Microgrid modeling for contribution to the frequency control of power system

Abstract

Recently, Distributed Energy Resources (DERs) are becoming more attractive to supply local loads under the concept of microgrids. These new parts of the power system have basically different dynamics compared with conventional power plants. Most of them are connected to the grid by power electronic interfaces, and their dynamic is determined by their controller. In this paper, the effect of the increased penetration of DERs on the load frequency problem of power systems is studied. The DERs of microgrids in each area are controlled to change their active power at Point of Common Coupling (PCC) after a disturbance in the power system. It is shown that with appropriate control of DERs in microgrids, the frequency deviation of the power system will decrease and the stability margin can be increased.

Keywords

Load frequency control microgrid smart grid droop control voltage source convertor

Introduction

Frequency is a common parameter throughout the power system and a change in active power generation or demand at one bus affects the whole system frequency (Nikzad et al., 2011). Automatic generation control (AGC) is a common solution for matching the area generation to the area load and controlling the power system frequency (Kundur, 1994). The load changes affect frequencies in all areas and tie-line power exchange among areas. The Load Frequency Control (LFC) should reduce the area frequency deviation to zero, and control the tie-line power exchange, so that the demand and generation reach to a new stable equilibrium point (Hemmati et al., 2011).

Considering environmental and technical merits, utilizing distributed energy resources (DERs) is being more attractive in the smart grid environment. The most of DERs are interfaced to the grid via controlled power converters. Increasing the integration of power electronics interfaced DERs makes them a flexible and important part of the power system (Elavarasi and Saravanan, 2014).

The microgrids facilitate high depth of penetration of DER units and they can be considered and exploited as the main building block of future smart grids (Aref et al., 2012). Utilizing generation and storage units in a microgrid, can transform the distribution system from a passive network to active one. In the grid-connected operation mode, the microgrid is connected to the grid at the point of common coupling (PCC), and each DER unit generates proper real and reactive power (Gao and Iravani, 2008). Figure 1 shows a microgrid consisting of energy units, storage devices, and loads.

Figure 1.

Sample microgrid system.

VSCs (voltage source converters) have been proposed for interfacing some types of DERs to distribution grid. Fast dynamic response, accurate performance, ease of implementation, and its inherent closed loop control are some of their advantages. VSCs are synchronized by Phase Lock Loop (PLL) with the grid frequency and follow the frequency variations (Mahdian et al., 2013).

Main differences of DERs in comparison with conventional power plants are; they are installed in distribution networks and the capacity of each DER is small. Storage systems can improve the performance of this system to supply required active power determined by the control system. It means that DERs may also be responsible for frequency control in power system in grid-connected mode and therefore, a control system should coordinate them with the conventional LFC.

The contribution of electric vehicles (EV) and controllable loads in AGC has been presented in Almeida et al. (2010) and Galus et al. (2011). Masuta and Yokoyama (2012) have focused on the application of EVs and heat pump water heaters in LFC. Basak et al. (2012), Bevrani and Daneshmand (2012), Dai et al. (2012), and Zhang et al. (2012), coordinated control has been applied to doubly fed induction generators (DFIGs) and high-voltage direct current systems to decrease the frequency deviation. The application of storage systems for AGC has been studied in Goya et al. (2011), Sheikh et al. (2009), and Zhu and Hug-Glanzmann (2013), respectively.

In this paper, the injected active power of microgrids is contributed to the LFC problem and the controller is implemented in two levels. The first level is the control of microgrids connected to the system in each area and the second level is the control of interconnected areas of the entire system. To solve the problem of the second level, microgrids are considered as dynamic systems and their reduced order model is used. Because of high dynamic order of microgrid contents, the exact model of the system including microgrids will be complicated for LFC. Therefore, the modeling method and model order reduction are very important for this study. Different modeling methods, especially for wind turbines, are studied in Singh and Sundaram (2020, 2021) and Singh et al. (2019a, 2019b, 2019c, 2021).

It should be noted that controllable loads can also contribute in frequency control with appropriate control actions (Molina-Garcia et al., 2011; Shao et al., 2011). The dynamic model of a microgrid can be achieved based on the number of DERs, loads, their location and feeders of the microgrid (Katiraei et al., 2007; Pogaku et al., 2007; Tang et al., 2014). High penetration of these frequency-dependent microgrids can affect LFC, significantly.

The brief outline of the paper is as follows: Section 2 presents the microgrid modeling and its reduced order model in grid connected mode. Section 3 describes the control strategy of the power system including microgrids. Section 4 contains the simulation results and the conclusion is drawn in the last section.

Microgrid modeling

To study the microgrid active power response, the dynamic model of a typical microgrid (IEEE 34 Bus distribution system) is derived in this section. Figure 2 shows the microgrid including feeders and DERs. The load and feeders data are given in Baughman et al. (2006). For the sake of simplicity, the equivalent three phase model of the microgrid proposed in Nwakabuta and Sekar (2007) and used in Mehrizi-Sani and Iravani (2012), is considered in this paper. Also, loads are assumed to be fixed and dynamic equations of VSCs and feeders are used. Based on these simplifications, the system in Figure 3 can be used to achieve the microgrid dynamic model.

Figure 2.

IEEE 34 bus test system.

Figure 3.

Simplified model of IEEE 34 bus test system.

Also, for VSC modeling, deviations of DC bus voltage are neglected and the circuit diagram of the VSC shown in Figure 4 can be used. Due to the fast switching of power electronic devices, it can be assumed that $V_{o} = V_{ref}$ .

Figure 4.

Circuit diagram of VSC.

Three DERs are installed at buses 1, 2, and 3 of Figure 3. These are connected to the Point of Coupling (PC) by VSCs. This microgrid should follow the active power changes set by the coordinated control of the system.

This microgrid is divided into three subsystems as shown in Figure 3. The state space equations of each subsystem are derived as follows:

Subsystem1 contains DER 1 with three feeders 1, 4, and 5.

Subsystem 2 contains DER 2 and feeder 2.

Subsystem 3 contains DER 3 and feeder 3.

The controller of VSCs is shown in Figure 5. This controller is implemented based on Grid Imposed Frequency method in dq frame (Yazdani and Iravani, 2010) to decouple active and reactive power control. The DER controller should follow the references of active power and PC voltage.

Figure 5.

Controller of VSCs in dq frame.

$i_{dref}$ is set by the reference active power of DERs $(P_{ref})$ as shown in equation (1) and its linearalized form is given by equation (2).

i_{d r e f} = \frac{2}{3} \frac{P_{r e f}}{V_{b d}}

(1)

Δ i_{d r e f} = \frac{2}{3} (\frac{1}{V_{b d 0}} Δ P_{r e f} - \frac{P_{r e f 0}}{{V_{b d 0}}^{2}} Δ V_{b d})

(2)

As the voltage control is not considered in this work, $Q_{ref}$ and $i_{qref}$ are set to zero. Using these assumptions, state space equations of each subsystem in the standard form of $\overset{\cdot}{x} = Ax + Bu$ , are derived in equations (3)–(6).

[\begin{matrix} {\overset{\cdot}{ρ}}_{1 d} \\ {\overset{\cdot}{ρ}}_{1 q} \\ {\overset{\cdot}{i}}_{1 d} \\ {\overset{\cdot}{i}}_{1 q} \\ {\overset{\cdot}{i}}_{l 1 d} \\ {\overset{\cdot}{i}}_{l 1 q} \end{matrix}] = [\begin{matrix} 0 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 \\ \frac{K_{i}}{L_{c}} & 0 & \frac{- (r_{c} + k_{p})}{L_{c}} & 0 & 0 & 0 \\ 0 & \frac{K_{i}}{L_{c}} & 0 & \frac{- (r_{c} + k_{p})}{L_{c}} & 0 & 0 \\ 0 & 0 & 0 & 0 & - \frac{r_{l 1}}{L_{l 1}} & ω \\ 0 & 0 & 0 & 0 & - ω & - \frac{r_{l 1}}{L_{l 1}} \end{matrix}] [\begin{matrix} ρ_{1 d} \\ ρ_{1 q} \\ i_{1 d} \\ i_{1 q} \\ i_{l 1 d} \\ i_{l 1 q} \end{matrix}] + [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ - \frac{1}{L_{c}} & 0 & 0 & 0 \\ 0 & - \frac{1}{L_{c}} & 0 & 0 \\ \frac{1}{L_{l 1}} & 0 & - \frac{1}{L_{l 1}} & 0 \\ 0 & \frac{1}{L_{l 1}} & 0 & - \frac{1}{L_{l 1}} \end{matrix}] [\begin{matrix} V_{b 1 d} \\ V_{b 1 q} \\ V_{b 4 d} \\ V_{b 4 q} \end{matrix}]

(3)

[\begin{matrix} {\overset{\cdot}{ρ}}_{2 d} \\ {\overset{\cdot}{ρ}}_{2 q} \\ {\overset{\cdot}{i}}_{2 d} \\ {\overset{\cdot}{i}}_{2 q} \\ {\overset{\cdot}{i}}_{ld 2} \\ {\overset{\cdot}{i}}_{lq 2} \end{matrix}] = [\begin{matrix} 0 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 \\ \frac{K_{i}}{L_{c}} & 0 & \frac{- (r_{c} + k_{p})}{L_{c}} & 0 & 0 & 0 \\ 0 & \frac{K_{i}}{L_{c}} & 0 & \frac{- (r_{c} + k_{p})}{L_{c}} & 0 & 0 \\ 0 & 0 & 0 & 0 & - \frac{r_{l 2}}{L_{l 2}} & ω \\ 0 & 0 & 0 & 0 & - ω & - \frac{r_{l 2}}{L_{l 2}} \end{matrix}] [\begin{matrix} ρ_{2 d} \\ ρ_{2 q} \\ i_{2 d} \\ i_{2 q} \\ i_{ld 2} \\ i_{lq 2} \end{matrix}] + [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ - \frac{1}{L_{c}} & 0 & 0 & 0 \\ 0 & - \frac{1}{L_{c}} & 0 & 0 \\ \frac{1}{L_{l 2}} & 0 & - \frac{1}{L_{l 2}} & 0 \\ 0 & \frac{1}{L_{l 2}} & 0 & - \frac{1}{L_{l 2}} \end{matrix}] [\begin{matrix} V_{b 2 d} \\ V_{b 2 q} \\ V_{b 5 d} \\ V_{b 5 q} \end{matrix}]

(4)

[\begin{matrix} {\overset{\cdot}{ρ}}_{3 d} \\ {\overset{\cdot}{ρ}}_{3 q} \\ {\overset{\cdot}{i}}_{3 d} \\ {\overset{\cdot}{i}}_{3 q} \\ {\overset{\cdot}{i}}_{l 3 d} \\ {\overset{\cdot}{i}}_{l 3 q} \end{matrix}] = [\begin{matrix} 0 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 \\ \frac{K_{i}}{L_{c}} & 0 & \frac{- (r_{c} + k_{p})}{L_{c}} & 0 & 0 & 0 \\ 0 & \frac{K_{i}}{L_{c}} & 0 & \frac{- (r_{c} + k_{p})}{L_{c}} & 0 & 0 \\ 0 & 0 & 0 & 0 & - \frac{r_{l 3}}{L_{l 3}} & ω \\ 0 & 0 & 0 & 0 & - ω & - \frac{r_{l 3}}{L_{l 3}} \end{matrix}] [\begin{matrix} ρ_{3 d} \\ ρ_{3 q} \\ i_{3 d} \\ i_{3 q} \\ i_{l 3 d} \\ i_{l 3 q} \end{matrix}] + [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ - \frac{1}{L_{c}} & 0 & 0 & 0 \\ 0 & - \frac{1}{L_{c}} & 0 & 0 \\ \frac{1}{L_{l 3}} & 0 & - \frac{1}{L_{l 3}} & 0 \\ 0 & \frac{1}{L_{l 3}} & 0 & - \frac{1}{L_{l 3}} \end{matrix}] [\begin{matrix} V_{b 3 d} \\ V_{b 3 q} \\ V_{b 5 d} \\ V_{b 5 q} \end{matrix}]

(5)

[\begin{matrix} {\overset{\cdot}{i}}_{l 4 d} \\ {\overset{\cdot}{i}}_{l 4 q} \\ {\overset{\cdot}{i}}_{l 5 d} \\ {\overset{\cdot}{i}}_{l 5 q} \end{matrix}] = [\begin{matrix} - \frac{r_{l 4}}{L_{l 4}} & ω & 0 & 0 \\ - ω & - \frac{r_{l 4}}{L_{l 4}} & 0 & 0 \\ 0 & 0 & - \frac{r_{l 5}}{L_{l 5}} & ω \\ 0 & 0 & - ω & - \frac{r_{l 5}}{L_{l 5}} \end{matrix}] [\begin{matrix} i_{l 4 d} \\ i_{l 4 q} \\ i_{l 5 d} \\ i_{l 5 q} \end{matrix}] + [\begin{matrix} - \frac{1}{L_{l 4}} & 0 & \frac{1}{L_{l 4}} & 0 & 0 & 0 \\ 0 & - \frac{1}{L_{l 4}} & 0 & \frac{1}{L_{l 4}} & 0 & 0 \\ \frac{1}{L_{l 5}} & 0 & 0 & 0 & - \frac{1}{L_{l 5}} & 0 \\ 0 & \frac{1}{L_{l 5}} & 0 & 0 & 0 & - \frac{1}{L_{l 5}} \end{matrix}] [\begin{matrix} V_{b 4 d} \\ V_{b 4 q} \\ V_{b 5 d} \\ V_{b 5 q} \\ V_{b 6 d} \\ V_{b 6 q} \end{matrix}]

(6)

In these equations, the bus voltages are inputs to the subsystems and determine the interactions between them. As each subsystem of Figure 3 contains a DER and some transmission lines, equations (3)–(5) have state-space equations of DERs and lines of these subsystems, respectively. Variables ρ and i, are the state variables of each DER and i_l, is the line current. Equations related to are derived from the KVL of lines. Also, equations related to are derived from the block diagrams of Figures 4 and 5. Considering a resistive load at each bus, the bus voltages and line currents are related by equation (7). Substituting bus voltages at equation (7) in equations (3)–(6), the system state space equations are derived.

\begin{matrix} V_{b 1 d} = r_{lo 1} (i_{1 d} - i_{l 1 d}) V_{b 1 q} = r_{lo 1} (i_{1 q} - i_{l 1 q}) V_{b 2 d} = r_{lo 2} (i_{2 d} - i_{l 2 d}) V_{b 2 q} = r_{lo 2} (i_{2 q} - i_{l 2 q}) \\ V_{b 3 d} = r_{lo 3} (i_{3 d} - i_{l 3 d}) V_{b 3 q} = r_{lo 3} (i_{3 q} - i_{l 3 q}) V_{b 4 d} = r_{lo 4} (i_{l 1 d} + i_{l 4 d} - i_{l 5 d}) V_{b 4 q} = r_{lo 4} (i_{l 1 q} + i_{l 4 q} - i_{l 5 q}) \\ V_{b 5 d} = r_{lo 5} (i_{l 2 d} + i_{l 3 d} - i_{l 4 d}) V_{b 5 q} = r_{lo 5} (i_{l 2 q} + i_{l 3 q} - i_{l 4 q}) V_{b 6 d} = r_{lo 6} (i_{l 5 d} - i_{lud}) V_{b 6 q} = r_{lo 6} (i_{l 5 q} - i_{luq}) \end{matrix}

(7)

where $r_{lok}$ is the resistive load of bus k.

Integrating these three subsystems, the microgrid dynamic equations of the entire system are derived in equation (8).

A = {[\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix}]}_{22 \times 22} B = {[\begin{matrix} B_{1} & 0 & 0 \\ 0 & B_{2} & 0 \\ 0 & 0 & B_{3} \end{matrix}]}_{22 \times 3}

(8)

where:

A_{11} = [\begin{matrix} 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{K_{i}}{L_{c}} & 0 & \frac{- (r_{c} + k_{p} + r_{lo 1})}{L_{c}} & 0 & \frac{r_{lo 1}}{L_{c}} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{K_{i}}{L_{c}} & 0 & \frac{- (r_{c} + k_{p} + r_{lo 1})}{L_{c}} & 0 & \frac{r_{lo 1}}{L_{c}} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{r_{lo 1}}{L_{l 1}} & 0 & \frac{- (r_{lo 1} + r_{lo 4} + r_{l 1})}{L_{l 1}} & ω & \frac{- r_{lo 4}}{L_{l 1}} & 0 & \frac{r_{lo 4}}{L_{l 1}} & 0 \\ 0 & 0 & 0 & \frac{r_{lo 1}}{L_{l 1}} & ω & \frac{- (r_{lo 1} + r_{lo 4} + r_{l 1})}{L_{l 1}} & 0 & \frac{- r_{lo 4}}{L_{l 1}} & 0 & \frac{r_{lo 4}}{L_{l 1}} \\ 0 & 0 & 0 & 0 & \frac{- r_{lo 4}}{L_{l 4}} & 0 & \frac{- (r_{lo 4} + r_{lo 5} + r_{l 4})}{L_{l 4}} & ω & \frac{- r_{lo 5}}{L_{l 4}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{- r_{lo 4}}{L_{l 4}} & - ω & \frac{- (r_{lo 4} + r_{lo 5} + r_{l 4})}{L_{l 4}} & 0 & \frac{- r_{lo 5}}{L_{l 4}} \\ 0 & 0 & 0 & 0 & \frac{r_{lo 4}}{L_{l 5}} & 0 & \frac{r_{lo 4}}{L_{l 5}} & 0 & \frac{- (r_{lo 4} + r_{lo 6} + r_{l 5})}{L_{l 5}} & ω \\ 0 & 0 & 0 & 0 & 0 & \frac{r_{lo 4}}{L_{l 5}} & 0 & \frac{r_{lo 4}}{L_{l 5}} & - ω & \frac{- (r_{lo 4} + r_{lo 6} + r_{l 5})}{L_{l 5}} \end{matrix}]

(9)

A_{22} = [\begin{matrix} 0 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 \\ \frac{K_{i}}{L_{c}} & 0 & \frac{- (r_{c} + k_{p} + r_{lo 2})}{L_{c}} & 0 & \frac{r_{lo 2}}{L_{c}} & 0 \\ 0 & \frac{K_{i}}{L_{c}} & 0 & \frac{- (r_{c} + k_{p} + r_{lo 2})}{L_{c}} & 0 & \frac{r_{lo 2}}{L_{c}} \\ 0 & 0 & \frac{r_{lo 2}}{L_{l 2}} & 0 & \frac{- (r_{lo 2} + r_{lo 5} + r_{l 2})}{L_{l 2}} & ω \\ 0 & 0 & 0 & \frac{r_{lo 2}}{L_{l 2}} & - ω & \frac{- (r_{lo 2} + r_{lo 5} + r_{l 2})}{L_{l 2}} \end{matrix}]

(10)

A_{33} = [\begin{matrix} 0 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 \\ \frac{K_{i}}{L_{c}} & 0 & \frac{- (r_{c} + k_{p} + r_{lo 3})}{L_{c}} & 0 & \frac{r_{lo 3}}{L_{c}} & 0 \\ 0 & \frac{K_{i}}{L_{c}} & 0 & \frac{- (r_{c} + k_{p} + r_{lo 3})}{L_{c}} & 0 & \frac{r_{lo 3}}{L_{c}} \\ 0 & 0 & \frac{r_{lo 3}}{L_{l 3}} & 0 & \frac{- (r_{lo 3} + r_{lo 5} + r_{l 3})}{L_{l 3}} & ω \\ 0 & 0 & 0 & \frac{r_{lo 3}}{L_{l 3}} & - ω & \frac{- (r_{lo 3} + r_{lo 5} + r_{l 3})}{L_{l 3}} \end{matrix}]

(11)

A_{12} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{r_{lo 5}}{L_{l 4}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{r_{lo 5}}{L_{l 4}} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

(12)

A_{13} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{r_{lo 5}}{L_{l 4}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{r_{lo 5}}{L_{l 4}} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

(13)

A_{21} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{r_{lo 5}}{L_{l 2}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{r_{lo 5}}{L_{l 2}} & 0 & 0 \end{matrix}]

(14)

A_{23} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{- r_{lo 5}}{L_{l 2}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{- r_{lo 5}}{L_{l 2}} \end{matrix}]

(15)

A_{31} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{r_{lo 5}}{L_{l 3}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{r_{lo 5}}{L_{l 3}} & 0 & 0 \end{matrix}]

(16)

A_{32} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{- r_{lo 5}}{L_{l 3}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{- r_{lo 5}}{L_{l 3}} \end{matrix}]

(17)

B_{1} = [\begin{matrix} \frac{2}{3 V_{bd 10}} \\ 0 \\ \frac{2}{3} \frac{k_{p}}{L_{c} V_{bd 10}} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}] B_{2} = [\begin{matrix} \frac{2}{3 V_{bd 20}} \\ 0 \\ \frac{2}{3} \frac{k_{p}}{L_{c} V_{bd 20}} \\ 0 \\ 0 \\ 0 \end{matrix}] B_{3} = [\begin{matrix} \frac{2}{3 V_{bd 30}} \\ 0 \\ \frac{2}{3} \frac{k_{p}}{L_{c} V_{bd 30}} \\ 0 \\ 0 \\ 0 \end{matrix}]

(18)

To incorporate the microgrid in the load-frequency control system, a reduced-order model is derived using Model Order Reduction (MOR) method. In this paper, the balanced realization method (Jamshidi, 1997) is used for MOR, which computes a similar transformation matrix S such that both controllability and observability Gramians become equal and diagonal, that is, balanced (Jamshidi, 1997). These matrices are defined in equation (19).

\begin{matrix} G_{c} = \int_{0}^{\infty} e^{At} B B^{T} e^{A^{T} t} dt \\ G_{o} = \int_{0}^{\infty} e^{At} C C^{T} e^{A^{T} t} dt \end{matrix}

(19)

These matrixes satisfy equation (20).

\begin{matrix} G_{c} A^{T} + A G_{c} + B B^{T} = 0 \\ G_{o} A^{T} + A G_{o} + C^{T} C = 0 \end{matrix}

(20)

It can be shown that if a transformation matrix S can be found to satisfy equation (21), $G_{o}$ satisfies equation (22).

S = L_{c} U Σ^{- \frac{1}{2}}

(21)

U^{T} ({L_{c}}^{T} G_{o} L_{c}) U = Σ^{2}

(22)

Then, ${\overset{⌢}{G}}_{o}$ and ${\overset{⌢}{G}}_{c}$ define as equation (23).

\begin{array}{l} {\overset{⌢}{G}}_{o} = S^{T} G_{o} S \\ {\overset{⌢}{G}}_{c} = S^{- 1} G_{c} S \end{array}

(23)

It can be seen that we have:

{\overset{⌢}{G}}_{c} = {\overset{⌢}{G}}_{o} = Σ = d i a g (σ_{1}, σ_{2}, \dots, σ_{n})

(24)

$L_{c}$ and $L_{o}$ are derived from Gramians, as follows :

\begin{matrix} G_{c} = L_{c} {L_{c}}^{T} \\ G_{o} = L_{o} {L_{o}}^{T} \end{matrix}

(25)

Using singular value decomposition of ${L_{o}}^{T} L_{c}$ , S can be determined by equations (26) and (27).

{L_{o}}^{T} L_{c} = V Σ U^{T}

(26)

S = L_{c} U Σ^{- \frac{1}{2}}

(27)

Applying the following transformation to the state space equations of the system, reduced order model of the system can be derived according to the Hankel norm deliberation as shown in equations (28)–(32).

\overset{⌢}{A} = S^{- 1} AS

(28)

\overset{⌢}{B} = S^{- 1} B

(29)

\overset{⌢}{C} = CS

(30)

\overset{⌢}{A} = [\begin{matrix} A' & {\overset{⌢}{A}}_{12} \\ {\overset{⌢}{A}}_{21} & {\overset{⌢}{A}}_{22} \end{matrix}] \overset{⌢}{B} = [\begin{matrix} B' \\ {\overset{⌢}{B}}_{2} \end{matrix}] \overset{⌢}{C} = [\begin{matrix} C' & {\overset{⌢}{C}}_{2} \end{matrix}] \hat{x} = [\begin{matrix} z \\ x_{2} \end{matrix}]

(31)

\begin{matrix} \overset{\cdot}{z} (t) = A' z (t) + B' u (t) \\ y (t) = c' z (t) \end{matrix}

(32)

The step response of the main system and the reduced order one is shown in Figure 6. It can be seen that the reduced first order system is an acceptable approximation of the main system.

Figure 6.

Step response of the main microgrid and the reduced order one.

The transfer function of the system with the reference active power of DERs (Pref) as input and the injected active power of the microgrid at PCC as output is presented by equation (33).

G_{MG} (s) = \frac{0.6}{1 + 0.2 s}

(33)

This transfer function is used in the second level of the hierarchical control.

Control of power system

In this section, the load frequency control of a two area power system is studied using the droop control method. In this work, each area of the power system is considered as a subsystem and interactions between them are determined by tie lines. Figure 7 shows the block diagram of the two area power system including microgrids.

Figure 7.

Block diagram of two area power system.

$Δ u_{1}$ and $Δ u_{2}$ are input signals to the microgrids in area 1 and 2, respectively. These signals are provided by droop control to determine the injected active power of microgrids to the system, as shown in Figure 8.

Figure 8.

Droop characteristic of microgrids.

The transfer function of the turbine and governor is shown in Figure 9 (Kundur, 1994).

Figure 9.

Transfer function of turbine and governor.

The state-space equations of the system are derived from the block diagram of Figure 7 and given in equation (34). As each area of the two-area power system of Figure 7 is a fifth-order dynamic system, the state space equation of the entire system has 10 state variables. These variables for area 1 are the rotor angle of the equivalent generator ( $Δ δ_{1})$ , the rotor speed of the generator ( $Δ ω_{1})$ , injected power of microgrid ( $Δ P_{MG 1})$ , the mechanical power of generator ( $Δ P_{m 1})$ , and valve position of turbine ( $Δ Y_{1})$ . Laplace transform of equation (34) results in the block diagram of Figure 7 and its transfer functions.

\begin{array}{l} [\begin{array}{l} Δ {\dot{δ}}_{1} \\ Δ {\dot{ω}}_{1} \\ Δ {\dot{P}}_{M G 1} \\ Δ {\dot{P}}_{m 1} \\ Δ {\dot{Y}}_{1} \\ Δ {\dot{δ}}_{2} \\ Δ {\dot{ω}}_{2} \\ Δ {\dot{P}}_{M G 2} \\ Δ {\dot{P}}_{m 2} \\ Δ {\dot{Y}}_{2} \end{array}] = [\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ - \frac{T}{M_{1}} & - \frac{D}{M_{1}} & - \frac{1}{M_{1}} & \frac{1}{M_{1}} & 0 & \frac{T}{M_{1}} & 0 & 0 & 0 & 0 \\ 0 & 0 & - \frac{1}{T_{M G 1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - \frac{1}{T_{C H 1}} & \frac{1}{T_{C H 1}} & 0 & 0 & 0 & 0 & 0 \\ 0 & - \frac{1}{R_{1} T_{G 1}} & 0 & 0 & - \frac{1}{T_{G 1}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ \frac{T}{M_{2}} & 0 & 0 & 0 & 0 & - \frac{T}{M_{2}} & - \frac{D}{M_{2}} & - \frac{1}{M_{2}} & \frac{1}{M_{2}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{1}{T_{M G 2}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{1}{T_{C H 2}} & \frac{1}{T_{C H 2}} \\ 0 & 0 & 0 & 0 & 0 & 0 & - \frac{1}{R_{2} T_{G 2}} & 0 & 0 & - \frac{1}{R_{2} T_{G 2}} \end{matrix}] [\begin{array}{l} Δ δ_{1} \\ Δ ω_{1} \\ Δ P_{M G 1} \\ Δ P_{m 1} \\ Δ Y_{1} \\ Δ δ_{2} \\ Δ ω_{2} \\ Δ P_{M G 2} \\ Δ P_{m 1} \\ Δ Y_{2} \end{array}] \\ + [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{1}{T_{M G 1}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & - \frac{1}{T_{G 1}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{T_{M G 2}} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - \frac{1}{T_{G 2}} \end{matrix}] [\begin{matrix} Δ u_{1} \\ Δ P_{r e f 1} \\ Δ u_{2} \\ Δ P_{r e f 2} \end{matrix}] \end{array}

(34)

Simulation results

The control method proposed in the previous section is applied to a two area power system given in Kundur (1994) and simulation results are presented in this section. It is assumed that the penetration of DERs in the power system generation capacity is 0.3 pu. Performance of the proposed method is studied after a disturbance (load change in area 1) and frequency deviation of 0.1 pu in area 1. Figures 10 and 11 show the frequency deviation of area 1 and 2 after the disturbance, respectively. It can be seen that the contribution of microgrids could improve the frequency oscillations considerably.

Figure 10.

Frequency deviations of area 1 after disturbance.

Figure 11.

Frequency deviations of area 2 after disturbance.

The other goal of LFC, besides minimizing frequency deviations, is to minimize the deviations of the power transfer between areas of the multi-area power system. This power transfer depends on the angle difference between sending and receiving end of the tie lines. Figures 12 and 13 show the angle deviation $Δ δ_{1}$ and $Δ δ_{2}$ of area 1 and 2, respectively.

Figure 12.

$Δ δ_{1}$ after disturbance.

Figure 13.

$Δ δ_{2}$ after disturbance.

It can be seen that the proposed controller could eliminate the angle deviations at both endings of the tie line.

Conclusions

In this paper, the effect of the high penetration of DERs on power system frequency control has been studied. Using the advantage of the VSC fast dynamic, the injected active power of DERs can be rapidly changed. So, they are good candidates to improve frequency control. As microgrids have various dynamic components, the exact dynamic model of microgrids may complicate the system analysis and control. To overcome the shortcoming of power system modeling, a proper model order reduction method has been used and the reduced order dynamic model of microgrids including DERs has been derived in this paper. Simulation results confirm the efficiency of the reduced-order model for the frequency control studied. Then, the proposed control method has been applied to a two-area power system including microgrids. The simulation results show the effectiveness of the proposed method to improve the power system frequency stability.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Alireza Ghafouri

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