Interval optimization for integrated electrical and natural-gas systems with power to gas considering uncertainties

Electrical Power and Energy Systems 119 (2020) 105906

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Interval optimization for integrated electrical and natural-gas systems with power to gas considering uncertainties Shouxiang Wang, Shuangchen Yuan

T

⁎

School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Interval optimization Integrated electrical and natural-gas systems Power to gas Uncertainties

This paper is aimed at the optimal operation of the integrated electrical and natural-gas systems (IENGS) with power to gas (P2G) considering uncertainties of the wind turbine (WT) and photovoltaic (PV) via interval mathematics. The optimization model of the IENGS is established under deterministic conditions. The model takes the operation and maintenance cost, environmental cost, purchasing electricity/gas costs and selling electricity/gas incomes into account. Based on the interval arithmetic, the outputs of WTs and PVs are represented in interval forms and the interval optimization model is established considering the uncertainties of WTs and PVs. The model is converted into a mixed-integer linear programming (MILP) model and solved by the interval linear programming method. In the numerical test, the simulation results under deterministic conditions and the influence of the P2G device are analyzed, then the interval outputs of components and the operation cost of the IENGS are discussed when the uncertainties of the WT and PV are considered. The simulation results show that a higher conversion rate of P2G will lower the operation cost of the IENGS and how the outputs of components change under different uncertain levels of the WT and PV.

1. Introduction In recent years, environmental pollution has become a vital issue. It is an excellent way to solve this problem by developing multi-energy utilization. Developing integrated energy systems (IES) is an ideal solution to utilize energies because IES can combine different kinds of energies [1]. The IES have many forms, among which the integrated electrical and natural-gas systems (IENGS) [2] are the typical ones because the natural gas has the advantage of economy, environmental protection, and flexibility. Hence, it is essential to study the operation of IENGS to get lower costs and higher efficiency. A lot of works have focused on the optimization of IENGS. As a common coupling component that links the electrical and gas networks, the optimization methods for IENGS with the gas-fired generator have been investigated in [3–9]. A coupled optimization method for the electricity and gas systems by using augmented Lagrangian and an alternating minimization is proposed in [3]. Multiple coordination scenarios for interdependent electric power and natural gas infrastructures are formulated in [4]. In these scenarios, tractable physically accurate computational implementations are developed. So dynamic gas flows on pipeline networks can be controlled to examine day-ahead scheduling and simulation to combined optimal control. A standard alternating direction method of multipliers (ADMM) approach and a ⁎

consensus-based ADMM approach are developed in [5]. The two approaches are used to solve the gas-electric integrated optimal power flow problem with and without a coordination operator. A multi-objective optimization model is developed and an improved objective reduction approach is proposed in [6]. A coordinated operation strategy for the gas-electricity integrated distribution systems, considering AC power flow is proposed in [7]. From a joint operator’s viewpoint, the coordinated scheduling of interdependent electric power and natural gas transmission systems is presented in [8]. In [9], the maximum profits that can be obtained in the steady-state coordinated operation of electricity networks and natural gas networks are studied considering demand response. With the popularity of IENGS, power to gas (P2G), as another kind of coupling component, is drawing wide attention [10]. Considering carbon capture systems and P2G, low-carbon economic dispatch for electricity and natural gas systems is investigated in [11]. A bi-level dispatch model considering P2G to minimize the total operation costs is proposed in [12]. The economic feasibility of P2G systems and gas storage options for both hydrogen and renewable methane is investigated in [13]. A robust co-optimization scheduling model to study the coordinated optimal operation of IENGS with P2G is proposed in [14]. The expected profit of P2G facilities without exceeding the grid capacities is maximized in [15]. P2G is optimally scheduled to convert

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

https://doi.org/10.1016/j.ijepes.2020.105906 Received 11 August 2019; Received in revised form 9 January 2020; Accepted 3 February 2020 0142-0615/ © 2020 Elsevier Ltd. All rights reserved.

Electrical Power and Energy Systems 119 (2020) 105906

S. Wang and S. Yuan

∼ HGF (t ) Pi(t) Ki, Mi

approximating function of HGF(t) at time t the ith piecewise point of PGF(t) at time t the slope of ith partition of HGF(t) and the equivalent intercept at time t Bi(t) state variable of HGF(t) at time t Bch(t), Bdis(t) charging and discharging state variables at time t Bbe(t), Bse(t) state variables of electricity purchasing and selling at time t Bbg(t), Bsg(t) state variables of gas purchasing and selling at time t

Nomenclature Abbreviations IENGS WT PV P2G ESS MILP

integrated electrical and natural-gas systems wind turbine photovoltaic power to gas energy storage system mix-integer linear programming

Parameters Variables

c3, GF , c2, GF , c1, GF , c0, GF , Pr fixed values of the gas-fired generator LHV low heating value of natural gas ηch, ηdis charging and discharging efficiencies of ESS Nt total periods of an optimal day KWT, KPV, KESS, KGF, KP2G operation and maintenance coefficients of WT, PV, ESS, gas-fired generator and P2G Cd,WT, Cd,PV, Cd,GF, Cd,ESS, Cd,P2G fixed costs of WT, PV, the gas-fired generator, ESS and P2G in an optimal day m number of pollutants βk,GF, βk,e emission factors of pollutants produced by the gas-fired generator and the outside power system αk cost coefficient that needs to treat the kth pollutant PGF,min, PGF,max minimum and maximum electrical outputs of the gas-fired generator LP2G,min, LP2G,max minimum and maximum electrical inputs of P2G Emin, Emax minimum and maximum capacity of ESS Pch,max, Pdis,max maximum charging and discharging power of ESS Pe,max, Hg,max maximum electricity and natural gas exchange with the outside UGF, DGF ramping up rate and ramping down rate of the gas-fired generator UP2G, DP2G ramping up rate and ramping down rate of P2G

ηGF power generation efficiency of the gas-fired generator PGF power generation of the gas-fired generator HGF input of natural gas by the gas-fired generator ηP2G conversion rate of P2G FP2G output of natural-gas by P2G LP2G input of electricity by P2G E(t) capacity of ESS at time t Pch(t), Pdis(t) charging and discharging power of ESS at time t PESS output of ESS Pgrid output of the electrical system in IENGS Fgas output of the gas system in IENGS CIENGS operation cost of IENGS Com, Cen, Ce, Cg peration and maintenance cost, environmental cost, electricity purchasing cost and gas purchasing cost Cunfix, Cfix unfixed cost and fixed cost of the operation and maintenance cost Ie, Ig electricity selling income and gas selling income Kbe, Kse electricity-purchasing price and electricity-selling price at time t Kbg, Ksg gas-purchasing price and gas-selling price at time t L(t), N(t) electrical load and gas load in the IENGS at time t [x] variable in the interval form x, x lower bound and upper bound of interval variable [x]

−

great number of scenarios [26]. On many occasions, the upper and lower bounds of uncertain variables can be obtained much easily. Interval mathematics [27], which only needs the upper and lower bounds of variables, is an effective tool to deal with this kind of uncertain optimization problem. Therefore, this paper addresses the optimization problem with uncertainties by interval mathematics. In this paper, firstly, models of the coupling components in IENGS, including the gas-fire generator and P2G device, and electrical energy storage system (ESS) are introduced. Then based on the deterministic objective function, the interval optimization function is built considering the uncertainties of WTs and PVs. To solve this model, the model is transformed into a mix-integer linear programming (MILP) model and solved by interval linear programming method. In the numerical test, the influence of P2G is analyzed and the outputs of components and the operation costs of the IENGS under different uncertain levels of the WT and PV are studied. The main contributions of this paper are summarized as follows:

waste/inexpensive electricity to synthetic natural gas for some useful operations at appropriate periods in [16]. Most of the above research mainly focuses on the optimization of IENGS under deterministic conditions. However, the uncertainties of the IENGS should be taken into consideration because the intermittent energies, such as WTs or PVs, are integrated to IENGS and increase the uncertainties of the IENGS [17]. So how to address the uncertainties is a vital issue. A comprehensive computational framework for quantification and integration of uncertainties by the scenario generation method is proposed in [18]. The uncertainty of wind power forecasting is captured by several scenarios in [19]. A robust optimization approach to accommodate the uncertainty of wind output under the worst wind output scenario is proposed in [20]. An optimization model considering an uncertainty-immunized solution in a unified framework is presented in [21]. A second-order cone (SOC) relaxation of Weymouth equation considering unfixed gas flow directions is proposed and reserve is used to manage renewable uncertainties in [22]. Fuzzy sets concept is used to model uncertainties and a three-stage optimization method is applied to find the optimal scheduling of IENGS in [23]. Considering the variability of the wind energy in IENGS, the stochastic security-constrained component commitment is applied in [24]. A coordinated stochastic model which considers the random outages of generating units, transmission lines and random errors in forecasting the dayahead hourly loads is proposed in [25]. Most of the above research has addressed uncertain variables by distribution functions and scenario generation. However, it is sometimes difficult to obtain the distribution functions of uncertain variables, and it is time-consuming to generate a

(1) Considering the uncertainties of WTs and PVs in the IENGS, interval mathematics is implemented in the deterministic optimization model which includes the operation and maintenance cost, environmental cost, energy-purchasing costs and energy-selling incomes. In the proposed interval model, the outputs of WTs and PVs are formulated in interval numbers instead of probability functions and other variables and constraints are also formulated in interval forms to reflect the uncertainties. (2) The method to solve the interval nonlinear optimization model is 2

Electrical Power and Energy Systems 119 (2020) 105906

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electrolyzed water. The output of natural gas of P2G, FP2G can be calculated by [37]:

proposed. By linearizing the power output of the gas-fired generator, ESS, energy exchange between IENGS and the outside power and gas systems, the method transforms the problem into a MILP model, and the optimization is implemented by the interval linear method. (3) The proposed method is compared with the Monte-Carlo simulation method, which is a conventional method to generate results with different power outputs of the WT and PV. The numerical results validate the effectiveness of the proposed method. (4) The outputs of P2G and the operation costs of the whole IENGS are analyzed for the different conversion rates of P2G. The investment year of P2G is presented to analyze the economy of P2G considering its capital cost and conversion rate. (5) The outputs of components in the IENGS are studied when the uncertainty of the WT and PV changes. The upper and lower bounds of the operation costs are compared with different uncertain levels of the WT and PV. Based on the interval results, the decision maker can get the information that the operation cost of the IENGS will fall in which interval under a certain level of the WT and PV uncertainties.

(iii) ESS The ESS is quite commonly used in the power system. Its capacity in time t is tightly related with its state in time t-1. The formulas of charging and discharging [38] in time t can be expressed as (4) and (5): (4)

E (t ) = E (t − 1)(1 − δ ) − ΔTPdis (t ) ηdis

(5)

2.1.2. Optimization model under deterministic conditions (i) Objective function The optimization model mainly takes the operation cost CIENGS of the IENGS in a day into consideration. The objective function can be expressed as:

2.1. Deterministic optimization model

min CIENGS = Com + Cen + Ce + Cg − Ie − Ig

2.1.1. Models of components in IENGS The models of components in IENGS are given in Fig. 1, including the gas-fired generator, P2G device and ESS. The WT and PV outputs, electrical and gas loads are obtained by forecast techniques [28–32]. In this paper, assume that the IENGS can either purchase electricity/gas from the outside power and gas systems or sell electricity/gas to the outside power and gas systems, but it cannot purchase and sell electricity/gas simultaneously.

Com = Cunfix + Cfix ⎧ Nt ⎪ ⎪ Cunfix = ∑ (K GF PGF (t ) + KESS (|Pdis (t )| + |Pch (t )|) + KP 2G LP 2G (t ) t=1

⎨ ⎪ ⎪ ⎩

The gas-fired generator is a kind of coupling components that can link the electrical system and natural-gas system of the IENGS. The primary function of the gas-fired generator is to convert natural gas to electric power. If the change of environment is not considered, the relationship of the power generation efficiency ηGF and output power PGF of the gas-fired generator can be fitted by a cubic function [33] via (1):

+ KWT PWT (t ) + KPV PPV (t )) Cfix = Cd, WT + Cd, PV + Cd, GF + Cd, ESS + Cd, P 2G Nt

Cen =

(7)

m

∑ ∑ αk (βk,GF PGF (t ) + βk,e Pb,e (t ))

(8)

t=1 k=1

Nt

Ce =

∑ Kbe (t ) Pbe (t )

(9)

t=1

2

P P P ηGF = c3, GF ⎛ GF ⎞ − c2, GF ⎛ GF ⎞ + c1, GF GF + c0, GF Pr ⎝ Pr ⎠ ⎝ Pr ⎠ ⎜

Power flow Gas flow

⎟

Outside power system

(1)

where c3, GF , c2, GF , c1, GF and c0, GF are positive constants, and Pr is a fixed value that is related to the type of the gas-fired generator. The gas consumed HGF by the gas-fired generator can be expressed as:

HGF =

(6)

In (6), the operation and maintenance cost Com, the environmental cost Cen, the electricity purchase cost Ce and the gas purchase cost Cg can be formulated by:

(i) Gas-fired generator

⎟

E (t ) = E (t − 1)(1 − δ ) + ΔTPch (t ) ηch

where E(t) and E(t-1) are the capacities of the ESS in time t and t-1, respectively. δ is the self-discharge rate of the ESS. Pch(t) and Pdis(t) are the power of charging and discharging of the ESS in time t, respectively. ηch and ηdis are the charging efficiency and discharging efficiency of the ESS, respectively.

2. Problem formulation

⎜

(3)

where LP2G is the input electric power of P2G, and ηP2G is its conversion rate.

The rest of the paper is organized as follows. Problem formulation is presented in Section 2. Section 3 proposes the solution method of the interval optimization problem. The case study is discussed in Section 4. Conclusions and the future work of this paper are summarized in Section 5.

3

LP 2G LHV

FP 2G = ηP 2G

PGF ηGF LHV

Electrical system WT

ESS

PV

Electrical load

IENGS

(2)

Gas-fired generator

P2G

where LHV is the low heating value of natural gas. Gas system

(ii) P2G

Outside gas system

As another type of coupling components in IENGS, P2G device is used to convert the electric power to natural gas by Sabatier reaction [34–36] with carbon dioxide and hydrogen that is produced by

Fig. 1. The structure of the IENGS. 3

Gas load

Electrical Power and Energy Systems 119 (2020) 105906

S. Wang and S. Yuan Nt

Cg =

∑ Kbg (t ) Hbg (t ) t=1

⎧ LP 2G (t ) − LP 2G (t − 1) ⩽ UP 2G ⎨ ⎩ LP 2G (t − 1) − LP 2G (t ) ⩽ DP 2G

(10)

In (7) the operation and maintenance cost Com can be divided into two parts: the unfixed cost Cunfix and fixed cost Cfix. The unfixed cost Cunfix varies with time and KWT, KPV, KESS, KGF and KP2G are the operation and maintenance coefficients of the WT, PV, ESS, gas-fired generator and P2G. Cfix is the fixed cost which is a constant cost and not subjected to time, and Cd,WT, Cd,PV, Cd,GF, Cd,ESS and Cd,P2G are fixed costs of the WT, PV, the gas-fired generator, ESS and P2G in a day. Nt is the total periods of an optimal day. In (8) m is the total number of pollutants including CO2, SO2 and NOx, βk,GF and βk,e are the emission factors of pollutants produced by the gas-fired generator and the outside power system, Pbe is the electric power which is purchased from the outside power system by IENGS, αk is the cost coefficient that needs to treat the kth pollutant. In (9) and (10) Kbe and Kbg are the prices of electricity and gas purchased from the outside power and gas systems, and Hbg is the gas purchased from the outside gas system. The electricity selling income Ie and gas selling income Ig in (6) can be expressed as:

(23)

(2) Equality constraints The energy balance in IENGS must be satisfied by electricity balance and gas balance, which can be donated by (24) and (25) respectively.

L (t ) + LP 2G (t ) − PGF (t ) − Pbe (t ) + Pse (t ) = PWT (t ) + PPV (t )

(24)

N (t ) − FP 2G (t ) − Hbg (t ) + Hsg (t ) = −HGF (t )

(25)

where L(t) and N(t) are the total electrical load and gas load in the IENGS at time t. Besides, to obtain the sustainable electricity supply and dispatch ESS easily, the electrical energy stored in ESS at the end of the optimal day is equal to the initial value [11,39,40], which can be expressed as:

E (0) = E (T )

(26)

2.2. Interval optimization model

Nt

Ie =

∑ Kse (t ) Pse (t ) t=1

The deterministic day-ahead optimization model of IENGS has been built in the above part. However, when intermittent energy resources like WTs and PVs are integrated into IENGS, the uncertainties should be considered and it will be difficult to implement optimization. Although WTs and PVs are only connected to the electrical system of IENGS, the coupling components like the gas-fired generator and P2G device will deliver these uncertainties to the natural-gas system. On many occasions, the distribution functions of the WT and PV cannot be obtained accurately. Only the upper and lower bounds of outputs are known quantities. Under this condition, interval mathematics is a proper means to analyze this kind of optimization problem.

(11)

Nt

Ig =

∑ Ksg (t ) Hsg (t ) t=1

(12)

where Pse and Hsg are the electricity and gas sold to the outside power and gas systems respectively. Kse and Ksg are the corresponding prices of Pse and Hsg. (ii) Constraints The operation constraints of IENGS can be divided into inequality constraints and equality constraints, which are shown below.

2.2.1. Interval mathematics The interval variable is donated as [x] in interval mathematics [41]. [x] means a set of variables that satisfy x ⩽ x ⩽ x . It can be expressed − as:

(1) Inequality constraints The components should operate within their minimum and maximum constraints, which can be expressed as:

PGF ,min ⩽ PGF (t ) ⩽ PGF ,max

(13)

LP 2G,min ⩽ LP 2G (t ) ⩽ LP 2G,max

(14)

[x ] = ⎡ x , ⎢− ⎣

(15)

0 ⩽ Pdis (t ) ⩽ Pdis,max

(16)

0 ⩽ Pch (t ) ⩽ Pch,max

(17)

2.2.2. Interval optimization model of IENGS This paper proposes an interval optimization model considering the uncertainties of WTs and PVs. At time t, considering the intermittent characteristics of WTs and PVs, their outputs are intervals and can be donated as [PWT(t)] and [PPV(t)] in interval forms. The upper and lower bounds of [PWT(t)] and [PPV(t)] are obtained by the minimum and maximum values in time t. Owing to the variations of WTs and PVs, the outputs of P2G device and ESS, the energy exchange between IENGS and the outside power and gas systems will fluctuate and they can be also converted to interval forms. Hence, the objective function is rewritten as:

The constraints of energy exchange in IENGS with the outside power and gas systems can be expressed as:

0 ⩽ Pse (t ) ⩽ Pe,max

(18)

0 ⩽ Pbe (t ) ⩽ Pe,max

(19)

0 ⩽ Hsg (t ) ⩽ Hg,max

(20)

0 ⩽ Hbg (t ) ⩽ Hg,max

(21)

min[CIENGS ] = [Com] + [Ce] + [Cg ] − [Ie] − [Ig ]

(28)

Meanwhile, the constraints are changed correspondingly as:

Besides, the outputs of the gas-fired generator and P2G are restricted by the ramping up/down rate limits, which can be expressed as:

⎧ PGF (t ) − PGF (t − 1) ⩽ UGF ⎨ ⎩ PGF (t − 1) − PGF (t ) ⩽ DGF

(27)

where x and x are the upper bound and lower bound of [x], and  is − the set of real numbers. The detailed operation arithmetic concerning interval mathematics can be found in [42].

The constraints of the capacity and discharging/charging power of ESS can be donated as:

Emin ⩽ E (t ) ⩽ Emax

x ⎤ = ⎧x ∈ | x ⩽ x ⩽ x ⎫ ⎥ ⎨ − ⎬ ⎭ ⎦ ⎩

(22) 4

PGF ,min ⩽ [PGF (t )] ⩽ PGF ,max

(29)

LP 2G,min ⩽ [LP 2G (t )] ⩽ LP 2G,max

(30)

Emin ⩽ [E (t )] ⩽ Emax

(31)

0 ⩽ [Pdis (t )] ⩽ Pdis,max

(32)

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S. Wang and S. Yuan

0 ⩽ [Pch (t )] ⩽ Pch,max

(33)

0 ⩽ [Pse (t )] ⩽ Pse,max

(34)

0 ⩽ [Pbe (t )] ⩽ Pbe,max

(35)

0 ⩽ [Hsg (t )] ⩽ Hsg,max

(36)

0 ⩽ [Hbg (t )] ⩽ Hbg,max

(37)

Since the working state of the ESS in time t is either charging/discharging or not working, the state variables Bch(t) and Bdis(t) can be introduced to reflect this kind of working state:

L (t ) + [LP 2G (t )] − [PGF (t )] − [Pbg (t )] + [Psg (t )] = [PWT (t )] + [PPV (t )]

1,charging Bch (t ) = ⎧ 0, ⎨ ⎩ not charging

(48)

1,discharging Bdis (t ) = ⎧ 0, ⎨ ⎩ not discharging

(49)

(38)

The capacity of the ESS can be expressed as:

N (t ) − [FP 2G (t )] − [Hbg (t )] + [Hsg (t )] = - [HGF (t )]

(39)

E (t ) = E (t − 1)(1 − δ ) + ΔTPch (t ) ηch − ΔTPdis (t ) ηdis

[E (0)] = [E (T )]

(40)

⎧[PGF (t )] − [PGF (t − 1)] ⩽ UGF ⎨ ⎩[PGF (t − 1)] − [PGF (t )] ⩽ DGF

(41)

⎧[LP 2G (t )] − [LP 2G (t − 1)] ⩽ UP 2G ⎨ ⎩[LP 2G (t − 1)] − [LP 2G (t )] ⩽ DP 2G

(42)

In (50), the charging/discharging power can be donated as:

(i) Linearization of the gas-fired generator One of the reasons why the proposed model is a nonlinear problem is that the relationship between the gas consumed by the gas-fired generator and its power output is not linear. Similar to [45], the nonlinear relationship is approximated by piecewise first-order function. The diagram of the approximation is shown in Fig. 2. The original function can be divided into N parts. Each part is described by a bool variable Bi(t) and a continuous variable Pi(t). Then the ∼ approximating function HGF (t ) can be expressed as:

⎧ 0 ⩽ Pbe (t ) ⩽ Pbe,max Bbe (t ) 0 ⩽ Pse (t ) ⩽ Pse,max Bse (t ) ⎨ ⎩ 0 ⩽ Bbe (t ) + Bse (t ) ⩽ 1

(54)

0 ⩽ Hbg (t ) ⩽ Hbg,max Bbg (t ) ⎧ ⎪ 0 ⩽ Hsg (t ) ⩽ Hsg,max Bsg (t ) ⎨ ⎪ 0 ⩽ Bbg (t ) + Bsg (t ) ⩽ 1 ⎩

(55)

3.2. Interval optimization method Owing to the uncertainties of WTs and PVs, the results of the objective function and outputs of components in the IENGS also vary in their intervals, which is difficult to calculate directly. To deal with this problem, the interval optimization problem can be converted to the

(43)

where Ki and Mi are calculated via (44) and (45): (44)

Mi = HGF , i − Pi Ki

(45) ∼ To ensure that HGF (t ) falls in one part, the bool variable Bi(t) and the continuous variable Pi(t) should satisfy the following constraints:

Pi Bi (t ) ⩽ Pi (t ) ⩽ Pi + 1 Bi (t )

(46)

N

0⩽

∑ Bi (t ) ⩽ 1 i=1

(53)

By the above linearization, the deterministic model is converted to a MILP model, which can be solved by CPLEX.

N

HGF , i + 1 − HGF , i Pi + 1 − Pi

(52)

In the operating IENGS, the energy exchange process between the IENGS and the outside power and gas systems is uni-directional. Purchasing or selling electricity/gas cannot happen simultaneously. To describe this characteristic, the state variables Bbe(t), Bse(t), Bbg(t) and Bsg(t) are introduced in this paper. Therefore, the electricity/gas exchange between the IENGS and the outside can be expressed as:

3.1. Linear optimization method

Ki =

0 ⩽ Pdis (t ) ⩽ Pdis,max Bdis (t )

(iii) Linearization of energy exchange

The interval nonlinear optimization problem that is proposed above is difficult to solve. To deal with this problem, this paper converts the deterministic model to a MILP model, which can be calculated by commercial software like CPLEX. The upper bound and lower bound of [CIENGS] is obtained by interval linear programming algorithm [43,44].

i=1

(51)

0 ⩽ Bch (t ) + Bdis (t ) ⩽ 1

3. Interval linear optimization method

∑ (Ki Pi (t ) + Mi Bi (t ))

0 ⩽ Pch (t ) ⩽ Pch,max Bch (t )

The bool variables Bch(t) and Bdis(t) should satisfy the following constraint:

From equation (28)–(42), it can be seen that the model that needs to be solved is a multi-constraint interval nonlinear optimization problem.

∼ HGF (t ) =

(50)

(47)

In the above equations, Ki is the slope of ith partition of HGF(t), and Mi is the corresponding equivalent intercept. Pi is the ith piecewise point, in which P1 is the minimum output of the gas-fired generator, and PN+1 is the maximum output. The bool variable Bi is a state variable whose value is chosen from 0 or 1. Fig. 2. Linearization of the gas-fired generator.

(ii) Linearization of the ESS 5

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some are absorbed by P2G to convert electric power to natural gas. For some periods (19th and 20th) when electrical load is higher than the power generated by the WT and the electrical price is relatively high, the gas-fired generator will generate electric power by consuming natural-gas. For the ESS, it is charged in some periods (1th-4th and 15th-17th) when the electrical price is relatively low and discharged in some periods (7th-11th and 18th-21th) when the electrical price is relatively high. The performance of these components accords with economical operation of the IENGS, which also reflects the effectiveness of the solution method.

deterministic form, which can be solved by the above method. ¯ ], This paper tries to obtain the interval [CIENGS ] = [ C̲ IENGS , CIENGS namely to calculate the best optimal objective C̲ IENGS and the worst ¯ . optimal objective CIENGS (i) Solving C̲ IENGS The interval optimization objective [CIENGS ] is replaced by the best optimal objective C̲ IENGS . Meanwhile every interval inequality constraint is replaced by the deterministic inequality constraint of its maximum range. Every interval equality constraint is replaced by two inequalities which come from its boundaries. Then utilize the above method to obtain a deterministic MILP model, and solve it by CPLEX.

4.2. Analysis of P2G in the IENGS

¯ (ii) Solving CIENGS

To analyze the influence of P2G, the conversion rate of P2G is changed from 49% to 81% in scenario 2. The electric power consumed by P2G are shown in Fig. 7. It can be learned that under different conversion rates, the consuming power of P2G will change. The P2G absorbs electricity in the periods (2th-4th) when the electrical price is relatively low. From Fig. 7, it can be concluded that the higher the conversion rate of P2G is, the more electric power the P2G device will consume. That is because when the electrical price is low and the conversion rate is high, the P2G tends to absorb more power to generate natural gas to reduce the operation cost of the IENGS. The costs of the IENGS under different conversion rates of P2G are shown in Fig. 8. P2G technology is a promising energy storage solution. To further study its economy, its capital cost and conversion efficiency are analyzed in this part. In Scenario 2, the conversion rate changes from 0.6 to 0.9. The investment return year is applied to analyze the economy of P2G. The investment return year nyear of IENGS can be calculated by:

The interval optimization objective [CIENGS ] is replaced by the worst ¯ . Each interval inequality constraint is replaced optimal objective CIENGS by the deterministic inequality constraint of its minimum range. Meanwhile, each interval equality constraint is replaced by two equalities which come from its boundaries. Every new equality constraint makes up a new model with those new inequalities. That is to say, if the original model owns eq equality constraints, then there will be 2eq new models. Every new model is converted to the deterministic MILP model. The maximum value of the results coming from these new ¯ . models is the worst optimal objective CIENGS The optimization model is implemented in Visual C++ with CPLEX 12.5 version on a PC with Intel Core i7 3.20 GHz CPU and 8 GB RAM. 4. Simulation results and discussion The operating day of the IENGS is divided into 24 h. The outputs of the WT and PV, gas load, electrical load of a typical day are shown in Fig. 3. The purchasing/selling prices of electricity and natural gas are shown in Fig. 4. The total capacity of the ESS is 10 MW. The minimum capacity of the ESS is 1 MW, and its discharging rate and charging rate are both 0.9. The conversion rate of P2G is 0.64. Other parameters of the gas-fired generator, P2G and ESS are shown in Table 1 and Table 2. This paper designs three scenarios that follow: Scenario 1: the IENGS operates under deterministic conditions. Scenario 2: the IENGS operates under deterministic conditions with different conversion rates of P2G. Scenario 3: the IENGS operates with different uncertain levels of the WT and PV.

(57)

PESS (t ) = Pdis (t ) − Pch (t )

(58)

Pgrid (t ) = Pse (t ) − Pbe (t )

(59)

Fgas (t ) = Hse (t ) − Hbe (t )

(60)

140

3000

120

2500

100

2000

80 1500

60

1000

40

500

20

It can be seen that since there exist periods that the total power produced by renewable power energy is more than the electrical load, the IENGS will sell power to the outside system. For some periods (2th and 3th), because the purchasing price is relatively low, and there is redundant power left, some of those power is sold to make profits, while

0

1

5

9 WT

time/h PV

13

17 Eletrical load

21 Gas load

Fig. 3. Curves of the WT, PV, electrical and gas load. 6

0

gas/m3

PP 2G (t ) = −LP 2G (t )

electricity/MW

The results of Scenario 1 are shown in Fig. 5 and Fig. 6. It should be noted that the electric power is positive if it produces power, so does it in the natural-gas part. Hence, the outputs of the gas-fired generator, the P2G, the ESS, the outside power and gas systems can be expressed as: (56)

(61)

where CIENGS_P2G is the operation cost of IENGS with P2G and CIENGS_0 is the operation cost without P2G. nod is the operational days of P2G in a year, and Cinvestment is total the investment cost of P2G. The investment return years of P2G under different conversion rates are shown in Fig. 9. From Fig. 9, it can be seen that the capital cost and the conversion rate of P2G have a large influence on its economy. With different investment costs(¥/MW), the investment return years range from 19 to 57 when the conversion rate is 80%. With different conversion rates, the investment return years range from 17 to 95 when investment cost (¥/MW) is 100000. The shortest investment return year can be obtained when the investment cost(¥/MW) is smallest and the conversion rate is highest. Note that when the investment cost(¥/MW) is relatively high and the conversion rate is relatively low, it takes hundreds of years to cover capital cost of P2G, which cannot satisfy the economic demand of P2G. In this paper, P2G works when the electrical price is relatively low, and the economy of P2G will be improved if there are more periods when the electrical prices are relatively low.

4.1. Results of deterministic conditions

FGF (t ) = −HGF (t )

Cinvestment (CIENGS _P 2G − CIENGS _0 ) nod

n year =

Electrical Power and Energy Systems 119 (2020) 105906

S. Wang and S. Yuan 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

1.4

¥/kW

1.2 1 0.8 0.6 0.4 0.2 0 1

5

9

13

time/h

purchaing price of electricity purchaing price of natural gas

17

Table 2 Operation and maintenance coefficients. ¥/m3

1.8 1.6

21

Components

Coefficients(¥/MW)

Gas-fired generator P2G ESS WT PV

25 23.1 83.2 29.6 9.6

selling prices of eletricity selling prices of natural gas

50

Fig. 4. Curves of purchasing/selling prices of electricity and natural gas.

40

4.3. Analysis of the proposed method power/MW

30

To analyze the effectiveness of the proposed method based on the interval mathematics, the proposed method is compared with MonteCarlo(MC) simulation method, the interval possibility degree method proposed in [26] under ± 5% uncertainty in Scenario 3. The MC simulation method randomly samples power outputs of the WT and PV for 104 trials. It is assumed that solutions of MC simulation are regarded as the real interval results if sufficient numbers of samples are applied. The interval possibility degree of the method in [26] is 0.9. The output of the gas-fired generator, power exchange with the outside power system, gas produced by P2G and gas exchange with the outside gas system are shown in Fig. 10. From the results it can be seen that both of the solutions obtained by the MC simulation method and the optimization method based on interval possibility degree are included in the solutions of the proposed method. It means the proposed method can estimate both the upper and lower bounds of the interval optimization problem, which can show the correctness of the proposed method. The MC simulation has the disadvantage of long-time computation, so the proposed method is applicable in practice.

20 10 0 -10

1

5

9

13

-20

17

21

time/h PESS

PP2G

PGF

Pgrid

Fig. 5. Outputs of the electrical system.

4.4. Analysis of uncertainty of the WT and PV This part aims at analyzing the changes of the operating IENGS when the uncertainty of the WT and PV increases in Scenario 3. When the outputs of the WT and PV variate under 10%, 20% and 30%, the outputs of components in the IENGS are drawn in Figs. 11, Figs. 12 and Figs. 13, respectively. When the uncertainty of the WT and PV increases, the outputs of the components in the IENGS will change. When the uncertainty of the WT and PV changes from −10% to −30%, the output range of the gas-fired generator will enlarge. Because the power generated by the WT and PV decreases, more electric power should be satisfied by the gas-fired generator which correspondingly consumes more natural gas. When the uncertainty of the WT and PV changes from 10% to 30%, the electric power supplied by the WT and PV will increase, there is more electric power left after satisfying the electrical load. P2G will absorb the redundant electric power and convert it to natural gas to satisfy the gas demand. Besides, the output of the ESS changes as the uncertain level changes. If there is still redundant power left, it will be sold to the outside power system to reduce the operation cost of the IENGS. Although the uncertainties come from the electrical system of the IENGS, Fig. 11b, Fig. 12b and Fig. 13b illustrate that these uncertainties will also make the gas system operate with variations. That is because

Fig. 6. Outputs of the gas system.

the coupling components (the gas-fired generator and P2G) will generate or consume electricity, which varies with variations of the WT and PV. Hence, the gas consumed by the gas-fired generator and produced by P2G will variate, making the gas system variate as well. With different uncertainty of the WT and PV, the operation costs of the IENGS are shown in Fig. 14. It can be seen that when the uncertainties of the WT and PV increase, the interval solution of the objective function will be wider. As the uncertainty of the WT and PV increases, the output of the components in IENGS and energy exchange with the outside systems will change, causing the operation and maintenance cost, the environmental cost, the electricity/gas purchase costs and electricity/gas selling incomes change as well. Hence, the objective will range in a more considerable interval. For the decision maker, the optimization results can show the interval that the operation cost of the IENGS will fall in under

Table 1 Parameters of components in the IENGS. Components

Minimum output (MW)

Maximum output (MW)

Ramping up power (MW)

Ramping down power (MW)

The gas-fired generator P2G ESS

0 0 0

15 10 2

5 5 /

5 5 /

7

Electrical Power and Energy Systems 119 (2020) 105906

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power/MW

40 30 20 10 0 -10

1

5

9

13

-20 PGF-MC simulation(+5%) PGF-based on interval possiblilty degree PGF-proposed method(-5%) Pgrid-proposed method(+5%) Pgrid-MC simulation(-5%)

time/h

17

21

PGF-proposed method(+5%) PGF-MC simulation(-5%) Pgrid-MC simulation(+5%) Pgrid-based on interval possiblilty degree Pgrid-proposed method(-5%)

Fig. 10a. Electrical system with ± 5% uncertain level. Fig. 7. Electricity consumed by P2G. 500

gas/m3

-500

1

5

9

13

17

21

-1500 -2500 -3500 -4500 FP2G-MC simulation(+5%) FP2G-based on interval possiblilty degree FP2G-proposed method(-5%) Fgas-proposed method(+5%) Fgas-MC simulation(-5%)

Fig. 8. The costs of the IENGS.

time/h

FP2G-proposed method(+5%) FP2G-MC simulation(-5%) Fgas-MC simulation(+5%) Fgas-based on interval possiblilty degree Fgas-proposed method(-5%)

Fig. 10b. Gas system with ± 5% uncertain level.

a certain level of WT and PV uncertainties.

(1) When there is redundant electricity left and the electrical price is relatively low, the high conversion rate of P2G can lower the cost of the IENGS. (2) The uncertainties which come from the electrical system of the IENGS will be delivered to the gas system, making the whole IENGS operate with uncertainties. (3) When the uncertainty of the WT and PV increases, the outputs of components will operate in a larger range, and the operation cost of the IENGS will fall in a wider interval.

5. Conclusion This paper builds the operating optimization model of the IENGS including ESS, P2G device and the gas-fired generator. The uncertain outputs of the WT and PV are taken into consideration and the interval optimization model is proposed. To solve this model, the problem is converted to a MILP model and solved by the interval linear programming method. The numerical test shows that the results of the IENGS under deterministic conditions. The influence of P2G and the uncertainty of the WT and PV are analyzed. The results lead to the following conclusions:

This paper only studies day-ahead scheduling of the IENGS. The studies for multi-time-scale optimization of the IENGS considering the temporary storage characteristics of the natural gas pipelines will be developed in our future work.

Fig. 9. Investment return years with different conversion rates and investment costs. 8

Electrical Power and Energy Systems 119 (2020) 105906

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100

50

80 60

30

power/MW

power/MW

40

20 10

20 0

0 -10

40

1

5

9

13

-20

17

21

-20

1

9

13

-40

time/h

17

21

time/h

PESS(+10%)

PESS(-10%)

PP2G(+10%)

PP2G(-10%)

PESS(+30%)

PESS(-30%)

PP2G(+30%)

PP2G(-30%)

PGF(+10%)

PGF(-10%)

Pgrid(+10%)

Pgrid(-10%)

PGF(+30%)

PGF(-30%)

Pgrid(+30%)

Pgrid(-30%)

Fig. 11a. Electrical system with ± 10% uncertain level.

Fig. 13a. Electrical system with ± 30% uncertain level. 500

1000

-500 1

0 1

5

9

13

17

21

5

9

13

17

21

-1500

-1000

-2500

gas/m 3

gas/m 3

5

-2000 -3000

-3500 -4500 -5500

-4000

-6500

-5000

-7500

time/h

time/h

FP2G(+10%)

FP2G(-10%)

Fmt(+10%)

FP2G(+30%)

FP2G(-30%)

Fmt(+30%)

Fmt(-10%)

Fgas(+10%)

Fgas(-10%)

Fmt(-30%)

Fgas(+30%)

Fgas(-30%)

Fig. 13b. Gas system with ± 30% uncertain level.

65

1000000

55

800000

45

600000

35

400000

25

200000

15

¥

power/MW

Fig. 11b. Gas system with ± 10% uncertain level.

0

5 -5 1

5

9

13

17

-200000

21

±10%

±20%

±30%

-400000

-15 -25

-600000

time/h

-800000

PESS(+20%)

PESS(-20%)

PP2G(+20%)

PP2G(-20%)

PGF(+20%)

PGF(-20%)

Pgrid(+20%)

Pgrid(-20%)

different uncertain levels lower bound of [CIENGS]

upper bound of [CIENGS]

Fig. 14. The costs under different uncertainty of the WT and PV.

Fig. 12a. Electrical system with ± 20% uncertain level. 1000

CRediT authorship contribution statement

0 1

5

9

13

17

Shouxiang Wang: Conceptualization, Validation, Resources, Writing - review & editing, Supervision, Project administration. Shuangchen Yuan: Methodology, Software, Formal analysis, Investigation, Data curation, Writing - original draft.

21

-1000

gas/m3

0

-2000 -3000 -4000

Declaration of Competing Interest

-5000 -6000

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.

time/h FP2G(+20%)

FP2G(-20%)

Fmt(+20%)

Fmt(-20%)

Fgas(+20%)

Fgas(-20%)

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Fig. 12b. Gas system with ± 20% uncertain level.

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