# A Building Block Method for Modeling and Small-Signal Stability Analysis of the Autonomous Microgrid Operation

^{*}

## Abstract

**:**

## 1. Introduction

- The presented modeling approach provides higher flexibility through modularity, less chance for error through a better model overview, and easier microgrid analysis with tested toolboxes compared to traditional state-space matrix obtainment.
- When models are created using the graphical building blocks method, elements in it are connected with other elements in the way to imitate the physical layout of the microgrid; thus, more natural modeling of the complex network configuration with different types of sources and loads is possible.
- More detailed AC/DC and DC/DC converter models than those available in the literature are presented and analyzed as a direct benefit of the proposed modeling approach.
- To support scientific collaboration and microgrid research, MATLAB/Simulink models from this paper, with accompanying documentation, are published on this paper’s MDPI webpage.

## 2. Autonomous Microgrid Small-Signal State-Space Modeling

- Boost converter subsystem (power model, voltage and current controllers, digital control emulator models);
- VSC subsystem (VSC with LCL filter model, power controller with virtual impedance loop, voltage and current controllers, dead-time, digital control emulator models);
- Network subsystem (a resistive/indictive nature of power lines);
- Load mode subsystem (a resistive/inductive load model).

#### 2.1. State-Space Model of the DC/DC Boost Converter Subsystem

_{dc}to a reference value. For a power input from RES, the voltage source is assumed. Figure 2 presents a complete block diagram of the boost converter with a cascaded proportional-integral (PI) voltage and current controllers and Digital Control Emulator (DCE) blocks.

#### 2.1.1. DC/DC Boost Converter Power Model

_{lb}and resistance R

_{lb}), DC-link capacitor (capacitance C

_{dc}), the resistance of the switching device during its conduction R

_{onb}, and the forward diode voltage drop U

_{Db}. It is represented as a combination of the turn-on and turn-off states [32].

_{inb}, u

_{dc}, i

_{inb}, i

_{dc}that average one switching cycle can be written as in Equations (1) and (2), respectively [33].

_{b}. Linearizing the average model described with Equations (1) and (2) around the operating point, the small-signal state-space model of the boost converter can be obtained as in Equation (3).

_{b}, U

_{dc}, and I

_{inb}represent the steady-state duty cycle, output DC-link voltage, and input inductor current, respectively. A similar principle is applied throughout the paper, where the upper-case symbols after linearization stand for the operating point of the system.

#### 2.1.2. Voltage and Current Controllers

_{pub}and K

_{iub}coefficients are proportional and integral terms of the boost voltage controller. Linearization of Equations (4) and (5) provides the following small-signal state-space model of the voltage controller:

_{b}, obtained according to the following set of algebraic equations:

_{pib}and K

_{iib}coefficients are proportional and integral terms of the boost current controller. By linearizing Equations (8) and (9) around the operating point, the small-signal state-space model of the current controller is obtained and presented in Equations (10) and (11) as:

#### 2.1.3. Digital Control Emulator Model

_{s}is the equivalent sampling period delay, and Laplace complex variable is denoted with s. The Pade approximation of Equation (12), as presented in [38], is used for the creation of a linear state-space model. In the boost converter subsystem, the DCE transfer function is applied on all input signals of the current controller.

#### 2.2. State-Space Model of the DC/AC Voltage Source Converter Subsystem

#### 2.2.1. Reference Frame Transformation

#### 2.2.2. Power Controller

_{c}and the grid current i

_{g}as in [15]. LPF with a cutoff frequency ω

_{c}is used to obtain filtered active and reactive power P and Q corresponding to the fundamental component.

_{n}, ω

_{g}, and ω

_{gcom}are frequencies during ideal no-load operation, when DGU is loaded with active power P, and the frequency of common reference frame, respectively. U

_{nd}is the d voltage component setpoint and ${u}_{cd}^{\prime}$, ${u}_{cq}^{\prime}$ are the output voltage d-q components when DGU is providing reactive power Q. The static droop gains for active power m

_{p}and reactive power n

_{q}are defined individually for every DGU according to its power capacity, allowed operating frequency range, and voltage deviation in the microgrid.

#### 2.2.3. Virtual Impedance

#### 2.2.4. VSC Voltage and Current Controllers

_{pu}and K

_{iu}are proportional and integral gain of the voltage controller, respectively, while C

_{f}represents the phase capacitance of the LCL filter.

_{pi}and K

_{ii}present the proportional and integral current gain, respectively, while L

_{i}corresponds to the converter side inductor. With linearization of Equations (31)–(33), the state-space model for the current controller in the small-signal form is obtained as:

#### 2.2.5. Three-Phase Two-Level Three-Wire VSC with Space Vector Modulation and LCL Filter

_{d}and d

_{q}as:

_{dc}and the d-q components of the VSC output current i

_{id}and i

_{iq}is achieved based on equality between the active power in DC-link p

_{dc}and active power at the output of the VSC p through:

_{id}and i

_{iq}as:

#### 2.2.6. Digital Control Emulator for VSC

#### 2.3. State-Space Model of the Equivalent Power Line Subsystem

^{th}line, between the two nodes x and k in Figure 9 could be described according to Equations (48) and (49).

_{n}is applied. A symbolic form of node voltages is given in [12,13,42] as follows:

#### 2.4. State-Space Model of the Load Subsystem

^{th}node of the microgrid are:

## 3. Proposed Building Block Modeling Method

#### 3.1. State-Space Model Creation

_{sys}[12,13,15,16,17,18,19,20,42]:

- The converter matrix creation only with an appropriate connection of consisted elements, in a physically logical way. It allows the complex system modeling through the creation of individual control and power stage blocks, and their appropriate graphical connection to form the models of the microgrid subsystems.
- The easy debugging procedure, since the inspection of the variable of interest requires an only connection with a scope, as in any MATLAB/Simulink simulation. Greater clarity and monitoring of the model input/output signals in the blocks reduce the possibility of modeling errors.
- The easy modification of the desired element and with it model complexity. Greater flexibility and modularity of the system modeling is allowed, and also the spent time is significantly reduced. It is possible to easily remove, modify, or insert new blocks of different complexity in a model.

#### 3.2. The Overview of the DGU Model

#### 3.2.1. Boost Converter Subsystem

_{dce}, B

_{dce}, C

_{dce}, and D

_{dce}that model DCE [43]. The next building block in the boost subsystem model, according to Figure 13, is the current controller block ③ modeled based on Equations (10) and (11). The boost duty cycle reference $\Delta {d}_{b}$ as output from the current controller block is together with the source voltage $\Delta {u}_{inb}$ and DC-link current $\Delta {i}_{dc}$ forwarded to the block that models physical boost converter ④. The boost converter model block is formed based on the state-space Equation (3). Its outputs are the DC-link voltage $\Delta {u}_{dc}$ and inductor current $\Delta {i}_{dc}$, which are used as feedback to the model. A selector next to this block gives a possibility to work with constant voltage $\Delta {u}_{dc}$ by disabling the boost subsystem model from DGU or enabling it by forwarding the voltage $\Delta {u}_{dc}$ to the VSC subsystem model. Based on the state of this selector, a boost subsystem model may or may not be included in the analysis.

#### 3.2.2. VSC Subsystem

_{idq}and the DC link current $\Delta {i}_{dc}$ is made with block ⑩ according to Equation (39).

#### 3.3. Network Subsystem

#### 3.4. Load Subsystem

^{th}node is formed according to the small-signal state-space Equation (55), and it is presented in Figure 15. The inputs of the block are voltage $\Delta {u}_{kDQ}$ in the k

^{th}node of the network and frequency of the common reference frame $\Delta {\omega}_{gcom}$, while the output is the k

^{th}node load current $\Delta {i}_{loadkDQ}$.

#### 3.5. A Complete Microgrid Model

- Fast parameter and control structure change, as can be seen from the example of DGU1 showed in Figure 16b.
- An easy selection of the elements and control structure blocks that should be included in the analysis. Model scalability is achieved through the simple selection of the desired elements.
- The switch between the dynamic model and linear model by a simple selection in the GUI. The dynamic model can be compared to the model made of SimPowerSystem components.

#### 3.6. Procedure for Small-Signal Model Analysis

## 4. Demonstration of the Applied Methodology

#### 4.1. Building Block Model Validation

#### 4.2. Eigenvalue Analysis Using LAT

#### 4.3. Participation Factor Analysis

#### 4.4. Singular Value Decomposition Analysis

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Detailed representation of the boxed areas from Figure 20b: (

**a**) boxed area 1, (

**b**) boxed area 2, (

**c**) boxed area 3, (

**d**) boxed area 4, (

**e**) boxed area 5.

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**Figure 3.**A DC/DC boost converter power model representation: (

**a**) Switch-on state; (

**b**) Switch-off state.

**Figure 7.**VSC controller realization with decoupling: (

**a**) Voltage controller; (

**b**) Current controller.

**Figure 8.**LCL-filter equivalent circuit: (

**a**) d-axis equivalent circuit; (

**b**) q-axis equivalent circuit.

**Figure 9.**The j

^{th}line equivalent circuit: (

**a**) d-axis equivalent circuit; (

**b**) q-axis equivalent circuit.

**Figure 10.**Comparison between voltage controller matrix definition: (

**a**) textual language; (

**b**) using the building block method.

**Figure 11.**Part of the converter state-space matrix creation: (

**a**) textual language; (

**b**) using the proposed method.

**Figure 12.**The general approach for element creation: (

**a**) The state-space representation of an element creation; (

**b**) The general representation of the MIMO block used in an element modeling.

**Figure 16.**The complete Simulink model of the microgrid: (

**a**) DGUs, network, and loads created using building blocks approach; (

**b**) Front-end mask of the DGU1.

**Figure 18.**Comparison of the dynamic response of the microgrid built using Simscape, model I, and model II (from Table 1) in the case of DGU in node 1: (

**a**) converter current; (

**b**) capacitor voltage; (

**c**) active and reactive power and; (

**d**) output frequency.

**Figure 20.**Participation factor analysis of the considered microgrid: (

**a**) microgrid model I; (

**b**) microgrid model II.

STATE-SPACE MICROGRID MODEL I (67 states) | ||

Blocks | Subsection in paper | Number of states |

Load in nodes 1,2,3,4,5 and 6 | 2.4 | 12 |

Network | 2.3 | 10 |

LCL-filter without dead-time effect and VSC model | 2.2.5 * | 6 per DGU |

Power controller | 2.2.2 | 3 per DGU |

Virtual impedance with LPF | 2.2.3 | 2 per DGU |

VSC current and voltage controllers | 2.2.4 | 4 per DGU |

STATE-SPACE MICROGRID MODEL II (247 states) | ||

Additional blocks to MODEL I | Subsection in paper | Number of states |

LCL-filter with dead-time effect and VSC model | 2.2.5 | 6 per DGU |

Digital control emulator | 2.2.6 | 42 per DGU |

Boost converter model | 2.1.1 | 2 per DGU |

Boost current and voltage controllers | 2.1.2 | 2 per DGU |

Digital control emulator of boost | 2.2.6 | 14 per DGU |

* Can be obtained by setting T_{dt} = 0 in Equations (41) and (42) |

DGU1 = DGU2 = DGU3 | |||
---|---|---|---|

Parameter | Value | ||

Nominal active power P_{n} | 100 kW | ||

Nominal phase grid voltage V_{n} | 230 V | ||

Nominal grid frequency f_{g} | 50 H_{Z} | ||

VSC parameters | Value | VSC parameters | Value |

Switching frequency f_{sw} | 10 kHz | Power coefficient m_{p} | 3.14 × 10^{−6} |

Dead time T_{dt} | 2 μs | Power coefficient n_{q} | 9 × 10^{−4} |

Equivalent sampling frequency T_{s} | 0.5*100 μs | LPF cut-off frequency ω_{c} | 62.8 rad/s |

Output voltage V_{c} | 230 V | VC proportional gain K_{pu} | 0.2475 |

Filter inductance L_{i} | 163 μH | VC integral gain K_{iu} | 437.5 |

Filter resistance R_{i} | 3 mΩ | CC proportional gain K_{pi} | 1.4224 |

Filter capacitance C_{f} | 70 μF | CC integral gain K_{ii} | 1241.3 |

Filter damping resistor R_{f} | 0.21 Ω | Virtual impedance Z_{vsc1} | (19.6 + j3.9) mΩ |

Filter inductance L_{g} | 34 μH | Virtual impedance Z_{vsc2} | (0 + j0) mΩ |

Filter resistance R_{g} | 1 mΩ | Virtual impedance Z_{vsc3} | (38.7 + j7.8) mΩ |

Virtual impedance LPF ω_{c} | 62.8 rad/s | ||

Boost parameters | Value | Boost parameters | Value |

DC-link voltage V_{dc} | 800 V | Switching resistance R_{onb} | 2 mΩ |

DC-link capacitor C_{dc} | 10 mF | Diode voltage drop V_{Db} | 1.1 V |

Input voltage V_{inb} | 540 V | VC proportional gain K_{pub} | 11.869 |

Filter inductance L_{lb} | 300 μH | VC integral gain K_{iub} | 364.07 |

Filter resistance R_{lb} | 1 mΩ | CC proportional gain K_{pib} | 0.0011 |

Switching frequency f_{swb} | 10 kHz | CC integral gain K_{iib} | 1.3229 |

Grid parameters | Value | Load parameters | Value |

Line impedance Z_{l14} | (0.1162 + j0.0233) Ω | Load impedance Z_{load1} | (7 + j2.198) Ω |

Line impedance Z_{l25} | (0.1356 + j0.0271) Ω | Load impedance Z_{load2} | (14 + j4.396) Ω |

Line impedance Z_{l36} | (0.0969 + j0.0194) Ω | Load impedance Z_{load3} | (4.85 + j1.445) Ω |

Line impedance Z_{l45} | (0.0193 + j0.0091) Ω | Load impedance Z_{load4} | (2.4 + j0.7536) Ω |

Line impedance Z_{l56} | (0.0231 + j0.011) Ω | Load impedance Z_{load5,6} | (1.6 + j0.5024) Ω |

Steady-state operating points (DGU1 = 1, DGU2 = 2, DGU3 = 3) | |||
---|---|---|---|

Parameter | Value | Parameter | Value |

I_{id1}, I_{id2}, I_{id3} [A] | [180, 177, 182] | I_{lb1,}I_{lb2} I_{lb3} [A] | [148, 148, 148] |

I_{iq1}, I_{iq2}, I_{iq3} [A] | [−51, −45, −52] | D_{b1}, D_{b2}, D_{b3} | [0.326, 0.326, 0.326] |

U_{cd1}, U_{cd2}, U_{cd3} [V] | [299, 304, 295] | I_{14D}, I_{14Q} [A] | [141, −42] |

U_{cq1}, U_{cq2}, U_{cq3} [V] | [0.5, 0, 1] | I_{25D}, I_{25Q} [A] | [156, −42] |

I_{gd1}, I_{gd2}, I_{gd3} [A] | [179, 175, 181] | I_{36D}, I_{36Q} [A] | [127, −40] |

I_{gq1}, I_{gq2}, I_{gq3} [A] | [−57, −51, −58] | I_{45D}, I_{45Q} [A] | [34.5, −11] |

U_{gD1}, U_{gD2}, U_{gD3} [V] | [296, 302, 292] | I_{56D}, I_{56Q} [A] | [32, −6] |

U_{gQ1}, U_{gQ2}, U_{gQ3} [V] | [3, 3, 3] | I_{load1D}, I_{load1Q} [A] | [39, −12] |

δ_{01}, δ_{02}, δ_{03} [rad] | [0, −3.88 × 10^{−4}, −3.3 × 10^{−3}] | I_{load2D}, I_{load2Q} [A] | [19.6, −6] |

D_{d1}, D_{d2}, D_{d3} | [0.375, 0.381, 0.37] | I_{load3D}, I_{load3Q} [A] | [55, −16] |

D_{q1}, D_{q2}, D_{q3} | [0.017, 0.017, 0.017] | I_{load4D}, I_{load4Q} [A] | [106, −31] |

U_{dc1}, U_{dc2}, U_{dc3}, [V] | [800, 800, 800] | I_{load5D}, I_{load5Q} [A] | [159, −47] |

I_{load6D}, I_{load6Q} [A] | [158, −47] |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Banković, B.; Filipović, F.; Mitrović, N.; Petronijević, M.; Kostić, V.
A Building Block Method for Modeling and Small-Signal Stability Analysis of the Autonomous Microgrid Operation. *Energies* **2020**, *13*, 1492.
https://doi.org/10.3390/en13061492

**AMA Style**

Banković B, Filipović F, Mitrović N, Petronijević M, Kostić V.
A Building Block Method for Modeling and Small-Signal Stability Analysis of the Autonomous Microgrid Operation. *Energies*. 2020; 13(6):1492.
https://doi.org/10.3390/en13061492

**Chicago/Turabian Style**

Banković, Bojan, Filip Filipović, Nebojša Mitrović, Milutin Petronijević, and Vojkan Kostić.
2020. "A Building Block Method for Modeling and Small-Signal Stability Analysis of the Autonomous Microgrid Operation" *Energies* 13, no. 6: 1492.
https://doi.org/10.3390/en13061492