple model schedule is $7578 higher than the expected system operation cost obtained from the simple model. It is mainly because the upward spinning reserve is executed when the discharging schedule of AA-CAES cannot be executed, and the upward power regulation can be provided by the units with relatively high operation cost. Note that, the potential economic loss in the situation that the system upward spinning reserve is insufficient and lead to power shortage is not considered in Table 3. Above all, in CHP dispatch problem involving AA-CAES and wind farms, following the AA-CAES schedule obtained from simple model can result in the infeasibility of its discharging schedule and the increase of system operation cost.

4.2.3. Analyzing the impact of AA-CAES’s off-design performance on CHP dispatch In order to analyze the impact of AA-CAES’s off-design performance on CHP dispatch, the mathematical model of AA-CAES without considering its off-design performance, hereafter called simple model, is used in CHP dispatch for comparison. Note that, in simple model, the scheduling parameters are obtained from the operation data that AACAES operates in its designed condition. Fig. 11(a) illustrates the schedule of the AA-CAES facility located at Bus 65 obtained from simple model. It is shown in Fig. 11(a) that, after using the simple model for CHP dispatch, the AA-CAES also operates in discharging mode during the periods of high electric demand and low wind power, and it also basically maintains its minimum charging power for providing downward spinning reserve. However, according to the actual AA-CAES schedule following simple model schedule, which is shown in Fig. 11(b), it can be observed that, the AA-CAES schedule obtained from the simple model in 13:00–15:00 cannot be executed in practical operation. It is mainly because ignoring AACAES’s off-design performance can result in the misjudgment of AACAES’s efficiency, and the expected AA-CAES’s state of charge is higher than the actual state of charge. The infeasibility of the discharging schedule can result in the execution of system upward spinning reserve. If, at the same time, the actual wind power output is less than its forecast value, the system spinning reserve might be insufficient for these emergencies, and users may suffer from power outage. Fig. 12(a) shows the expected and actual pressure variation following simple model schedule. It can be seen from Fig. 12(a) that, the air pressure in the air reserve has obvious differences between expected value and actual value, and the actual air pressure will reach its lower limit in advance. It is mainly because AA-CAES usually operates at its minimum charging power to provide downward spinning reserve for wind accommodation, and the decrease of AA-CAES’s efficiency when it operates in off-design conditions is ignored in the simple model. Fig. 12(b) shows the expected and actual stored heat variation following simple model schedule. Fig. 12(b) demonstrates that, the actual

A MILP-based AA-CAES mathematical model with considering its off-design characteristic has been developed. A CHP dispatch model with considering the coordination of AA-CAES facilities, CHP units, non-CHP thermal units and wind farms has been developed for mitigating wind curtailment and decreasing system operation cost in IEHS. Case studies have been implemented for assessing the benefits of AACAES in CHP dispatch and studying the importance of considering the AA-CAES off-design performance. The conclusions can be drawn:

5. Conclusion

• The participation of AA-CAES in CHP dispatch can lead to an ap• • •

parent reduction of system operation cost and wind curtailment, due to its ability of flexible integration with various kinds of EHS and storing and generating both electricity and heat. The heat production ability of AA-CAES can increase the power regulation flexibility of CHP units. The participation of AA-CAES in CHP dispatch can significantly increase the downward spinning reserve preserved for wind accommodation. To the CHP dispatch involving AA-CAES and wind farms, the AACAES off-design performance should be considered in detail; otherwise it can cause the misjudgment of AA-CAES’s state of charge, which in turn results in the system operation cost increase.

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. Acknowledgements The authors would like to thank the research grant support from The National Key Research and Development Program of China (2017YFB0903601) and The National Natural Science Foundation of China (51777088).

14

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Appendix A. . Supplementary data See Tables A1–A3, Fig. A4 and Tables A5 and A6.

Table A1 Schedule parameters of CHP units [66]. Parameters

CHP1–2

CHP3–5

CHP6

Unit

Power to heat ratio Lower bound on fuel consumption Upper bound on fuel consumption Maximum heat generation power Maximum electric generation power Minimum electric generation power Fuel consumption per electricity generation Fuel consumption per heat generation Ramp up/down rates Minimum ON/OFF time Cost coefficient of CHP Bus Number Start up cost

0.85 352 1008 290 420 125 2.4

0.85 528 1200 350 500 150 2.4

0.85 704 1560 455 650 200 2.4

/ MW MW MW MW MW /

0.95 4.2 10 10.2 65, 66 250

0.95 5 10 10.2 10,80,89 400

0.95 6.5 10 10.2 87 440

/ MW/min hour MW/$ / $

Table A2 Schedule parameters of AA-CAES facility [6]. Parameters

Value

Unit

Maximum/minimum charging power Maximum/minimum discharging power Maximum heat releasing power Number of compressors/turbines Rated inlet temperature of compressors Rated inlet temperature of turbine Ambient temperature Lower/upper limits of pressure in air reservoir Lower/upper limits of stored heat Volume of air reservoir

50/20 30/12 30 4 312 363 298 40–55 0–300 180,000

MW MW MW / K K K bar MW.h m3

Table A3 Value of ratios k mPc, t , k mPd,t , kQc m,t and kQd m, t after piecewise approximation. Air pressure

Compression/generation level 100–90%

90–80%

80–70%

70–60%

60–50%

50–40%

kmPc,t kg/(s.MW)

40–43 bar 43–46 bar 46–49 bar 49–52 bar 52–55 bar

2.43 2.35 2.29 2.24 2.19

2.32 2.25 2.19 2.14 2.10

2.17 2.10 2.05 2.00 1.96

1.98 1.92 1.87 1.82 1.78

1.74 1.68 1.64 1.60 1.57

1.33 1.29 1.26 1.23 1.20

kQc m, t MW/(kg/s)

40–43 bar 43–46 bar 46–49 bar 49–52 bar 52–55 bar

0.396 0.412 0.420 0.432 0.44

0.417 0.432 0.441 0.452 0.462

0.446 0.463 0.474 0.484 0.497

0.493 0.510 0.521 0.534 0.546

0.561 0.580 0.595 0.609 0.623

0.738 0.764 0.781 0.798 0.816

kmPd, t MW/(kg/s)

/

3.45

3.61

3.86

4.24

4.82

6.29

kQd m, t MW/(kg/s)

/

0.277

0.268

0.252

0.238

0.216

0.180

15

Energy Conversion and Management 200 (2019) 112091

Direct normal irradiance (W/m2)

Y. Li, et al.

1200 1000 800 600 400 200 0

1

3

5

7

9

11

13

15

17

19

21

23

Time(h) Fig. A4. Forecast curve of direct normal irradiance. Table A5 Schedule parameters of solar collectors. Parameters

Value

Unit

Transfer efficiency between solar and heat Efficiency of storing heat from solar collectors Area of solar multiple

83 90 35,000

% % m2

Table A6 Schedule parameters of electric heater. Parameters

Value

Unit

Upper limit of power consumption Lower limit of power consumption Energy conversion efficiency of electrical heater Efficiency of storing heat from and electrical heater

22 2 98 90

MW MW % %

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