Interval optimization‐based scheduling of interlinked power, gas, heat, and hydrogen systems
2021; Institution of Engineering and Technology; Volume: 15; Issue: 6 Linguagem: Inglês
10.1049/rpg2.12101
ISSN1752-1424
AutoresNima Nasiri, Ahmad Sadeghi Yazdankhah, Mohammad Amin Mirzaei, Abdolah Loni, Behnam Mohammadi‐Ivatloo, Kazem Zare, Mousa Marzband,
Tópico(s)Renewable energy and sustainable power systems
ResumoIET Renewable Power GenerationVolume 15, Issue 6 p. 1214-1226 ORIGINAL RESEARCH PAPEROpen Access Interval optimization-based scheduling of interlinked power, gas, heat, and hydrogen systems Nima Nasiri, Nima Nasiri Department of Electrical Engineering, Sahand University of Technology, Tabriz, IranSearch for more papers by this authorAhmad Sadeghi Yazdankhah, Corresponding Author Ahmad Sadeghi Yazdankhah sadeghi@sut.ac.ir Department of Electrical Engineering, Sahand University of Technology, Tabriz, Iran Correspondence Ahmad Sadeghi Yazdankhah, Department of Electrical Engineering, Sahand University of Technology, Tabriz, Iran. Email: sadeghi@sut.ac.irSearch for more papers by this authorMohammad Amin Mirzaei, Mohammad Amin Mirzaei Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, IranSearch for more papers by this authorAbdolah Loni, Abdolah Loni Faculty of Mathematics Sciences and Computer, Allameh Tabataba'i University, Tehran, IranSearch for more papers by this authorBehnam Mohammadi-Ivatloo, Behnam Mohammadi-Ivatloo Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran Department of Energy Technology, Aalborg University, Aalborg, DenmarkSearch for more papers by this authorKazem Zare, Kazem Zare Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, IranSearch for more papers by this authorMousa Marzband, Mousa Marzband Northumbria University, Electrical Power and Control Systems Research Group, Newcastle upon Tyne, UK Center of Research Excellence in Renewable Energy and Power Systems, King Abdulaziz University, Jeddah, Saudi ArabiaSearch for more papers by this author Nima Nasiri, Nima Nasiri Department of Electrical Engineering, Sahand University of Technology, Tabriz, IranSearch for more papers by this authorAhmad Sadeghi Yazdankhah, Corresponding Author Ahmad Sadeghi Yazdankhah sadeghi@sut.ac.ir Department of Electrical Engineering, Sahand University of Technology, Tabriz, Iran Correspondence Ahmad Sadeghi Yazdankhah, Department of Electrical Engineering, Sahand University of Technology, Tabriz, Iran. Email: sadeghi@sut.ac.irSearch for more papers by this authorMohammad Amin Mirzaei, Mohammad Amin Mirzaei Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, IranSearch for more papers by this authorAbdolah Loni, Abdolah Loni Faculty of Mathematics Sciences and Computer, Allameh Tabataba'i University, Tehran, IranSearch for more papers by this authorBehnam Mohammadi-Ivatloo, Behnam Mohammadi-Ivatloo Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran Department of Energy Technology, Aalborg University, Aalborg, DenmarkSearch for more papers by this authorKazem Zare, Kazem Zare Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, IranSearch for more papers by this authorMousa Marzband, Mousa Marzband Northumbria University, Electrical Power and Control Systems Research Group, Newcastle upon Tyne, UK Center of Research Excellence in Renewable Energy and Power Systems, King Abdulaziz University, Jeddah, Saudi ArabiaSearch for more papers by this author First published: 23 February 2021 https://doi.org/10.1049/rpg2.12101Citations: 5AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract The combined heat and power (CHP) plant is one of the emerging technologies of gas-fired units, which plays an important role in reducing environmental pollutants and delivering high energy efficiency. Moreover, the hydrogen energy storage (HES) system with extra power storage from wind turbine via power to hydrogen technology allows the injection of stored energy into the power grid by reverse hydrogen to power services, offsetting in this way the uncertainty of wind power. Consequently, simultaneous usage of CHP and HES units not only makes the maximum use of wind power distribution but also increases flexibility and reduces the operating costs of the entire network. Therefore, this paper proposes an interval optimization technique for managing the uncertainty of wind power generation in the integrated electricity and natural gas (NG) networks considering CHP–HES. Moreover, to enhance the flexibility of the NG network, a linearized Taylor series-based model is proposed for modelling linepack of gas pipelines in the proposed scheduling framework that is formulated mixed-integer linear programming and solved using the Cplex solver. The obtained results indicate that the simultaneous use of CHP–HES in the day-ahead scheduling reduces the operating cost and increases the flexibility of the whole network. 1 INTRODUCTION The desire to provide safe, efficient, and sustainable energy calls for dramatic changes in energy networks. In line with this, with the technological advancement of multiple energy systems (MES) across a spectrum range of disciplines, it is possible to establish a physical connection between various energy networks such as electricity, natural gas (NG), hydrogen, and local heating. Such an initiative will lessen the barriers to traditional non-integrated networks. As a result, the entire energy supply chain in modern society has undergone a rapid transition to an integrated energy network. One of the most important technologies in the integration of energy networks is the combined heat and power (CHP) units. These units are utilized in industries to provide electricity and heat at the same time. The heat generated by the recovery of waste heat is obtained in the process of producing electrical energy. This method leads to a decrease in the cost of supplying electrical and heat demand, as well as reducing greenhouse gas emissions. Reports demonstrate that using CHP units instead of conventional production units results in gaining a maximum efficiency of up to 90% [1]. Also, CHP units reduce the emission of pollutants by 13–18% [2]. In recent years, many studies have been conducted on the coordination of integrated electricity and NG networks. The authors in [3] have investigated the impact of NG network constraints on the unit commitment (UC) in power grids. A mixed-integer linear programming (MILP) model has been considered in [4] to study the impact of applying the electric storage system to integrated electricity and NG systems with the aim of increasing system reliability and pressure control in NG network pipelines. In [5], a two-step multi-objective problem has been investigated on the unit's commitment of integrated electricity and gas networks, taking into consideration the flexible energy sources such as the power to gas (P2G) and demand response (DR) program, as well as high permeability level of the wind energy source. A coordinate-decomposition-based framework is proposed to study the optimization performance of integrated electricity and NG systems in [6]. In this framework, a robust distributed optimization model is presented based on the existing data to solve the problem of power system scheduling, considering wind energy uncertainty. A robust security-constrained UC model based on info-gap diction theory (IGDT) has been provided in [7]. Numerical analysis shows that flexible resources such as compressed air energy storage (CAES) and DR lead to a reduction in operating costs and management of wind power uncertainty. A stochastic day-ahead scheduling approach has been proposed for the hourly dispatch of power plants and deploys flexible ramping for the management of renewable energy sources in integrated electricity and NG networks [8]. The conducted research studies show that the real-time distribution of NG can directly affect the hourly distribution of deploying flexible ramping and the operating costs of networks. In [9], a market-clearing model constrained by the restrictions of the electricity and NG networks with a two-step stochastic unit commitment approach has been discussed considering the impact of CAES to increase network flexibility. A bi-level scheduling is suggested in [10] for integrated NG and electricity systems. The purpose of this bi-level problem is to minimize the costs of investing in wind farms, P2G equipment, NG storage units, as well as day-ahead market operating costs. The authors in [11] provided a context to evaluate the impact of different types of economic, environmental, security, and sustainability indicators on the integrated performance of integrated energy systems considering the constraints of electricity, NG, and district heating networks. In [12], to improve the system performance and optimize energy flow, a coordinated strategy has been proposed based on a non-probabilistic optimization model considering the DR program for integrated electricity and NG systems. The authors in [13] have investigated a non-linear scheduling problem for electricity and NG systems, where the uncertainty of the electricity price is managed by applying the IGDT method. In [14], a hybrid IGDT-stochastic approach for integrated power and NG systems has been presented to reduce the total operating costs of the integrated system and increase the permeability of wind turbines by applying P2G technology. In [15], a two-stage iterative-based algorithm for the interaction of integrated electricity and natural gas networks in the presence of the energy hub system under the approach of stochastic uncertainty is presented. A two-stage stochastic approach to the operation of integrated power and natural gas networks considering interconnected hubs is presented in [16]. Many researchers have been focused on the optimal operation of CHP units in heat- and power-based energy systems. The impact of CHPs in the UC problem has been analyzed in [17]. An IGDT approach has been presented to evaluate the profit-oriented strategy for CHP units in an electricity market [18]. A market-clearing model for integrated electricity and NG networks considering CHP and P2G technologies was provided in [19] to minimize the expected operating costs. In [20], a DC power flow has been utilized in the problem of energy pricing in electricity, NG, and heat networks in the presence of CHP units and limits of pollutant emissions. In [21], robust scheduling to optimize the performance of CHP with a demand response program aimed at reducing operating costs was presented. A unit commitment problem for CHP units with the aim of reducing pollutant emission was presented in [22]. In [23], a non-linear approach has been provided for optimizing photovoltaic heating systems and the CHP system with the aim of maximizing profits using a demand retrospective program. The authors in [24] have presented multi-objective scheduling for optimal performance of CHP system and energy storage systems (ESSs) in the presence of DR program with the goals of minimizing CHP operation costs and minimizing pollutant emission costs. In [25], mixed-integer non-linear programming was presented to optimize the day-ahead integrated electrical–water–heat systems to minimize the operating costs of the CHP and the fuel cost for freshwater. It also evaluates the impact of the hybrid vehicle and DR program on the target system. Hydrogen energy storage (HES) technology plays a major role in strengthening the balance between generation and consumption of energy. Much research has been done on the optimal operation of HES technology, for example, the authors in [26] proposed the optimal scheduling for an intelligent parking lot (IPL) considering the demand response program and the uncertainty derived from the energy price of the upstream network. In [27], risk-averse stochastic exploitation of HES in the presence of the wind energy sources has been presented. In addition, the demand response program is considered utilizing a scenario-based stochastic approach. In [28], a stochastic UC problem has been modelled considering security constraints, HES system, and price-based DR program. In [29] a multi-objective approach to the optimal scheduling of hybrid renewable energy systems, including wind turbines, solar panels, fuel cells, electrolysis, hydrogen storage system, and electrical storage systems, is presented. The authors in [30] have proposed optimal stochastic scheduling to study the coordination impact of hydrogen storage systems, diesel generators, solar panels, water electrolyzer, fuel cell (FC), and electric vehicle. The results show that most of the solar energy is consumed by hydrogen storage and reduces the operating costs. The authors in [31] have proposed a stochastic approach in IPL integrated with the HES to minimize the cost of purchasing energy from the upstream grid using a particle swarm optimization (PSO) algorithm. In [32], an energy management system for optimal operation of photovoltaic, battery, and hydrogen storage systems using PSO algorithm is presented. In [33], optimal scenario-based management was presented for a grid-connected microgrid with various RESs such as FC, wind turbine, microturbine, and electrical storage system to improve energy management and reduce microgrid costs. To the best knowledge of the authors, none of the reviewed works have examined the synergy between the HES system and the integrated electricity and NG networks in the presence of the CHP unit and linepack flexibility. The main gaps in the reviewed literature can be summarized as follows: In some works, for example, [3-9, 11-14], the problem of optimal scheduling of integrated electricity and NG systems without considering the linepack system has been investigated. The existence of linepack system in natural gas networks is very useful and increases the flexibility of NG systems and generation units, especially in critical times of the NG network. In addition, the linepack system reduces the total operating costs of the integrated electricity and NG system. In some studies, for example, [16-19, 21-24], the problem of optimal generation scheduling of CHP units has been evaluated without considering the constraints of the NG network. Constraints of the NG network have a significant impact on the commitment of units in the power grid. Ignoring the constraints of the NG network in scheduling the commitment of units leads to unrealistic and careless results. In some literature, for example, [25, 26, 28-34], the problem of optimal scheduling of HES systems without considering the constraints of the power and NG grids has been investigated. Ignoring such constraints cannot completely describe the benefits of HES in optimal scheduling of the integrated energy systems. To cover these gaps, here, an interval optimization technique is proposed for the day-ahead scheduling of integrated electricity and NG networks considering HES and CHP units. In addition, the linepack technology is applied to increase the flexibility of the power and NG system. The main contributions of this paper are as follows: Investigating the impact of the HES system on the day-ahead scheduling of integrated electricity and NG networks aiming to minimize the cost of operating costs of both networks with CHP unit and wind energy sources. Performing an interval optimization technique to handle the uncertainty of wind energy production and its impact on the operating costs of the whole network. The proposed interval approach is formulated as a multi-objective optimization problem in which the average cost and cost deviation are minimized simultaneously. Evaluation of gas system flexibility equipped with linepack technology on power dispatch of gas-fired and non-gas-fired units in critical times of NG network. The remaining is organized as follows: Section 2 introduces HES. Section 3 presents the problem description and formulation. Section 4 revolves around an interval optimization technique to estimate the existing uncertainties. Section 4 describes the results and discussion regarding the proposed model. Ultimately, Section 5 concludes the paper. 2 HYDROGEN STORAGE TECHNOLOGY The HES technology, in addition to emissions reduction, can play an important role in securing network demand–supply. As shown in Figure 1, HES technology converts electrical energy into hydrogen by electrolyzer in periods of off-peak and high wind energy generation, then stores it in a hydrogen storage tank. In this way, during periods of on-peak and low wind energy production, the stored energy can be converted to electric power by the fuel cell and is injected into the grid. This operation, while optimally managing wind uncertainty, can play an important role in reducing the generation power of expensive power units. A unique feature of HES compared to other ESSs is that it can be used in hydrogen-dependent industries or injected into the NG network for residential gas consumers [24]. FIGURE 1Open in figure viewerPowerPoint HES system performance diagram 3 PROBLEM DESCRIPTION AND FORMULATION In this research, it is assumed that the optimal scheduling of the integrated energy system is the responsibility of a central system operator (CSO). The CSO holds comprehensive information on the operation of the power grid and NG network. Based on the available data, the CSO performs the optimal scheduling of the integrated system in a day-ahead time horizon. As illustrated in Figure 2, three types of power generating units are considered in this study: (a) CHP unit, (b) gas-fired power plant (GFPP), and (c) non-gas-fired power plant (NGFPP). Physically, power and NG networks are connected via CHP and GFPP. In this research, linepack technology has been used to increase system reliability and pressure constraint security in NG pipelines. Linepack technology enhances the flexibility of the NG system by storing some of the gas in the network pipelines. In addition, we have used HES technology to increase the security of supply and demand in the power grid as well as to absorb the wind power overcapacity. FIGURE 2Open in figure viewerPowerPoint Integrated power and NG networks considering wind–HES–CHP 3.1 Objective function The objective in problem formulation is to minimize the costs of (i) NGFPP cost and startup/shutdown of NGFPP, (ii) NG producers, (iii) HES costs. min ∑ t ∑ i ∈ C U ( F C i , t + S U i , t + S D i , t ) + ∑ s p γ s p g a s V s p , t + ∑ h ρ h H E S P h , t H 2 P (1) The first term of Equation (1) concerns the operating cost and startup/shutdown of power plants resulting from the electricity generation cost of NGFPP. The second term deals with the producer costs of NG (NG wells). The third term is the cost of HES in discharge mode. The various sets of constraints are presented below. 3.2 Generating unit constraints The constraint in Equation (2) relates to the limitation of power units generation, and Equations (3) and (4) are related to the startup/shutdown cost of NGFPP. Furthermore, Equations (5) and (6) revolve around the startup/shutdown cost of GFPP, and Equations (7) and (8) set the startup/shutdown modes of all units. Equations (9) and (10) are related to the rate of ramp-up and ramp-down in the units' generation power.The linearized constraints in Equations (11) and (12) represent the number of hours required by generation unit i startup and shutdown at the beginning of the study horizon. Equation (13) applies the minimum ON time requirement if generation unit i is on-line at the beginning of the study horizon. Equation (14) applies the minimum ON time requirement for all consecutive sets of hours of cardinality T i o n . Equation (15) applies the minimum ON time requirement for the final T i o n hours of the study horizon. Equation (16) applies the minimum OFF time requirement if generation unit i is off-line at the beginning of the study horizon. Equation (17) applies the minimum OFF time requirement for all consecutive sets of hours of cardinality T i o f f . Equation (18)applies the minimum OFF time requirement for the final T i o f f hours of the study horizon. P i M i n I i , t ≤ P i , t ≤ P i M a x I i , t ∀ i ∈ C U , G U , ∀ t (2) S U i , t ≥ C i S U y i , t ∀ i ∈ C U , ∀ t (3) S D i , t ≥ C i S D z i , t ∀ i ∈ C U , ∀ t (4) G S U i , t ≥ C i G S U y i , t ∀ i ∈ G U , ∀ t (5) G S D i , t ≥ C i G S D z i , t ∀ i ∈ G U , ∀ t (6) y i , t − z i , t = I i , t − 1 − I i , t ∀ i , ∀ t (7) y i , t + z i , t ≤ 1 ∀ i , ∀ t (8) P i , t − P i , t − 1 ≤ ( 1 − y i , t ) R i U P + y i . t P i M i n ∀ i , ∀ t (9) P i , t − 1 − P i , t ≤ ( 1 − z i , t ) R i D N + z i , t P i M i n ∀ i , ∀ t (10) L i o n = min T , ( T i o n − T i , 0 o n ) I i , 0 (11) L i o f f = min T , ( T i o f f − T i , 0 o f f ) ( 1 − I i , 0 ) (12) ∑ t ∈ L i o n ( 1 − I i , t ) = 0 ∀ i (13) ∑ t = r t + T i o n − 1 I i , r ≥ T i o n ( I i , t − I i , t − 1 ) ∀ i , ∀ t ∈ L i o n + 1 , … . , T − T i o n + 1 (14) ∑ t = r T ( I i , r − ( I i , t − I i , t − 1 ) ) ≥ 0 ∀ i , ∀ t ∈ T − T i o n + 2 , … . , T (15) ∑ t ∈ L i o f f I i , t = 0 ∀ i (16) ∑ t = r t + T i o f f − 1 ( 1 − I i , r ) ≥ T i o f f ( I i , t − 1 − I i , t ) ∀ i , ∀ t ∈ L i o f f + 1 , … . , T − T i o f f + 1 (17) ∑ t = r T ( 1 − I i , r − ( I i , t − 1 − I i , t ) ) ≥ 0 ∀ i , ∀ t ∈ T − T i o f f + 2 , … . , T (18) 3.3 Constraints of electricity grid Equation (19) indicates the constraint of electricity grid balance and Equation (20) describes the limitation of power flow on lines. Further, Equation (21) concerns DC power flow in the power grid and Equation (22) defines the phase angle of the slack bus, also Equation (23) relates to the production constraint of the wind power plant. ∑ j ∈ T r f b , j , t + ∑ h ∈ A b H E S P h , t P 2 H + P b , t L o a d = ∑ i ∈ A b i P i , t + ∑ w ∈ A b w P W w , t + ∑ h ∈ A b H E S P h , t H 2 P ∀ b , ∀ t (19) − f b M a x ≤ f b , j , t ≤ f b M a x ∀ ( b , j ) ∈ T r , ∀ t (20) f b , j , t = ( δ b , t − δ j , t ) / X L ∀ ( b , j ) ∈ T r , ∀ t (21) δ r e f , t = 0 ∀ t (22) 0 ≤ P W w , t ≤ P W w max ∀ w , ∀ t (23) 3.4 Constraints of CHP unit The day-ahead scheduling constraints for CHP system are presented Equations (24) and (25).The amount of electric power and heat energy production in CHP unit is interdependent and is calculated by the feasible CHP operation region. As shown in Figure 4, the operating area of a CHP unit can be described by a polyhedron characteristic. Equations (24) and (25), respectively, show how CHP generates electricity and heat depending on the characteristic of combined points in the CHP operating area. The non-negative coefficient of a t k , constrained by (26) and (27), expresses the CHP unit commitment. In addition, Equation (28) demonstrates the balance of heating energy that is fully supplied by the CHP [35]. P i , t = ∑ k = 1 N K α t k P k ∀ i ∈ C H P , ∀ t (24) H i , t = ∑ k = 1 N K α t k Q k ∀ i ∈ C H P , ∀ t (25) ∑ k = 1 N K α t k = I i , t ∀ i ∈ C H P , ∀ t (26) 0 ≤ α t k ≤ 1 ∀ i ∈ C H P , ∀ t (27) ∑ i = 1 H i , t = H t l o a d ∀ i ∈ C H P , ∀ t (28) FIGURE 3Open in figure viewerPowerPoint Modified 6-bus power grid with 6-node NG network FIGURE 4Open in figure viewerPowerPoint Operation area of the CHP unit 3.5 The constraints of nodes and NG flow The pressure constraints of NG network nodes are given in Equation (29). According to Equation (30), NG flow can be expressed as a function of the square of pressure and the characteristics of pipe such as length, diameter, and friction coefficient. Equation (30) is referred to as the general flow [36] that can be approximated by Weymouth equations under certain conditions. This model, given the function of sgn (mentioned in Equation (31)), allows the flow to be bidirectional. It is important to mention that Equation (30) is non-convex in addition to being non-linear. P r n M i n ≤ P r n , t ≤ P r n M a x (29) q n , m , t = s g n ( P r n , P r m ) K n , m f P r n , t 2 − P r m , t 2 ∀ n , ∀ t (30) sgn ( Pr n , Pr m ) = 1 , Pr n ≥ Pr m − 1 , Pr n ≤ Pr m ∀ ( n , m ) ∈ z (31) The non-linearity and non-convexity of the gas flow equation make the pricing of NG more difficult. Therefore, we use an outer approximation approach based on the Taylor series at the fixed pressure points to linearize the Weymouth equation [36] and present a globally optimal solution. q n , m , t ≤ K n , m f P R n , u P R n , u 2 − P R m , u 2 Pr n , t − K n , m f P R m , u P R n , u 2 − P R m , u 2 Pr m , t ∀ ( n , m ) ∈ z , ∀ t (32) Here, u is set of pressure fixed points ( PR n , u ′ , PR m , u ) [37]. However, the limitation of the gas flow is approximated by Equation (32). The sgn function is ignored in (31) because of non-linearity. Hence, to guarantee the bidirectional flow of gas in the pipeline, defining an equation is vital. Consequently, to this end, inequalities Equations (33)–(36) are used to ensure the bidirectional flow of the network [36]. q n , m , t = q n , m , t + − q n , m , t − (33) q n , m , t − = M ( 1 − y n , m , t ) ∀ ( n , m ) ∈ z , ∀ t (34) q n , m , t + = M y n , m , t ∀ ( n , m ) ∈ z , ∀ t (35) y n , m , t ∈ 1 , 0 ∀ ( n , m ) ∈ z , ∀ t (36)where q ( n , m , t ) + denotes the gas flow in the pipeline from node n to m and vice versa for q ( n , m , t ) − . The parameter M is a large constant number and Equations (33) fulfills the function of sgn. Equations (34) and (35) ensure that only one of the two variables q ( n , m , t ) − , q ( n , m , t ) + is non-zero. In addition to the above mentioned limitations, the following inequalities should be defined [36]: q n , m , t + ≤ K n , m f P R n , u P r n , t P R n , u 2 − P R m , u 2 − K n , m f P R m , u P r m , t P R n , u 2 − P R m , u 2 + M ( 1 − y n , m , t ) ∀ ( n , m ) ∈ z ∀ ( n , m ) ∈ z m < n m < n , ∀ u , ∀ t (37) q n , m , t − ≤ K n , m f P R m , u P r m , t P R m , u 2 − P R n , u 2 − K n , m f P R n , u P r n , t P R m , u 2 − P R n , u 2 + M ( y n , m , t ) ∀ ( n , m ) ∈ z ∀ ( n , m ) ∈ z m > n m > n , ∀ u , ∀ t (38) The linear Equations (37) and (38) state the direction of gas flow, specified by binary variables. In addition, two positive variables q ( n , m , t ) + , q ( n , m , t ) − are determined for the flexibility of linepacks to specify inflow and outflow [36]. q n , m , t + = q n , m , t i n − q n , m , t o u t 2 ∀ ( n , m ) ∈ z , ∀ t (39) q n , m , t − = q m , n , t i n − q m , n , t o u t 2 ∀ ( n , m ) ∈ z , ∀ t (40) One of the unique features of NG networks is linepack that can serve as temporary storage (an economical way to store energy). The linepack system indicates the ability to store a certain amount of NG in the pipeline and is very important for short-term NG network operation [36]. The linepack system indicates the ability to store a certain amount of NG in the pipeline and is very important for short-term NG network operation. h n , m , t = K n , m f P r n , t + P r m , t 2 ∀ ( n , m ) ∈ z , ∀ t (41) h n , m , t = h n , m , t − 1 + q n , m , t i n − q n , m , t o u t ∀ ( n , m ) ∈ z , ∀ t ≥ 1 (42) h n , m , t = h n , m , 0 + q n , m , t i n − q n , m , t o u t ∀ ( n , m ) ∈ z , ∀ t = 1 (43) h n , m , s , t ≤ h n , m , s , 0 ∀ ( n , m ) ∈ z , ∀ t (44) Equation (41) shows that the linepack system is directly related to the average pressure in the pipeline. Therefore, increasing the pressure in a pipeline node leads to an increase in the linepack and vice versa. Moreover, Equations (42) and (43) show that the linepack, in addition to, Equation (39) is equal to the difference between the pipeline's inflow and outflow. Furthermore, the initial value of linepack is represented by Equation (44). 3.6 Other technical constraints of the NG network The constraint in Equation (45) relates to the limitation of gas generated by NG wells, whereas, Equation (46) specifies the energy balance in NG production and consumption. Furthermore, Equation (47) and (48) indicate the coupling constraints of electricity and NG
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