# Stochastic Coordinated Management of Electrical–Gas–Thermal Networks in Flexible Energy Hubs Considering Day-Ahead Energy and Ancillary Markets

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

- Coordination between sources and active loads (ESS, EV, and DRP) in the form of energy hubs can provide technical and economic benefits for each element. For example, the presence of ESSs in an EH including renewable resources can increase the flexibility of the EH in energy networks.
- Appropriate energy management for EHs can provide the proper capabilities for the presence of hubs in energy networks such as electricity, gas, and thermal networks in terms of technical indexes (operation, reliability, flexibility, security, and economic).

#### 1.2. Literature Review

#### 1.3. Research Gaps

- The energy hubs can participate in the ancillary services and energy markets to make profits; however, this aspect has been less considered in most research, which has focused on how to minimize the cost of EH operation within the networks. Moreover, studies that deal with this issue have also focused on the participation of EHs in the energy market. However, there are different sources, storages, and DRPs that can control their active and reactive power. In addition, EH can store its excess power production relative to the network energy consumption; therefore, it can play a role in reserve regulation [8]. Therefore, EHs can also participate in the ancillary services market and obtain the desired financial benefit. However, this issue has rarely been discussed in other studies.
- The presence of RES in EH reduces its flexibility in the electrical sector. The dependence of the CHP thermal power on active power also reduces the flexibility of EH in the thermal sector. The low flexibility of EH leads to an unequal conclusion between DA and real-time (RT) operation. It is possible that the balance between production and supply in the energy network, including EHs, is not achieved in RT operation. Therefore, in various studies [1,5], the use of flexible sources such as storage, DRP, and renewable sources (CHP for RES, boiler for CHP) in EH has been suggested. These sources can enhance EH flexibility by RES active and CHP thermal power fluctuations’ compensation in RT operation relative to DA scheduling operation. This has been stated in most studies, but fewer studies, such as [11], have concentrated on system flexibility modeling. Obtaining the values of an index needs the application of its mathematical model to the problem. By achieving the values of the index, an accurate and reliable evaluation of its pros and cons can be provided.
- In the issue of grid-connected EHs management, there are different uncertainty parameters such as load, market price, reserve demand, mobile storage energy demand, and renewable power. Therefore, it has a high number of uncertainties. In addition, the precise calculation of some indicators, such as flexibility, is required to examine different uncertainty scenarios; therefore, a reliable optimal solution can be achieved using stochastic programming. This is also not achieved by robust modeling such as ARO and IGDT, which have only the worst-case scenario. Energy management of EH is an operation problem of the power system. In this type of problem, the execution step is low and, in some cases, it reduces up to 15 min. Therefore, it is of particular importance to achieve the optimal solution in the shortest computational time. In proportion to these two cases, uncertainty modeling methods obtaining the minimum number of scenarios are required. The choice of point estimation methods (PEM) is appropriate in this regard; however, its usage in the proposed problem has been considered in less research.

#### 1.4. Contributions

- The evaluation of flexible EH ability on energy networks’ operation status.
- Simultaneous participation of flexible energy hubs in reactive power, DA reserve regulation, and energy markets to obtain high financial benefit for resources, storage devices, and aggregated responsive loads in EH format.
- Presenting a two-level problem for energy management modeling of different networks in the presence of EHs in order to simultaneously model EHOs and ENOs’ objectives.
- Using PEM for many uncertainties modeling in network-connected EHs’ energy management problems to achieve a reliable optimal solution in low computational time.

#### 1.5. Paper Organization

## 2. Proposed Scheme Formulation

- (A)
- Upper-level problem

^{ES}, Q

^{ES}, H

^{HS}, and G

^{GS}are non-zero just for the bus (node) o.

- (B)
- Lower-level problem

^{EH}, Q

^{EH}, H

^{EH},and G

^{EH}. The objective function (15) refers to the EHs’ maximization in DA reserve regulation, reactive power, and energy markets. In the first line of this equation, the EHs’ financial benefit from the energy market (first, third, and fourth item) [6] and DA reactive power (second item) have been formulated. The second level of Equation (15) calculates the EHs’ financial benefit obtained from the electrical (first item), thermal (second item), and gas (third item) DA reserve regulation market. In this equation, the profit in each section and hour (t) is equal to the market price and power of its section. This equation then calculates the profit (expense) of that section if the power has a positive (negative) value. According to [24], the reactive power price (K

_{Q}) has been considered as a factor of electrical energy price (λ

^{E}) in this equation.

^{R}) [1]. This is because they have negligible operating costs. In this article, RES cost has been considered as zero. In addition, they are generally connected to the grid using an electronic power converter. Therefore, RESs’ active and reactive power can be controlled simultaneously using the mentioned converter [25]. Therefore, in the RESs’ model, it is necessary to model their reactive power limit, as achieved by Equation (25). The energy storage [26,27] operation model has been considered in Equations (26)–(29) [9,11], in which (26)–(28) are the same for EES and TES, and (29) can be used only for EES. Equations (26) and (27) express storage devices’ charge and discharge rate limits, respectively. Equations (28) and (29) model the limitation of storable energy in storage devices and controllable reactive power by EES charger, respectively. In this paper, it is assumed that the EES charger has an active front-end rectifier [25]. Thus, it is able to control EES active and reactive power simultaneously.

^{EES}, DR

^{EES}, and so on [3,6]. Because it is possible that different EVs can be connected to the EH at any time and scenario, CR

^{EES}at hour t is equal to the total EVs’ charge rate connected to the network at this hour. The same equation holds for DR

^{EES}, $\underset{\xaf}{E}$, $\overline{E}$, ${\overline{Q}}^{E}$, and ${\underset{\xaf}{Q}}^{E}$, but EI at hour t is equal to the total new EVs’ primary energy connected to the grid at this hour [3,6]. Equations (30) and (31) express the DRP operation model for electrical, thermal, and gas consumers in EH [3,11]. In this problem, DRP is based on encouragement. The consumers participating in DRP shift their energy consumption from hours with a high price (corresponding to peak hours) to hours with a low price (corresponding to off-peak hours). Therefore, in accordance with this explanation, Equation (30) refers to the power control range in the DRP scheme. Equation (31) also ensures that the total reduced energy consumption of consumers for peak hours is provided by EH in off-peak hours. The value of the reserve variable must always be positive based on Equation (32). Finally, EHs’ flexibility limitation in the electrical and thermal section has been expressed by Equation (33). Low flexibility leads to the unequal result of DA operation and real-time. As a result, it is possible to have an imbalance between production and consumption in real-time operation. To compensate for this, flexible resources such as storage, DRP, and renewable resources have to compensate or eliminate the fluctuations of RESs’ active power and CHPs’ thermal power in each scenario compared to the corresponding scenario with the deterministic model (first scenario with the predicted value of uncertainty parameter). Under these conditions, it is expected that the difference between EHs’ active (thermal) power in each scenario compared to the first scenario is low and within the flexibility tolerance (ΔF). The 100% flexibility is obtained if ΔF is equal to 0 per unit. It is noteworthy that uncertain power generation of renewable resources makes the day-ahead and real-time operation results different. Therefore, generation and consumption in real-time operation might be unbalanced. This condition is known as a low flexibility case. To compensate for this, there is a need for elements capable of power control, such as storage devices, so that they can deal with the power fluctuations of renewable resources in real-time operation compared to day-ahead operation. In this situation, flexibility is enhanced. In the thermal sector, CHP reduces flexibility as its thermal part does not have independent control from the electrical sector.

- (C)
- Uncertainty modeling

^{ED}, Q

^{ED}, H

^{HD}, G

^{GD}, market price, λ

^{E}, λ

^{H}, λ

^{G}, λ

^{ER}, λ

^{HR}, λ

^{GR}, renewable power, P

^{R}, EVs’ aggregation parameters, CR

^{EES}, DR

^{EES}, EI, $\underset{\xaf}{E}$, ${\overline{Q}}^{E}$, and ${\underset{\xaf}{Q}}^{E}$. In this article, stochastic programming is used to model the uncertainties. However, due to a large number of uncertainty parameters, the point estimation method (PEM) is used to model these parameters [2]. This method extracts the least possible number of scenarios compared to the other stochastic programming methods. Hence, the confident optimal solution in low computational time is obtained. It is noteworthy that the proposed plan is a type of operation problem. Since the executive step is generally less than 1 h in operation problem, it is necessary to have a low problem-solving computational time. This article uses the PEM method based on the model of 2n + 1, in which n and 2n + 1 are equal to the number of uncertainty parameters and the number of extracted scenarios, respectively. The details of this method are as follows [2]:

- Step 1: definition of n.
- Step 2: define $E({S}_{}^{a})=0(a=1,2)$, a as output torque index, S as an optimization problem.
- Step 3: selection of uncertainty parameter (Z1).
- Step 4: calculation of skewness (${\lambda}_{{z}_{l},3}$) and kurtosis (${\lambda}_{{z}_{l},4}$) of uncertainty parameter (Z1) using Equations (34) and (35), respectively:

- Step 5: calculation of two standard locations (ξ) based on Equation (38):

- Step 6: calculation of two standard locations (ξ) based on Equation (38):

- Step 7: solving the confident problem in the presence of estimated locations as follows:

- Step 8: calculating the impact factor of (ω) using Equation (41):

- Step 9: updating first and second output torque using Equation (42):

- Step 10: repeat steps 3 to 9, until all stochastic variables are entered in the calculations.
- Step 11: calculating the impact coefficient of the mean point using Equation (43):

- Step 12: updating the first and second output torques using Equation (44):

- Step 13: calculation of mean and standard deviation of the stochastic variable using Equations (45) and (46):

## 3. Single-Level Modeling of the Proposed Problem

## 4. Numerical Result and Discussion

#### 4.1. Problem Data

_{Q}is equal to 0.08. The system in Figure 1 has eight EHs. EH location in the different networks, peak load, and the elements in each EH are reported in Table 2. According to this table, RES in EHs 1–6 are photovoltaic plants (PV) and wind turbines (WT). The RES peak active power is equal to 0.2 MW and 0.25MW, respectively. The active power generation daily profile of these sources is obtained by multiplying the active peak power by a daily curve of their generated power rate taken from [3]. Each of these sources is also able to control their reactive power between −0.1 MVAr and 0.1 MVAr. Regarding DRP, it is assumed that the consumers in EH participate in the DRP scheme at a rate of 40%. EHs 1–6 have two types of EES, static and mobile. The static EES is a battery (B), and the mobile type is related to the aggregation of EVs. In each of these aforementioned EHs, it is assumed that there are 80 electric vehicles. The number of connected EVs to the EH per hour is equal to the multiplication of the total number of EVs in the EH and a daily curve of their penetration rate [3]. The curve has been taken from [3], and EVs characteristics such as charge/discharge rate, charge capacity, consumption energy, and other items are reported in [3,16]. A battery with a capacity of 1.5 MWh with 90% charging and discharging efficiency has been used in the mentioned EHs. Its charge and discharge rate is equal to 0.8 MW, and the initial and minimum stored energy is 0.2 MW and 0.2 MW, respectively [6]. The battery charger can control reactive power between −0.2 MVAr and 0.2 MVAr. There are similar specifications for TES, except that its charge and discharge efficiency is 80%. In EHs 5–8, a boiler with a capacity (maximum thermal power) of 0.2 MW with an efficiency of 80% has been used. In each of these EHs, the CHP has a maximum (minimum) active, reactive, and thermal power equal to 0.5 MW, 0.2 MVAr, and 0.3 MW (0 MW, 0.2 MVAr, and 0 MW), respectively. Electricity, loss, and thermal efficiency in CHP are 40%, 8%, and 40%, respectively [2]. Finally, the standard deviation of uncertainty parameters is assumed to be equal to 10%, and the flexibility tolerance (ΔF) for achieving flexible EHs is equal to 0.05 per unit.

#### 4.2. Results

- (A)
- The convergence evaluation of the proposed problem-solving

^{C}, Q

^{C}, H

^{B}, Q

^{R}, P

^{CH}, P

^{DCH}, H

^{CH}, H

^{DCH}, P

^{D}, H

^{D}, G

^{D}, and Q

^{E}are defined by EA according to (22), (24)–(27), (29), and (30). Therefore, the value of dependent variables, including P

^{EH}, Q

^{EH}, H

^{EH}, G

^{EH}, ER

^{EH}, HR

^{EH}, GR

^{EH}, H

^{C}, G

^{C}

^{,}, and G

^{B}are calculated using (16)–(21) and (23), and the value of dependent variables P

^{ES}, Q

^{ES}, H

^{HS}, G

^{GS}, P

^{EL}, Q

^{EL}, H

^{HL}, G

^{GL}, V, T, σ,and ξ, are calculated using (2)–(9). The backward–forward load flow method for radial-network and Newton–Raphson method for ring-network is used to solve the constraints (2)–(9). The dependent variables ρ and μ are then calculated using (54) and (57). In the following, the fitness function value is evaluated. The fitness function is equal to the sum of the main objective function, (1), and the sum of the penalty function of constraints (10)–(14), H

^{C}limitation in (22), (28), and (31)–(33). In other words, the penalty function methods [13] have been used to estimate the mentioned constraints. The penalty function for the limitation of a ≤ b and constraint of a = b are expressed as δ·max(0, a − b) and α·(b − a), respectively, where δ ≥ 0 and α ∈ (−∞, +∞) represent the Lagrangian multipliers [11], the values of which are determined by the EA as a decision variable. The solution process continues to the point of convergence. In EA, it is generally assumed that the convergence point is reached after the maximum number of iterations, I

_{max}. The population size and I

_{max}have been considered to be equal to 80 and 4000, respectively. Other regulation parameters of these algorithms have been selected based on [22]. Finally, to evaluate statistical indexes in problem-solving, the problem is solved 30 times by each EA and classical mathematical algorithm. Therefore, the standard deviation of the final response is calculated.

- (B)
- Evaluation of EH performance

- (C)
- Assessing energy networks’ operation status

- Power flow analysis;
- Proposed scheme includes only EHs 1–6 that are contained only RESs;
- Case 2 adding electrical DRP;
- Case 3 adding EESs;
- Case 4 adding CHP;
- Case 5 adding thermal DRP;
- Case 6 adding a boiler;
- Case 7 adding TES;
- Proposed scheme includes EHs 1–8 considering all sources, storages, and DRPs.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

- 1.
**Acronyms**

ARO | Adaptive robust optimization |

CHP | Combined heat and power |

CSA | Crew Search Algorithm |

DA | Day-ahead |

DG | Distributed generation |

DRP | Demand response program |

EA | Evolutionary Algorithm |

EES | Electrical energy storage |

EH | Energy hub |

EHO | Energy hub operator |

EMS | Energy management system |

ENO | Energy network operator |

ESS | Energy storage system |

EV | Electric vehicle |

FRO | Flexible-reliable operation |

FS | Flexible source |

GA | Genetic Algorithm |

GWO | Grey Wolf Optimization |

IDRP | Incentive-based demand response program |

IGDT | Information-gap decision theory |

KKT | Karush–Kuhn–Tucker |

LP | Linear programming |

MCS | Monte Carlo Simulation |

MES | Multi-energy system |

MINLP | Mixed-integer non-linear programming |

MOP | Maximum overpressure |

MOT | Maximum overtemperature |

MOV | Maximum overvoltage |

MPD | Maximum pressure drop |

MTD | Maximum temperature drop |

MVD | Maximum voltage drop |

PEM | Point estimation method |

PV | Photovoltaic |

RES | Renewable energy source |

RT | Real-time |

SBSP | Scenario-based stochastic programming |

TES | Thermal energy storage |

TLBO | Teaching–Learning Based Optimization |

WT | Wind turbine |

- 2.
**Variables**

EEL | Expected energy losses in MWh |

ER^{EH}, HR^{EH}, GR^{EH} | Electric, thermal, and gas reserve power (p.u.) |

H^{B}, G^{B} | Boiler thermal and gas power (p.u.) |

H^{CH}, H^{DCH} | Thermal power of thermal energy storage (TES) in charge and discharge mode (p.u.) |

P^{C}, Q^{C}, H^{C}, G^{C} | Active, reactive, thermal, and gas power of combined heat and power (CHP) system (p.u.) |

P^{CH}, P^{DCH} | Active power of electrical energy storage (EES) in charge and discharge mode (p.u.) |

P^{D}, H^{D}, G^{D} | Active, thermal, and gas power in demand response program (DRP) |

P^{EH}, Q^{EH}, H^{EH}, G^{EH} | Active, reactive, thermal, and gas power of energy hub (EH) (p.u.) |

P^{EL}, Q^{EL}, H^{HL}, G^{GL} | Active and reactive power flow through electric distribution line; thermal and gas power flow through distribution pipes (p.u.) |

P^{ES}, Q^{ES}, H^{HS}, G^{GS} | Active, reactive, thermal, and gas power passing through distribution substation (p.u.) |

Profit | Total EHs’ expected profits in reserve regulation, energy, and reactive markets ($) |

Q^{R}, Q^{E} | Reactive power of renewable energy source (RES) and EES (p.u.) |

T | Temperature (p.u.) |

V, σ | Voltage magnitude (p.u.) and angle (radians) |

ξ | Gas pressure (p.u.) |

- 3.
**Constants**

B^{EL}, G^{EL} | Susceptance and conductivity of electrical distribution line (p.u.) |

CR^{EES}, DR^{EES} | Charge and discharge rate of EES (p.u.) |

CR^{TES}, DR^{TES} | Charge and discharge rate of TES (p.u.) |

$\underset{\xaf}{E},\overline{E}$ | Minimum and maximum storable energy in the energy storage system (ESS) in p.u. |

EI | Initial energy of ESS (p.u.) |

${\overline{G}}^{GL},{\overline{G}}^{GS}$ | Maximum gas power flow through pipeline and gas substation (p.u.) |

${\underset{\xaf}{H}}^{B},{\overline{H}}^{B}$ | Minimum and maximum boiler thermal power (p.u.) |

${\underset{\xaf}{H}}^{C},{\overline{H}}^{C}$ | Minimum and maximum thermal power of CHP (p.u.) |

${\overline{H}}^{HL},{\overline{H}}^{HS}$ | Maximum thermal power flow through the pipeline and thermal substation (p.u.) |

K_{Q} | Rate of reactive power price to energy price |

I^{E}, I^{H}, I^{G} | Incidence matrix of electric bus and EH, thermal node and EH, and gas node and EH |

J^{E}, J^{H}, J^{G} | Incidence matrix of electric bus and line, thermal node and pipeline, and gas node and pipeline |

${\underset{\xaf}{P}}^{C},{\overline{P}}^{C}$ | Minimum and maximum active power of CHP (p.u.) |

P^{ED}, Q^{ED}, H^{HD}, G^{GD} | Active, reactive, thermal, and gas load (p.u.) |

P^{R} | Generated active power of RES (p.u.) |

${\underset{\xaf}{Q}}^{C},{\overline{Q}}^{C}$ | Minimum and maximum reactive power of CHP (p.u.) |

${\underset{\xaf}{Q}}^{E},{\overline{Q}}^{E}$ | Minimum and maximum reactive power of EES (p.u.) |

${\underset{\xaf}{Q}}^{R},{\overline{Q}}^{R}$ | Minimum and maximum reactive power of RES (p.u.) |

${\overline{S}}^{EL},{\overline{S}}^{ES}$ | Maximum apparent power flow through the electric distribution line and substation (p.u.) |

sign(a, b) | Sign function, having a value of 1 for a ≥ b; otherwise, it is equal to −1 |

β | Consumer participation rate in DRP |

$\underset{\xaf}{\chi},\overline{\chi}$ | Minimum and maximum limits of voltage magnitude, pressure, or temperature (p.u.) |

η^{B} | Boiler efficiency |

η^{CH}, η^{DCH} | Charge and discharge efficiency of ESS |

η^{T}, η^{L}, η^{H} | Efficiency of electricity, loss, and thermal in CHP |

λ^{E}, λ^{H}, λ^{G} | Energy price in electric, thermal, and gas market ($/MWh) |

λ^{ER}, λ^{HR}, λ^{GR} | Reserve price in the electric, thermal, and gas reserve regulation market ($/MWh) |

π | Scenario occurrence probability |

ω | Gas pipeline constant (p.u.) |

ΔF | Flexibility tolerance (p.u.) |

ϑ | Thermal pipeline constant (p.u.) |

- 4.
**Sub-indexes**

e, h, g | Electric bus, thermal node, and gas node |

i | Energy hub |

m | Corresponding sub-index to the bus (node) |

t | Operating hours |

w o | Scenario Slack (node) bus |

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**Figure 2.**Expected daily curve of (

**a**) active power and (

**b**) reactive power of EHs and their elements in ΔF = 0.05 p.u.

**Figure 3.**Expected daily curve of (

**a**) heating power and (

**b**) gas power of EHs and their elements in ΔF = 0.05 p.u.

**Figure 4.**Expected daily EHs’ reserve power curve in the electrical and district heating networks in ΔF = 0.05 p.u.

**Figure 5.**Expected curve of EHs’ profit in ΔF for (

**a**) DA energy market, (

**b**) DA reactive power market, (

**c**) DA reserve market, and (

**d**) all markets.

Ref. | Flexibility Model | Market Model | Uncertainty Modeling | ||
---|---|---|---|---|---|

Energy | Reactive Power | Reserve Regulation | |||

[6] | No | Yes | No | No | SBSP |

[7] | No | No | No | No | ARO |

[8] | No | No | No | No | SBSP |

[9] | No | Yes | No | No | Deterministic |

[10] | No | Yes | No | No | IGDT |

[11] | Yes | No | No | No | SBSP |

[12] | No | No | No | No | Hybrid SBSP/IGDT |

[13] | No | No | No | No | Deterministic |

[14] | No | No | No | No | SBSP |

[15] | No | No | No | No | Scenario-based/interval/IGDT |

PM | Yes | Yes | Yes | Yes | PEM |

EH | Location (e, h, g) | Source | ESS | DRP | P^{ED} (MW) | Q^{ED} (MVAr) | H^{HD} (MW) | G^{GD} (MW) |
---|---|---|---|---|---|---|---|---|

1 | 6, -, - | PV, WT | B, EVs | Electrical | 0.6 | 0.3 | 0 | 0 |

2 | 13, -, - | PV, WT | B, EVs | Electrical | 0.4 | 0.2 | 0 | 0 |

3 | 23, -, - | PV, WT | B, EVs | Electrical | 0.6 | 0.3 | 0 | 0 |

4 | 26, -, - | PV, WT | B, EVs | Electrical | 0.4 | 0.2 | 0 | 0 |

5 | 17, 5, 2 | CHP, boiler, PV, WT | TES, B, EVs | Electrical and thermal | 0.8 | 0.4 | 0.4 | 0 |

6 | 31, 11, 4 | CHP, boiler, PV, WT | TES, B, EVs | Electrical and thermal | 0.8 | 0.4 | 0.4 | 0 |

7 | 21, 2, 3 | CHP, boiler | TES | Thermal | 0.4 | 0.2 | 0.3 | 0 |

8 | 10, 8, 3 | CHP, boiler | TES | Thermal | 0.4 | 0.2 | 0.3 | 0 |

Solver | EEL (MWh) | Profit ($) | Convergence Iteration | Convergence Time (min) | Standard Deviation of Final Solution (%) | Model State |
---|---|---|---|---|---|---|

GA | 5.95 | 3578.1 | 2046 | 10.5 | 4.76 | Feasible |

TLBO | 5.64 | 3696.8 | 1469 | 7.0 | 2.22 | Feasible |

GWO | 5.78 | 3649.2 | 1722 | 8.4 | 3.65 | Feasible |

CSA | 5.54 | 3752.3 | 1274 | 5.5 | 2.01 | Feasible |

CONOPT | 5.50 | 3767.5 | 423 | 5.1 | 0 | Feasible |

IPOPT | 5.35 | 3839.6 | 137 | 4.3 | 0 | Feasible |

LGO | - | Infeasible | ||||

MINOS | 5.71 | 3687.4 | 522 | 7.3 | 0 | Feasible |

OQNLP | - | Infeasible |

Case | Network (ΔF = ∞) | Network (ΔF = 0.05 p.u.) | Network (ΔF = 0 p.u.) | ||||||
---|---|---|---|---|---|---|---|---|---|

Electrical | Thermal | Gas | Electrical | Thermal | Gas | Electrical | Thermal | Gas | |

1 | 4.42 | 3.06 | 0 | Infeasible | |||||

2 | 3.42 | 3.06 | 0 | Infeasible | |||||

3 | 3.28 | 3.06 | 0 | 3.32 | 3.06 | 0 | 3.37 | 3.06 | 0 |

4 | 3.07 | 3.06 | 0 | 3.10 | 3.06 | 0 | 3.14 | 3.06 | 0 |

5 | 2.52 | 2.54 | 0.78 | Infeasible | |||||

6 | 2.52 | 2.41 | 0.78 | 2.56 | 2.43 | 0.78 | 2.60 | 2.46 | 0.78 |

7 | 2.52 | 1.97 | 1.04 | 2.56 | 2.01 | 1.04 | 2.60 | 2.04 | 1.04 |

8 | 2.52 | 1.84 | 1.04 | 2.56 | 1.87 | 1.04 | 2.60 | 1.90 | 1.04 |

9 | 2.43 | 1.69 | 1.23 | 2.47 | 1.72 | 1.23 | 2.51 | 1.75 | 1.23 |

**Table 5.**Maximum deviation of (maximum over-) voltage, temperature, and pressure (p.u.) for various values of ΔF.

Case | (ΔF = ∞) | (ΔF = 0.05 p.u.) | (ΔF = 0 p.u.) | ||||||
---|---|---|---|---|---|---|---|---|---|

MVD/MOV | MTD/MOT | MPD/MOP | MVD/MOV | MTD/MOT | MPD/MOP | MVD/MOV | MTD/MOT | MPD/MOP | |

1 | 0.087/0 | 0.072/0 | 0/0 | Infeasible | |||||

2 | 0.068/0 | 0.072/0 | 0/0 | Infeasible | |||||

3 | 0.065/0 | 0.072/0 | 0/0 | 0.065/0 | 0.072/0 | 0/0 | 0.066/0 | 0.072/0 | 0/0 |

4 | 0.061/0 | 0.072/0 | 0/0 | 0.061/0 | 0.072/0 | 0/0 | 0.061/0 | 0.072/0 | 0/0 |

5 | 0.054/0.04 | 0.064/0.05 | 0.032/0 | Infeasible | |||||

6 | 0.054/0.04 | 0.061/0.05 | 0.032/0 | 0.054/0.04 | 0.061/0.05 | 0.032/0 | 0.055/0.03 | 0.062/0.04 | 0.032/0 |

7 | 0.054/0.04 | 0.055/0.06 | 0.045/0 | 0.054/0.04 | 0.055/0.06 | 0.045/0 | 0.055/0.03 | 0.056/0.05 | 0.045/0 |

8 | 0.054/0.04 | 0.053/0.06 | 0.045/0 | 0.054/0.04 | 0.053/0.06 | 0.045/0 | 0.055/0.03 | 0.054/0.05 | 0.045/0 |

9 | 0.053/0.04 | 0.052/0.06 | 0.047/0 | 0.053/0.04 | 0.052/0.06 | 0.047/0 | 0.054/0.03 | 0.053/0.05 | 0.047/0 |

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**MDPI and ACS Style**

Parhoudeh, S.; Eguía López, P.; Kavousi Fard, A.
Stochastic Coordinated Management of Electrical–Gas–Thermal Networks in Flexible Energy Hubs Considering Day-Ahead Energy and Ancillary Markets. *Sustainability* **2023**, *15*, 10744.
https://doi.org/10.3390/su151310744

**AMA Style**

Parhoudeh S, Eguía López P, Kavousi Fard A.
Stochastic Coordinated Management of Electrical–Gas–Thermal Networks in Flexible Energy Hubs Considering Day-Ahead Energy and Ancillary Markets. *Sustainability*. 2023; 15(13):10744.
https://doi.org/10.3390/su151310744

**Chicago/Turabian Style**

Parhoudeh, Sina, Pablo Eguía López, and Abdollah Kavousi Fard.
2023. "Stochastic Coordinated Management of Electrical–Gas–Thermal Networks in Flexible Energy Hubs Considering Day-Ahead Energy and Ancillary Markets" *Sustainability* 15, no. 13: 10744.
https://doi.org/10.3390/su151310744