# Novel Hybrid Modified Modal Analysis and Continuation Power Flow Method for Unity Power Factor DER Placement

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## Abstract

**:**

## 1. Introduction

- Type 3 DER unit (0 < pf
_{DERi}< 1 with lagging power factor). Type 3 DERs can provide active power and reactive power to the network. Examples of type 3 DER are a synchronous generator operated in cogeneration and gas-fired DER; - Type 4 DER unit (0 < pf
_{DERi}< 1 with leading power factor). This type of DER provides active power to the system but attracts reactive power. An example of type 4 DER is primarily an induction generator in a wind farm, such as doubly fed induction generators (DFIG) [6].

- developing a new hybrid scheme to compute optimal DER placement. This technique is a hybrid approach between the modified modal analysis (MMA) and continuation power flow (CPF) or MMA–CPF method. This approach combines the key features of both techniques. The MMA incorporates eigenvalue computation and the correlated eigenvectors of the reduced modified voltage–active power Jacobian matrix. MMA uses eigenvectors to compute the bus active participation factor (APF). The APF provides an indication of the participation of a certain bus in solving the instability problem of the network. On the other hand, the CPF reformulates the equation of power flow by using a prediction–correction stepping algorithm to reach the solution track and computes the tangent vector sensitivity (TVS). Both APF and TVS provide indications about the bus that has the largest influence in improving the system stability directly. Thus, the load bus that has the largest APF/TVS is chosen as the place for the DER unit;
- delivering a complete evaluation of the impact of DER allocation on system losses and assessment of voltage stability, which, in this case, are the smallest eigenvalues for the system as they are a common indicator for assessing the performance of system stability;
- enhancing the objective functions based on the power losses and eigenvalues to conclude the most suitable DER site when a difference between APF and TVS occurs. Formulation of this objective function provides a calculation in which bus will provide the least losses and the most stable system.

## 2. Modified Modal Analysis

## 3. Continuation Power Flow (CPF)

## 4. Proposed Methodology, System Constraints, Objective Function, and Evaluation Parameters

#### 4.1. Proposed Hybrid MMA–CPF Technique

- Step 1
- Input the system data.
- Step 2
- Execute power flow with Equation (1) and evaluate voltage stability for the original state (no DER unit) to compute voltage profile, system losses, and eigenvalue.
- Step 3
- (a) Execute MMA to compute APF at each load bus to define the most influential bus (bus
_{j-MMA}), and(b) execute CPF to compute TVS at each load bus to define the most sensitive bus (bus_{j-CPF}). - Step 4
- Compare the outcomes of MMA and CPF. If bus
_{j-MMA}≠ bus_{j-CPF}, go to Step 5; otherwise, go to Step 7. - Step 5
- Compute $\Delta {P}_{Loss}$, $\Delta {Q}_{Loss}$, and VSI.
- Step 6
- Compute the OF. The bus that has the highest objective function is recommended as the DER location.
- Step 7
- Set up DER in the designated bus.
- Step 8
- Execute power flow and evaluate voltage stability assessment to compute voltage profile, system losses, and eigenvalue.
- Step 9
- Assess the performance of the system as to if all the voltages are within the voltage limit constraints.
- Step 10
- If the bus voltage magnitudes are not fulfilled, then adjust the system data to acquire a new DER place and go back to Step 3.
- Step 11
- Once the voltage stability constraints are fulfilled, the program is terminated.

#### 4.2. Voltage Stability Constraints

#### 4.3. Eigenvalue Evaluation

^{th}modal voltage’s degree of stability.

#### 4.4. Network Power Losses

#### 4.5. Objective Function

## 5. Test Results and Analysis

#### 5.1. APF and TVS Computation for DER Location

#### 5.2. Voltage Profile Enhancement

#### 5.3. The System Smallest Eigenvalue (${\epsilon}_{min}$)

#### 5.4. Network Power Losses

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

APF | Active Participation Factor |

TVS | Tangent Vector Sensitivity |

RDN | Radial Distribution Network |

DG | Distributed Generation |

DER | Distributed Energy Resources |

MMA | Modified Modal Analysis |

CPF | Continuation Power Flow |

MMA–CPF | Modified Modal Analysis–Continuation Power Flow |

OF | Objective Function |

$\Delta P$ | Active power variations |

$\Delta Q$ | Reactive power variations |

∆𝜃 | Voltage angle variations |

∆𝑉 | Voltage magnitude variations |

J | Jacobian Matrix |

${J}_{R}^{*}$ | Reduced Modified Jacobian Matrix |

${\Re}^{*}$ | Right eigenvector matrix of ${J}_{R}^{*}$ |

${\mathsf{\upsilon}}^{*}$ | Left eigenvector matrix of ${J}_{R}^{*}$ |

${\mathsf{\phi}}^{*}$ | Diagonal eigenvalue matrix of ${J}_{R}^{*}$ |

${\epsilon}_{i}^{*}$ | ith eigenvalue of ${J}_{R}^{*}$ |

${\zeta}_{i}^{*}$ | ith column right eigenvector of ${J}_{R}^{*}$ |

${\varrho}_{i}^{*}$ | ith row left eigenvector of ${J}_{R}^{*}$ |

$\varpi $ | Load parameter |

${P}_{Gi0}$ | Base case active power generation at bus i |

${P}_{Li0}$ | Initial active power load at bus i |

${P}_{Ti}$ | Injected active power at bus i |

${Q}_{Gi0}$ | Base case reactive power generation at bus i |

${Q}_{Li0}$ | Initial reactive power load at bus i |

${Q}_{Ti}$ | Injected reactive power at bus i. |

${S}_{\mathsf{\Delta}base}$ | A specified amount of complex power that is selected to offer suitable $\varpi $ scaling |

${k}_{Gi}$ | Constant assigned for the degree of generation variation at bus i as $\varpi $ varies |

${k}_{Li}$ | Constant assigned for the degree of load variation at bus i as $\varpi $ varies |

${\theta}_{i}$ | Power angle changes at bus i |

$\overline{\delta}$ | Vector of generator angle |

$\overline{V}$ | Vector of the bus voltage magnitude vector |

${V}_{i}\angle {\delta}_{i}$ | Complex voltages at bus i |

${V}_{j}\angle {\delta}_{j}$ | Complex voltages at bus j |

${R}_{ij}+j{X}_{ij}={Z}_{ij}$ | ijth component of ${Z}_{bus}$ impedance matrix |

${P}_{i}$ | Active power generation at bus i |

${P}_{j}$ | Active power generation at bus j |

${Q}_{i}$ | Reactive power injection at bus i |

${Q}_{j}$ | Reactive power injection at bus j |

${P}_{Loss}$ | Active power losses at initial conditions without DER integration |

${P}_{Loss}^{DG}$ | Active power losses after integration of DER |

${Q}_{Loss}$ | Reactive power losses at initial conditions without DER integration |

${Q}_{Loss}^{DG}$ | Reactive power losses after integration of DER |

$\Delta {P}_{Loss}$ | Reduction in active power losses |

$\Delta {Q}_{Loss}$ | Reduction in reactive power losses |

$\%\Delta {P}_{Loss}$ | Reduction percentage of active power losses |

$\%\Delta {Q}_{Loss}$ | Reduction percentage of reactive power losses |

$VSI$ | Voltage stability index, indicating voltage stability improvement after DER placement |

${\epsilon}_{min}^{DG}$ | The smallest eigenvalue with DER unit(s) |

${\epsilon}_{min}$ | The smallest eigenvalue without any DER unit |

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**Figure 2.**IEEE 34-bus distribution network test system [19].

**Figure 13.**Voltage magnitude after DER integration with the CPF method and the proposed hybrid method.

**Table 1.**Summary of the results of the highest APF, TVS, and OF calculations to determine the DER location.

Iteration | Highest APF | Highest TVS | Highest OF | DER Location |
---|---|---|---|---|

1 | 26 | 26 | - | 26 |

2 | 33 | 33 | - | 33 |

3 | 11 | 17 | 11 | 11 |

**Table 2.**Results comparison in terms of the system smallest eigenvalue, and network losses reduction percentage.

High Values of APF/TVS (Recommended) | Small Values of APF/TVS | Average Values of APF/TVS | |
---|---|---|---|

DER Locations | 26, 33, and 11 | 1, 2, and 12 | 5, 15, and 29 |

${\mathsf{\epsilon}}_{\mathrm{min}}$ | 1.625 | 1.579 | 1.589 |

$\%\Delta {P}_{Loss}$ | 42.25 | 4.44 | 7.06 |

$\%\Delta {Q}_{Loss}$ | 44.59 | 8.88 | 14.27 |

Iteration | CPF Method [19] | Proposed Method Hybrid MMA–CPF |
---|---|---|

1 | 26 | 26 |

2 | 33 | 33 |

3 | 17 | 11 |

DER Placement | Approach | ${\mathit{\epsilon}}_{\mathit{m}\mathit{i}\mathit{n}}$ | $\mathit{\%}\mathit{\Delta}{\mathit{P}}_{\mathit{L}\mathit{o}\mathit{s}\mathit{s}}$ | $\mathit{\%}\mathit{\Delta}{\mathit{Q}}_{\mathit{L}\mathit{o}\mathit{s}\mathit{s}}$ | VSI (%) | OF |
---|---|---|---|---|---|---|

26, 33, and 17 | CPF [19] | 1.610 | 36.36 | 41.29 | 2.430335 | 80.08033 |

26, 33, and 11 | MMA–CPF | 1.618 | 42.25 | 44.59 | 2.939305 | 89.77931 |

26 and 33 | Both | 1.6067 | 30.34 | 35.33 | 2.220384 | 67.89038 |

26 | Both | 1.5988 | 8.87 | 18.97 | 1.717776 | 29.55778 |

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**MDPI and ACS Style**

Arief, A.; Nappu, M.B.
Novel Hybrid Modified Modal Analysis and Continuation Power Flow Method for Unity Power Factor DER Placement. *Energies* **2023**, *16*, 1698.
https://doi.org/10.3390/en16041698

**AMA Style**

Arief A, Nappu MB.
Novel Hybrid Modified Modal Analysis and Continuation Power Flow Method for Unity Power Factor DER Placement. *Energies*. 2023; 16(4):1698.
https://doi.org/10.3390/en16041698

**Chicago/Turabian Style**

Arief, Ardiaty, and Muhammad Bachtiar Nappu.
2023. "Novel Hybrid Modified Modal Analysis and Continuation Power Flow Method for Unity Power Factor DER Placement" *Energies* 16, no. 4: 1698.
https://doi.org/10.3390/en16041698