#
An Enhanced Second-Order Cone Programming-Based Evaluation Method on Maximum Hosting Capacity of Solar Energy in Distribution Systems with Integrated Energy^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- (1)
- Model: An optimization model of the maximum hosting capacity evaluation of solar energy in IEDS is proposed, in which the maximization of PV capacity and solar collector (SC) capacity are fully considered.
- (2)
- Mechanism: IEDS’s potential in multi-energy coordinated optimization is fully exploited to enhance the hosting capacity of solar energy in which the electric distribution network, heating network, and natural gas network constraints are fully modeled.
- (3)
- Method: An enhanced-SOCP-based solving method is developed to solve the proposed maximum hosting capacity model, which can output a satisfactory solution and reduce the computation time.

## 2. Model of Maximum Hosting Capacity

#### 2.1. Objective Function

_{PV}is the output power of the PV, f

_{SC}is the output power of the SC, f

_{loss}

_{1}is the distribution system power loss, and f

_{loss}

_{2}is the heating system power loss. The quantities ϕ

_{1}and ϕ

_{2}are the coefficient of the output power of the PV and the coefficient of the output power of the SC, respectively. The distribution system power losses f

_{loss}

_{1}and the heating system power losses f

_{loss}

_{2}are formulated in (2)–(5), respectively.

_{b}is the set of all branches of the DN; r

_{ij}is the resistance of branch ij; i

_{ij,t}is the current magnitude square of branch ij at time t.

_{c}is the set of all pipelines of the HN; H

_{ij,t}and H

_{ji,t}are heat power of the pipeline from node i to node j and heat power of the pipeline from node j to node i in the HN at time t, respectively; η

_{ij}is heat loss ratio of pipeline ij in the HN.

#### 2.2. Constraints of the Distribution Network

#### 2.3. Constraints of the Heating Network

_{i,t}is the heat power injected into node i at time t; δ is the heat loss ratio per unit length in the HN; l

_{ji}is the length of the pipeline ij in the HN; u

_{ij,t}is equal to 1, and u

_{ji,t}is equal to 0 if the heat power direction of the pipeline ij is from node i to node j in the HN at time t. $\overline{{H}_{ij}}$ is the maximum heat power of the pipeline ij in the HN, whose detailed expression can be referred to in [29].

#### 2.4. Constraints of the Natural Gas Network

_{l,i,t}and p

_{l,j,t}are the begin node pressure of node i and the end node pressure of node j of gas pipeline l at time t, respectively; s

_{l,ij,t}is the 0-1 integer variable to represent the pipeline flow direction; q

_{l,ij,t}is the gas pipeline l volume flow at time t; q

_{i,t}is the volume flow injected into node i in the NGN at time t; F

_{l,ij}is the resistance coefficient of gas pipeline l, whose detailed expression can be referred to in [23]; p

^{max}and p

^{min}are the allowable maximum pressure and minimum pressure in the NGN.

#### 2.5. Constraints of the Energy Station

**C**is the energy station energy conversion matrix,

**C**

_{n}is the energy converters’ conversion vector;

**I**is the energy input power vector of the energy converters;

**L**is the load vector of the energy station;

**I**

_{max}is the capacity vector of the energy converters.

_{i,t}and Q

_{i,t}are the active and reactive power injected into node I in the DN, respectively; ${P}_{j,t}^{line}$, ${Q}_{j,t}^{line}$, ${G}_{j,t}^{line}$, and ${H}_{j,t}^{line}$ are the active power, the reactive power, the gas power, and the heat power injected into the jth energy station, respectively; G

_{CV}is the gross calorific value of gas.

## 3. Solution Methodology

_{t,ij,k}, Q

_{t,ij,k}, and i

_{t,ij,k}are the active power flow, the reactive power flow, and the current magnitude square of branch ij in the kth iteration, respectively; u

_{t,i,k}is the voltage magnitude square of node i in the kth iteration.

_{n}is the set of all pipelines in the NGN; ${\overline{v}}_{l,t,k}$ is the larger of ${p}_{l.i.t}$ and ${p}_{l,j,t}$ in the kth iteration; ${\underset{\xaf}{v}}_{l,t,k}$ is the smaller of ${p}_{l.i.t}$ and ${p}_{l,j,t}$ in the kth iteration; q

_{l,ij,t,k}is the gas pipeline l volume flow in the kth iteration.

- ①
- Basic data inputting;
- ②
- Initialization parameters setting;
- ③
- Check whether k is fewer than or equal to k
_{max}. If so, continue to step 4. Otherwise, terminate the process; - ④
- Model constructing;
- ⑤
- Model converting;
- ⑥
- Model solving;
- ⑦
- Check whether $r{\mathrm{Gap}}_{\mathrm{DN}}\le {\epsilon}_{1}r{\mathrm{Gap}}_{\mathrm{GN}}\le {\epsilon}_{2}$. If so, move to step 9. Otherwise, continue to step 8;
- ⑧
- Cutting planes adding and move to step 3;
- ⑨
- Results outputting.

## 4. Case Study

#### 4.1. Case Introduction

^{2}. The predicted value of irradiance intensity is taken as 700 W/m

^{2}[37]. The maximum allowable branch current is 250 A. The allowable range of the DN voltage is 0.9–1.1 p.u. and the allowable pressure of the NGN is 35–75 mbar. We assume that the maximum acceptable water velocity in pipelines is 2 m/s, the temperature difference between water at the inlet and outlet of the pipe is 25 °C, and the heat loss ratio per unit length is 0.15/km. The predefined precision with regard to the SOCP relaxation deviation of the DN and the NGN are set to 1 × 10

^{−6}and 1 × 10

^{−2}, respectively. The G

_{CV}of natural gas is 41.04 MJ/m

^{3}.

#### 4.2. The Single-Period Case (Case 1)

_{1}is set to 1, the same as the value of the quantity ϕ

_{2}. Based on the above data, five scenarios are set as follows:

- Scenario I: Only PV are considered based on the DN.
- Scenario II: PV, CHP, GB, and EB are considered based on the IEDS.
- Scenario III: Based on Scenario II, P2G is considered.
- Scenario IV: Based on Scenario II, SC is considered.
- Scenario V: Based on Scenario II, SC and P2G are considered.

- Step 1: Input the basic network data and parameters of the devices.
- Step 2: The predefined precision about the SOCP relaxation deviation of the DN and the NGN are set to 1 × 10
^{−6}and 1 × 10^{−2}, respectively. - Step 3: The maximum number of iteration steps is set to 30. Initialize the iteration step k = 1. Check whether k is fewer than or equal to 30. If so, proceed to Step 4. Otherwise, the process terminates.
- Step 4: Build the optimization model for the maximum hosting capacity evaluation of solar energy.
- Step 5: Convert this model into an MISOCP model through SOCP relaxation and linearization.
- Step 6: Solve the MISOCP model to obtain the maximum relaxation deviation of the DN and the NGN.
- Step 7: Check whether $r{\mathrm{Gap}}_{\mathrm{DN}}\le 1\times {10}^{-6}r{\mathrm{Gap}}_{\mathrm{NGN}}\le 1\times {10}^{-2}$. If so, move to Step 9. Otherwise, continue to Step 8.
- Step 8: Update k = k + 1. Add the cutting plane constraint (14) and (15), and return to Step 3.
- Step 9: Output the optimization results and end the solving process.

_{1}and the quantity ϕ

_{2}in Case 1 are listed in Table 6 and Table 7, respectively. Because the total alternative installation area of PV and SC in the system is limited, the results of the proposed model are affected by the quantity ϕ

_{1}and the quantity ϕ

_{2}. From Table 6 and Table 7, we can see that as the quantity ϕ

_{1}becomes larger, the hosting capacity of PV is increased and the hosting capacity of SC is decreased. Because the quantity ϕ

_{1}becomes larger, more space can be provided for the installation area of PV to maximize the objective function.

#### 4.3. The 24 h Period Case (Case 2)

_{1}is set to 1 and the quantity ϕ

_{2}is set to 1 in Case 2. Only PV considered based on the DN is set as Scenario VI. PV, SC, CHP, GB, EB, and P2G considered based on the IEDS is set as Scenario VII. Because the utilization of P2G and SC can support the pressure management of the NGN, the allowable pressure of the NGN is set to 50–75 mbar.

#### 4.4. Algorithm Validation

^{−6}and 1 × 10

^{−2}, respectively.

## 5. Conclusions

- (1)
- The optimization results show that the maximum hosting capacity of solar energy is improved significantly by realizing the coordination of integrated multi-energy system and the optimal utilization of electricity, heat, and gas energy. With the utilization of gas energy (P2G, etc.), the hosting capacity of PV increases from 3.795 MW in Scenario II to 4.341 MW. With the utilization of gas energy (SC, etc.), the hosting capacity of PV increases from 3.795 MW in Scenario II to 4.379 MW and the hosting capacity of SC is increased from 0 MW to 1.810 MW.
- (2)
- Meanwhile, the distribution system power losses and the voltage fluctuations are effectively decreased with the optimal utilization of multiple energy. The distribution system active power loss over a day reduced from 1106.21 kWh in Scenario VI to 696.95 kWh, and a flat voltage profile was attained through the IEDS operation optimization.
- (3)
- By applying SOCP relaxation, linearization, and adding increasingly tight linear cuts of distribution system and natural gas system to the SOCP relaxation, the proposed model can be solved accurately and efficiently.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## List of Symbols and Abbreviations

Abbreviations | |

IEDS | integrated energy distribution system |

DN | distribution network |

NGN | natural gas network |

HN | heating network |

ADN | active distribution network |

MG | micro-grid |

NLP | nonlinear programming |

SOCP | second-order cone programming |

MISOCP | Mixed-integer second-order cone programming |

PV | photovoltaic |

SC | solar collector |

CHP | combined heat and power |

EB | electrical boiler |

GB | gas boiler |

P2G | power to gas |

Symbols | |

ϕ_{1} | the coefficient of the output power of the PV |

ϕ_{2} | the coefficient of the output power of the SC |

T | the total periods of time horizon |

N | the total number of nodes in the DN |

${\eta}_{i,t}^{\mathrm{SE}}$ | the actual irradiance intensity of the system at node i |

η_{PV-P} | the efficiency of PV |

$S{E}_{i}^{PV}$ | the installation area of PV at node i |

η_{SC} | the efficiency of SC |

$S{E}_{i}^{\mathrm{SC}}$ | the installation area of SC at node i |

Ω_{b} | the set of all branches in the DN |

r_{ij} | resistance of branch ij |

i_{ij,t} | the current magnitude square of the branch ij at time t |

Ω_{c} | the set of all pipelines in the HN |

H_{ij,t} | heat power of the pipeline from node i to node j at time t |

η_{ij} | heat loss ratio of pipeline ij in the HN |

j | the set of nodes which can be directly connected to i |

H_{i,t} | the heat power injected into node i at time t |

δ | heat loss ratio per unit length in the HN |

l_{ji} | the length of the pipeline ij in the HN |

u_{ij,t} | the 0-1 integer variable to represent the pipeline flow direction in the HN |

$\overline{{H}_{ij}}$ | the maximum heat power of the pipeline ij in the HN |

p_{l,i,t} | the head node pressure of gas pipeline l at time t |

p_{l,j,t} | the end node pressure of gas pipeline l at time t |

s_{l,ij,t} | the 0-1 integer variable to represent the pipeline flow direction in the NGN |

q_{l,ij,t} | gas pipeline l volume flow at time t |

q_{i,t} | the volume flow injected into node i in the NGN at time t |

F_{l,ij} | the resistance coefficient of gas pipeline l |

p^{max} | the allowable maximum pressure in the NGN |

p^{min} | the allowable minimum pressure in the NGN |

C_{n} | the energy converters conversion vector |

I | the energy input power vector of the energy converters |

L | the load vector of energy station |

C | the energy station energy conversion matrix |

I_{max} | the capacity vector of the energy converters |

P_{i,t} | the active power injected into node i in the DN |

Q_{i,t} | the reactive power injected into node i in the DN |

${P}_{j,t}^{line}$ | the active power injected into the jth energy station |

${Q}_{j,t}^{line}$ | the reactive power injected into the jth energy station |

${G}_{j,t}^{line}$ | the gas power injected into the jth energy station |

${H}_{j,t}^{line}$ | the heat power injected into the jth energy station |

G_{cv} | the gross calorific value of gas |

${\overline{v}}_{l,t}$ | the larger one of node pressure of gas pipeline l at time t |

${\underset{\xaf}{v}}_{l,t}$ | the smaller one of node pressure of gas pipeline l at time t |

M | an arbitrarily large positive number that is not infinite |

${F}_{l,ij}^{\prime}$ | the square root of F_{l,ij} |

x_{l,t} | the 0-1 integer variable to represent the size relationship of the node pressure |

y_{l,t} | the 0-1 integer variable to represent the size relationship of the node pressure |

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**Figure 1.**Flow chart of the maximum hosting capacity evaluation method of solar energy in electrical–natural gas–thermal IEDS.

**Figure 4.**Prediction curves of typical daily solar irradiance and load data. (

**a**) Prediction curve of typical daily solar irradiance. (

**b**) Prediction curve of typical daily load.

**Figure 8.**Optimal dispatch results of multiple energy power in Scenario VII. (

**a**) Optimal dispatch results of electrical power in Scenario VII. (

**b**) Optimal dispatch results of thermal power in Scenario VII. (

**c**) Optimal dispatch results of gas power in Scenario VII.

**Figure 12.**SOCP relaxation deviation of different scenarios in each iteration (Case 1). (

**a**) Distribution system. (

**b**) Natural gas system.

**Figure 13.**SOCP relaxation deviation in scenarios VI and VII (Case 2). (

**a**) Distribution system. (

**b**) Natural gas system.

Ways | Features |
---|---|

MG [4,5,6] | The integration and coordination of distributed generators, distributed storage systems, controllable loads, etc. |

ADN [7,8,9,10,11] | The network reconfiguration, the power factor control strategy, the flexible interconnection technology, etc. |

IEDS [12,13,14,15] | The optimal utilization of multiple types of energies. |

Converter | Capacity/kW | Efficiency |
---|---|---|

CHP | ${G}_{i,t}^{CHP}=300/0.3$ | ${\eta}_{\mathrm{CHP}-\mathrm{P}}=0.3\hspace{0.33em}{\eta}_{\mathrm{CHP}-\mathrm{Q}}=0\hspace{0.33em}{\eta}_{\mathrm{CHP}-\mathrm{H}}=0.39$ |

EB | ${P}_{i,t}^{EB}=200$ | ${\eta}_{\mathrm{EB}}=0.95$ |

GB | ${G}_{i,t}^{GB}=300/0.85$ | ${\eta}_{\mathrm{GB}}=0.85$ |

P2G | ${P}_{i,t}^{P2G}=200$ | ${\eta}_{\mathrm{P}2\mathrm{G}}=0.7$ |

PV | ${\eta}_{i,t}^{\mathrm{SE}}S{E}_{i}^{PV}=1873.5$ | ${\eta}_{\mathrm{PV}-\mathrm{P}}=0.175\hspace{0.33em}{\eta}_{\mathrm{PV}-\mathrm{Q}}=0$ |

SC | ${\eta}_{i,t}^{\mathrm{SE}}S{E}_{i}^{SC}=5250$ | ${\eta}_{\mathrm{SC}}=0.5$ |

Node Number | Energy Demand (kJ/s) | Pressure (mbar) |
---|---|---|

1 (Source Node) | 0 | 75 |

2 | 1250 | / |

3 | 1100 | / |

4 | 1000 | / |

5 | 1300 | / |

6 | 900 | / |

7 | 250 | / |

8 | 1175 | / |

9 | 275 | / |

10 | 237.5 | / |

11 | 175 | / |

Branch | From–To | Pipe Length (m) | Pipe Diameter (mm) |
---|---|---|---|

1 | 1–2 | 50 | 160 |

2 | 2–3 | 500 | 160 |

3 | 2–4 | 500 | 110 |

4 | 2–5 | 500 | 110 |

5 | 3–6 | 600 | 110 |

6 | 3–7 | 600 | 110 |

7 | 3–8 | 500 | 110 |

8 | 7–9 | 200 | 80 |

9 | 9–10 | 200 | 80 |

10 | 10–11 | 200 | 80 |

Scenario | PV Capacity/MW | SC Capacity/MW |
---|---|---|

I | 3.795 | 0.000 |

II | 3.739 | 0.000 |

III | 4.341 | 0.000 |

IV | 4.379 | 1.810 |

V | 4.854 | 1.882 |

Capacity/MW | ${\mathit{\varphi}}_{1}=0.2,\hspace{0.33em}{\mathit{\varphi}}_{2}=0.8$ | ${\mathit{\varphi}}_{1}=0.4,\hspace{0.33em}{\mathit{\varphi}}_{2}=0.6$ | ${\mathit{\varphi}}_{1}=0.5,\hspace{0.33em}{\mathit{\varphi}}_{2}=0.5$ | ${\mathit{\varphi}}_{1}=0.6,\hspace{0.33em}{\mathit{\varphi}}_{2}=0.4$ | ${\mathit{\varphi}}_{1}=0.8,\hspace{0.33em}{\mathit{\varphi}}_{2}=0.2$ |
---|---|---|---|---|---|

Total PV | 3.796 | 3.796 | 4.398 | 4.398 | 4.398 |

Total SC | 2.313 | 2.313 | 1.743 | 1.743 | 1.743 |

Total PV + SC | 6.109 | 6.109 | 6.141 | 6.141 | 6.141 |

Capacity/MW | ${\mathit{\varphi}}_{1}=0.2,\hspace{0.33em}{\mathit{\varphi}}_{2}=0.8$ | ${\mathit{\varphi}}_{1}=0.4,\hspace{0.33em}{\mathit{\varphi}}_{2}=0.6$ | ${\mathit{\varphi}}_{1}=0.5,\hspace{0.33em}{\mathit{\varphi}}_{2}=0.5$ | ${\mathit{\varphi}}_{1}=0.6,\hspace{0.33em}{\mathit{\varphi}}_{2}=0.4$ | ${\mathit{\varphi}}_{1}=0.8,\hspace{0.33em}{\mathit{\varphi}}_{2}=0.2$ |
---|---|---|---|---|---|

Total PV | 4.399 | 4.399 | 4.854 | 4.854 | 5.000 |

Total SC | 2.313 | 2.313 | 1.881 | 1.881 | 1.465 |

Total PV + SC | 6.712 | 6.712 | 6.735 | 6.735 | 6.465 |

Capacity/MW | Scenario VI | Scenario VII |
---|---|---|

Total PV | 4.684 | 4.911 |

Total SC | 0.000 | 1.718 |

Scenario | The Objective Function (MWh) | Time (s) | ||
---|---|---|---|---|

BONMIN | Proposed Method | BONMIN | Proposed Method | |

I | 3.715 | 3.715 | 0.218 | 0.244 |

II | 3.490 | 3.500 | 6.138 | 0.581 |

III | 3.940 | 4.105 | 5.588 | 0.494 |

IV | 5.909 | 5.908 | 15.223 | 2.375 |

V | 6.502 | 6.502 | 6.718 | 0.165 |

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**MDPI and ACS Style**

Wang, C.; Luo, F.; Jiao, Z.; Zhang, X.; Lu, Z.; Wang, Y.; Zhao, R.; Yang, Y.
An Enhanced Second-Order Cone Programming-Based Evaluation Method on Maximum Hosting Capacity of Solar Energy in Distribution Systems with Integrated Energy. *Energies* **2022**, *15*, 9025.
https://doi.org/10.3390/en15239025

**AMA Style**

Wang C, Luo F, Jiao Z, Zhang X, Lu Z, Wang Y, Zhao R, Yang Y.
An Enhanced Second-Order Cone Programming-Based Evaluation Method on Maximum Hosting Capacity of Solar Energy in Distribution Systems with Integrated Energy. *Energies*. 2022; 15(23):9025.
https://doi.org/10.3390/en15239025

**Chicago/Turabian Style**

Wang, Chunyi, Fengzhang Luo, Zheng Jiao, Xiaolei Zhang, Zhipeng Lu, Yanshuo Wang, Ren Zhao, and Yang Yang.
2022. "An Enhanced Second-Order Cone Programming-Based Evaluation Method on Maximum Hosting Capacity of Solar Energy in Distribution Systems with Integrated Energy" *Energies* 15, no. 23: 9025.
https://doi.org/10.3390/en15239025