# Second-Order Cone Approximation for Voltage Stability Analysis in Direct-Current Networks

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## Abstract

**:**

## 1. Introduction

## 2. Non-Linear Programming Formulation

#### Objective function:

#### Set of constraints:

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

#### Convexity Test

## 3. Second-Order Cone Programming Formulation

**Remark**

**4.**

**Theorem**

**1.**

**Proof.**

**Objective function:**

**Set of constraints:**

**Remark**

**5.**

## 4. Graphical Example

**O**is the solution of the classical power flow problem when all ${\lambda}_{i}$ are fixed as zero; this point is $({v}_{1},{v}_{2})=(0.9312,0.8783)$. From this initial point, we evaluate the evolution of the voltage collapse in the DC grid when its constant-power loads increase. Trajectory

**O**–

**A**shows the evolution of the voltage at load nodes when it is increased only at the load connected at node 1, with the load at node 2 being fixed as $0.15$ p.u. Please note that point

**A**presents the maximum reduction in both load voltages at the same time, i.e., $({v}_{1},{v}_{2})=(0.4802,0.4265)$; both voltages are observed to be lower than $0.5000$ p.u. In addition, these points represent the maximum objective function possible in this numerical example, which is $z=3.4304$ p.u, when ${\lambda}_{1}=15.4021$. Trajectory

**O**–

**B**shows the evolution of the voltage in the DC system when both loads are increased by the same magnitude, i.e., ${\lambda}_{1}={\lambda}_{2}=5.5319$. These increments in the loads produce a maximum objective function of $2.2862$ p.u, where one voltage is higher than $0.6500$ p.u (see node 2), and the other node is lower than $0.4500$ p.u (see node 3). This behavior implies that node 3 conditioned the stability margin behavior of this test system since it is more sensitive to load changes than node 2 is. On the other hand, trajectory

**O**–

**C**presents the voltage evolution of the numerical example when the load connected to node 2 is increased, with the load at node 1 fixed as $0.20$ p.u. This trajectory shows that the voltage at node 2 decreases until $0.4534$, while the voltage at node 1 remains upper that $0.7600$ p.u. This behavior confirms that node 2 has a lower possibility of increasing its load consumption since the voltage collapse point is reached when the total load of the DC system is $1.4611$ p.u, which is the minimum objective functions across the three cases analyzed. Please note that in Table 1 the numerical behavior of the voltage stability problem in DC networks resumes when constant-power loads start to increase.

**Remark**

**7.**

## 5. Test Systems and Simulation Results

#### 5.1. Test System Configurations

#### 5.2. Numerical Validation

- 🗸
- The HVDC test feeder with its meshed structure maintains voltages higher than $0.50$ p.u. until the voltage collapse scenario. Please note that node 4 presents the lower voltage profile with $0.5114$ p.u., which is a radial extension of this HVDC system.
- 🗸
- The voltage collapse in the MVDC test feeder is evident long after node 57 and onward. This situation occurs in this part (node 57 and onward) of the test feeder since the total load is more significant than regarding routes. Voltage collapse occurs when the maximum voltage drop is $0.4610$ p.u. at node 69.
- 🗸
- In both test systems, the voltage collapse occurs when voltages are lower than $0.55$ p.u; while the total load consumptions increase at least three times. This behavior implies that the power system protection disconnects this system before the voltage collapse occurs because of the high currents flowing through the branches.

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Voltage collapse trajectories followed by different increments in the constant power consumptions per node.

**Figure 5.**Voltage behavior in the test system for the initial state of load and voltage collapse: (

**a**) HVDC test system, and (

**b**) MVDC test feeder.

Trajectory | ${\mathit{\lambda}}_{1}$ | ${\mathit{\lambda}}_{2}$ | Collapse Point $({\mathit{v}}_{1},{\mathit{v}}_{2})$ | z [p.u.] |
---|---|---|---|---|

O–A | 15.4021 | 0 | A (0.4802,0.4265) | 3.4304 |

O–B | 5.5319 | 5.5319 | B (0.6776,0.4151) | 2.2862 |

O–C | 0 | 7.4076 | C (0.7613,0.4534) | 1.4611 |

Test System | NR-DJM | IP-NLP | SDP | SOCP |
---|---|---|---|---|

HVDC | 5.6588 | 5.6588 | 5.6588 | 5.6588 |

MVDC | 3.0200 | 3.0200 | 3.0067 | 3.0200 |

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**MDPI and ACS Style**

Montoya, O.D.; Gil-González, W.; Molina-Cabrera, A.
Second-Order Cone Approximation for Voltage Stability Analysis in Direct-Current Networks. *Symmetry* **2020**, *12*, 1587.
https://doi.org/10.3390/sym12101587

**AMA Style**

Montoya OD, Gil-González W, Molina-Cabrera A.
Second-Order Cone Approximation for Voltage Stability Analysis in Direct-Current Networks. *Symmetry*. 2020; 12(10):1587.
https://doi.org/10.3390/sym12101587

**Chicago/Turabian Style**

Montoya, Oscar Danilo, Walter Gil-González, and Alexander Molina-Cabrera.
2020. "Second-Order Cone Approximation for Voltage Stability Analysis in Direct-Current Networks" *Symmetry* 12, no. 10: 1587.
https://doi.org/10.3390/sym12101587