Distributed Control of Multiple-Bus Microgrid With Paralleled Distributed Generators

Ⅰ. INTRODUCTION

Microgrid generally comprises a collection of loads, energy storage systems, and distributed generators (DGs), operating in either grid-connected mode or islanded one [1], [2]. Different types of DGs such as photovoltaics and wind turbines can be connected to a microgrid through power converters [3]. The power electronic-based converters make the DGs more flexible in their control and operation than conventional synchronous generators (SGs).

However, due to the significant differences between the traditional large-scale power systems and power converter- interfaced microgrids, especially in terms of low inertia and uncertainties, control solutions for traditional power grid cannot be introduced to microgrids directly [3]. Recent researches on microgrid controls can be classified into two types. The first one is to make the DGs in a microgrid behave similarly to conventional SGs with the idea of virtual synchronous generator [4], [5]. Since the virtual inertias of microgrids are increased, traditional control schemes that have been proved to be effective for traditional power grids can be introduced to microgrids [6]. However, such solutions cannot fully unlock the advantages of modern power electronic techniques, especially in terms of flexibility and response speed [7].

Alternatively, the second type of solutions model a microgrid as a cluster of decoupled subsystems, each of which is composed of a DG, a power converter, and an output filter [8]. In the hierarchical control scheme, a secondary control algorithm is developed on top of the traditional primary control system [9], [10]. Since the microgrid system is formulated similarly to that of unmanned vehicles that have no physical connections among the subsystems, various control techniques for multiagent systems (MASs) [11], including optimal control [12], cooperative control [13], [14], and game theory [15], can be introduced for the secondary control design. Such secondary control algorithms require continuous communications for implementation. Also, the limited performance of traditional primary control algorithms (droop and inner PI control loops) makes it hard to achieve desired dynamic performance.

In most studies on microgrid control, the microgrid is modeled as a multiple-bus system with each bus having only one DG connected [16], [17]. In many applications, multiple local DGs are connected to one bus, such as PV farms [18] and wind farms [19]. Sometimes, multiple DGs in a microgrid, which are located close enough, can also be modeled together as a one-bus subsystem. Since the dynamics of the parallel-DGs connected to one bus are not specifically considered during control designs, most existing solutions cannot be directly applied to such microgrids. This is mainly due to the different models and parameters of the parallel-DGs. If the controls of the paralleled DGs are not well coordinated, large load current sharing errors among DGs will degrade the energy efficiency and cause unexpected load sharing even system failures [20]. Thus, the control of multiple-bus microgrid with parallel-DGs should be well studied.

In previous literature, the control of a one-bus multiple-DG system, which is a subsystem of the above-mentioned microgrid, has been studied with the application for paralleled uninterruptible power supplies (UPSs) [21]. Over the past decades, many control strategies have been proposed for such systems. The first type of solutions require communications among DGs including centralized control [22], master-slave control [23], circular chain control [20], and etc. The requirement of communications lowers not only flexibility but also reliability due to the imperfections with the communication system [24]. To relax the communication requirement, droop control, which has been the most popular decentralized control solution for microgrids [25]-[27], was introduced. However, droop control strategies are static proportional rule-based and suffer from low response speed and static deviations, which could be problematic for high- performance applications [28]. Although new control algorithms are available for paralleled UPSs, their applications for multiple-bus microgrids are incapable. To achieve good control performance over the targeted general class of microgrids, hierarchical control schemes with advanced primary control algorithms need to be developed.

In this study, an advanced two-level distributed control solution is presented for microgrids that consist of multiple converter-interfaced parallel-DG subsystems. At the upper level, the objective is to obtain the desired voltage trajectory reference for each parallel-DG subsystem. At the lower level, two control objectives are addressed: 1) The bus voltage is well regulated under different operating conditions; 2) The load current is proportionally shared among multiple local DGs. Since the control reference for primary control is updated periodically and held unchanged between updates, algorithm designs of the two control levels can be decoupled. This control strategy not only simplifies algorithm designs but also lowers communication requirement for control implementations. Firstly, a distributed consensus-based power sharing algorithm is introduced to determine the generations of subsystems. Secondly, a discrete-time droop equation is used to periodically adjust the subsystem frequency at a smaller time scale to fight against uncertainties with line losses, generations, load predictions, and etc. Finally, a Lyapunov-based decentralized control algorithm is designed for bus voltage regulation and proportional load current sharing in the subsystem. Extensive simulations are conducted to demonstrate the advantages of the proposed control scheme.

The rest of this paper is organized as follows. In Section Ⅱ, the mathematical model of a multiple-bus microgrid with paralleled DGs is presented. Section Ⅲ introduces the proposed control design along with the stability analysis. Simulations with microgrid models of different levels of detail are presented in Section Ⅳ. Concluding remarks are given in Section Ⅴ.

Ⅱ. PROBLEM FORMULATION

The topology of a multiple-bus microgrid consisting of several parallelDG subsystems is illustrated in Fig. 1. Each one-bus subsystem comprises multiple DGs and a lumped load. The DGs are connected to a common bus through power converters and $LC$-filters The average model is used for control design and the detailed switch-level model is used for control evaluation [29].

Figure  1.  A multiple-bus microgrid with parallel-DGs

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In this study, a microgrid consisting of $N$ parallel-DG subsystems is considered. The average model of the $p$th subsystem ($p=1, 2, \ldots , N)$ comprising $n_p$ DGs is given as [21]

where $v_{p, o}$ is the bus voltage, $v_{p, q}$ and $i_{p, q}$ are the output voltage and input current to the filter of the $q$th DG, respectively, $i_{p, oq}$ is the output current, and $R_{p, q}$, $L_{p, q}$ and $C_{p, q}$ are the parameters of the $q$th $LC$-filter.

Ⅲ. DISTRIBUTED CONTROL DESIGN FOR MULTIPLE-BUS MICROGRIDS

In this section, a two-level control scheme is introduced for the real-time coordination of subsystems within a large-scale microgrid [3]. In the proposed design, the objective of the upper-level is to obtain the desired voltage trajectory references for parallel-DG subsystems. First, fair power sharing is achieved by a distributed consensus-based algorithm. The root mean square (RMS) value of the subsystem bus voltage references, $V$, is fixed and only the desired bus frequency $f_p^\ast$ for each subsystem is maneuvered by a discrete-time droop equation within a small range, periodically. As a result, the subsystem's bus phase angle $\delta _p$ can be adjusted indirectly to achieve fair power sharing.

For the lower-level control of each subsystem, two control objectives are considered. The first one is to maneuver the bus voltage of the $p$th subsystem, $v_{p, o}$, to track a desired trajectory $v_{p, o}^\ast = V\sin (2\pi f_p^\ast t+\delta _p )$ with $f_p^\ast$ set by the upper-level controller. The second control objective is to achieve proportional load current sharing among DGs. In this way, the bus voltage $V$ and system frequency $f$ can be well regulated together. Details of the proposed control solution are presented in the following subsections.

A Upper-Level Control Design

The upper-level controller is introduced to find the desired bus voltage trajectory $v_{p, o}^\ast$ of the $p$th subsystem with the bus frequency references $f_p^\ast$, based on the operational restrictions of fair power sharing in a two-step manner.

Firstly, the distributed consensus algorithm [30] is employed to determine the power generation references ($P_p^\ast )$ of subsystems. In order to achieve fair power sharing, the power generation of each subsystem is assigned based on a common utilization level, which is decided by the overall demand including the total load demands and estimated system-wide losses, and the overall generation determined by the predicted intermittent sources and the capability of the non-intermittent sources. Next, the distributed consensus algorithm is utilized to find the local utilization level with a relatively large time step $T_U$, which is the same for the entire microgrid. Thus, the fair power sharing can be achieved by synchronizing the utilization level and the impact of inaccurate generation prediction can be minimized as well.

Secondly, once the power generation reference $P_p^\ast$ is obtained, the corresponding bus frequency reference $f_p^\ast$ can be decided through a discrete-time droop equation with a relatively small time step of $T_D$, which is expressed as

where $f_{\rm rate}$ is the nominal frequency, $P_p$ is the power generation of the $p$th subsystem, and $\eta _p$ is the active droop slope of the $p$th subsystem. Thus, the desired bus voltage trajectory can be calculated by $v_{p, o}^\ast =V\sin (2\pi f_p^\ast t+\delta _p )$.

B Lower-Level Control Design

To facilitate the control development, for the $p$th subsystem, define

where $C_p =\sum\nolimits_{q=1}^{n_p } {C_{p, q} }$ is the total capacitance, with $u_{p, q}$, $x_{p, q}$, $d_p$ being the control input, the system state, and the disturbance, respectively. Therefore, the original system model (1) can be rewritten in a compact form as

Before proceeding, the following assumption is given [31].

Assumption 1: $d_p$, and its first- and second-time derivatives are unknown but bounded, i.e., $| {d_p } |\le D_0$, $| {\dot {d}_p } |\le D_1$ and $| {\ddot {d}_p } |$ $\le$ $D_2$, where $D_0$, $D_1$ and $D_2$ are unknown positive constants.

1)Dynamics of the Filtered Tracking Error: Consider a parallel-DG subsystem (3). Define the tracking error as

Next, define the filtered tracking error [32] as

where $k_p$ is a positive constant. Furthermore, define

with $k_{p, r}$ being a positive constant. By recalling (3), the time derivative of $\bar {r}_p$ is

Apparently, systems (5}) and (6) are Hurwitz. If the filtered tracking error $r_p$ or $\bar {r}_p$ can be controlled to approach zero, the original tracking error $e_{p, o}$ will converge to zero, too [33].

2)Controller Development and Stability Analysis: Based on the dynamics of the filtered tracking error (7), design the decentralized control law as

whose time derivative is equivalently given as

where $\rho _p$ is a positive constant, and $0 <m_{p, q} <1$, $q=$ $1, 2, \ldots , n_p$ is the load current sharing ratio of DG #$p, q$, satisfying $\sum\nolimits_{q=1}^{n_p } {m_{p, q} } =1$. Noticed that the control law (8) is essentially decentralized for subsystem #$p$, in the sense that each individual DG only utilizes its own signals $i_{p, oq}$, $v_{p, o}$, and the information of $v_{p, o}^\ast$ obtained from the upper-level controller. No information is required to be shared among multiple DGs.

Subsequently, with the help of the definitions of $r_p$ and $\bar {r}_p$, substituting (9) into (7) yields

Now the theoretical results of the proposed controller are stated in the following theorem.

Theorem 1:Consider a parallel-DG system characterized by (1) satisfying Assumption 1, and a decentralized control law developed as in (8). The following facts hold: 1) the load current sharing errors, defined as $e_{p, qs} =i_{p, oq} /m_{p, q} -i_{p, os} /m_{p, s}$ for all $q,$ $s=1, 2, \ldots , n_p$, $q\ne s$, can be eliminated; 2) the voltage tracking error $e_{p, o}$ together with all the other signals of the closed-loop system are bounded.

Proof:The proof of Theorem 1 is divided into two parts.

Proof of Fact 1: With the help of the Kirchhoff's current law, one has

Recalling the second differential equation in (6), and substituting (8) into the time derivative of $i_{p, oq}$ yield

Thereafter, define a class of Lyapunov functions as

for $q,$ $s=1, 2, \ldots , n_p$, $q\ne s$. Taking the time derivative of $V_{p, qs}$ gives

By resorting to the standard Lyapunov synthesis [34], we have

for $q,$ $s=1, 2, \ldots , n_p$, $q\ne s$, where $e_{p, qs} (0)$ is the initial value of the load current sharing error, which will converge to zero exponentially. Thus, the first fact holds.

Proof of Fact 2: Define a Lyapunov function as

Since $\dot {r}_p =\bar {r}_p -k_{p, r} r_p$, (10) can be borrowed to obtain the time derivative of (16) as

With the help of Assumption 1, one has

where $0 <\varepsilon _p <k_{p, r}$ is a constant. Furthermore, according to Lemma 1.1 in [35], we have

and $r_p$, $\bar {r}_p$ are bounded for all time.

From the definition of $r_p$ in (5), $e_{p, o}$ and $\dot {e}_{p, o}$ can be treated as the outputs of a stable linear system with the bounded input $r_p$. Therefore, $e_{p, o}$ and $\dot {e}_{p, o}$ are bounded. From (6), one has that $\dot {r}_p =\bar {r}_p -k_{p, r} r_p$ is bounded, which results in that $\ddot {e}_{p, o} =\dot {r}_p -k_p \dot {e}_{p, o}$ is also bounded. Notice that the desired trajectory $v_{p, o}^\ast$ is a sinusoidal function, thus the first and the second time derivatives of $v_{p, o}^\ast$, $\dot {v}_{p, o}^\ast$, and $\ddot {v}_{p, o}^\ast$, are also bounded. With the boundedness of $e_{p, o}$, $\dot {e}_{p, o}$, and $\ddot {e}_{p, o}$, one can prove that $v_{p, o} =v_{p, o}^\ast +e_{p, o}$, $\dot {v}_{p, o} =\dot {v}_{p, o}^\ast +\dot {e}_{p, o}$, and $\ddot {v}_{p, o} =\ddot {v}_{p, o}^\ast +\ddot {e}_{p, o}$ are all bounded. By recalling (15), one has that $i_{p, oq} /m_{p, q} -i_{p, os} /m_{p, s}$ is bounded for all time. With the help of Assumption 1, $\sum\nolimits_{q=1}^{n_p } {i_{p, oq} }$ is also bounded. Furthermore, standard linear analysis methods can be applied to prove that $i_{p, oq}$ is bounded. Thereafter, $i_{p, q} =i_{p, oq} +C_{p, q} \dot {v}_{p, o}$ is bounded, and thus the second fact holds.

Finally, with the proposed lower-level controller, the two control objectives of the lower-level are achieved as illustrated in Theorem 1.

Remark 1: Notice that the input transformation from $v_{p, q}$ to $u_{p, q}$ is invertible. The actual control law for the parallel-DG system (1) is

with $u_{p, q}$ given in (8).

Remark 2: According to (8), the information of the time derivatives of the bus voltage $v_{p, o}$ is required in actual controllers, which can be obtained by the high-gain observers in [36] in the implementation of the proposed control.

C Control Implementation

Fig. 2 shows the control block diagram of one DG. There is no communication among DGs. The lower-level controllers receive the same voltage reference signal to make sure their control activities are coordinated. To construct the sinusoidal control reference signal, the lower-level controller only needs the frequency reference obtained from the upper-level.

Figure  2.  The control block diagram of one DG

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The overall flowchart diagram for the implementation of the proposed microgrid scheme is presented in Fig. 3. During the initialization of the consensus-based power sharing algorithm, each local upper-level controller estimates the maximum generation and the local load of the corresponding subsystem. Through the distributed communications among the upper-level controllers, the synchronized utilization level can be found. Thus, the power generation references of subsystems can be calculated locally. Thereafter, the frequency references of the bus voltages can be acquired via a discrete-time droop control. The detailed upper-level control process can be found in [30] and is omitted here.

Figure  3.  The flowchart diagram of the proposed microgrid control scheme

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During the regulation process, the bus voltage references are firstly determined by the RMS value of bus voltages together with their frequency references obtained from upper-level controllers. Then, with the proposed decentralized control strategy in Section Ⅲ-B, the tracking of the desired bus voltage trajectory, as well as the sharing of load current among DGs of each subsystem can be realized. Finally, PWM signals are generated based on the final control signals for the power converters.

Ⅳ. SIMULATION RESULTS

The performance of the proposed control scheme is testified on both mathematical and detailed switch-level models using MATLAB/Simulink. The microgrid adopted in the simulation is shown in Fig. 4. Each bus contains several DGs and a local load. The system and control parameters are listed in TABLEⅠ and Ⅱ, respectively. To better illustrate the effectiveness of the proposed control scheme, different load profiles are considered, i.e., Loads 1, 2 and 3 are switched on at 2.1 s, 0.7s, and 2.2s, respectively, and then switched off at 4.3s, 4.3s, 6.2 s, respectively.

Figure  4.  The topology of the simulated microgrid

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Table  Ⅰ.  MICROGRID SYSTEM PARAMETERS

	Quantity	Value
Subsystem 1	Output inductors($L_{1, 1}, L_{1, 2}, L_{1, 3})$	1.8, 2.0, 2.0mH
	Output resistors ($R_{1, 1}, R_{1, 2}, R_{1, 3})$	0.20, 0.18, 0.20$\Omega$
	Output capacitors ($C_{1, 1}, C_{1, 2}, C_{1, 3})$	25.0, 25.0, 27.5$\mu$F
	Active power rating	2.4kW
Subsystem 2	Output inductor ($L_{2, 1})$	2.1mH
	Output resistor ($R_{2, 1})$	0.20$\Omega$
	Output capacitor ($C_{2, 1})$	26mF
	Active power rating	1.2kW
Subsystem 3	Output inductors ($L_{3, 1}, L_{3.2})$	2.0, 2.0mH
	Output resistors ($R_{3, 1}, R_{3, 2})$	0.20, 0.20$\Omega$
	Output capacitors ($C_{3, 1}, C_{3, 2})$	25.0, 25.0$\mu$F
	Active power rating	2.4kW
Line impedances	Line inductors($L_{\ell 12}, L_{\ell 13}, L_{\ell 23})$	0.10, 0.11, 0.09mH
Line impedances	Line resistors ($R_{\ell 12}, R_{\ell 13}, R_{\ell 23})$	0.25, 0.24, 0.26$\Omega$
	Nominal frequency ($f_{rate})$	60Hz
	Output AC bus voltage ($V)$	110V
	Local loads ($R_{1}, R_{2}, R_{3})$	6.05, 12.10, 6.05 $\Omega$
	Sampling period	0.1ms
	Droop control time step ($T_{D})$	0.2s
	Upper-level control time step ($T_{U})$	1.5s

 | Show Table

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Table  Ⅱ.  CONTROL PARAMETERS

Parameter	Value
$k_{1}, k_{2}, k_{3}$	4.0$\times$10$^{3}$
ρ1ρ2ρ3	1.5$\times$10$^{7}$
$k_{r, 1}, k_{r, 2}, k_{r, 3}$	1.0$\times$10$^{3}$
$\eta$$_{1}$	0.199
$\eta$$_{2}$	0.398
$\eta$$_{3}$	0.199
$m_{11}$	0.167
$m_{12}$	0.333
$m_{13}$	0.5
$m_{21}$	1
$m_{31}$	0.5
$m_{32}$	0.5

 | Show Table

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A Test With Mathematical Microgrid Model

In this test, the proposed control scheme is testified with the mathematical microgrid model. The system responses are shown in Figs. 5-9.

Figure  5.  Generation powers of the mathematical model

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Figure  6.  Bus voltages of the mathematical model

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Figure  7.  Bus frequencies of the mathematical model

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Figure  8.  Load current sharing errors of Subsystem 1 of the mathematical model

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Figure  9.  Load current sharing errors of Subsystem 3 of the mathematical model

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Fig. 5 illustrates the trajectories of the generation power. At 0.7s, Load 2 is switched on, with the help of the well-designed upper-level controller, the total generated power can be allocated according to their rating powers during steady-state (between 1.5s and 2.1s). To test the stability of the upper-level controller and its robustness, two sudden load increasing events are simulated, i.e., Loads 1 and 3 are switch on at 2.1s and 2.2, s with a total changing power of 0.67, p.u. In this case, the generated powers respond quickly. The total power is shared among subsystems with a proportion of 2:1:2 between 3s and 4.3s. Furthermore, at 4.3s, Loads 1 and 2 are switched off. Again, the total power is shared proportionally. Finally, at 6.2s, Load 3 is switched off and the generation powers of all 3 subsystems go back to zero.

The performances of the bus voltage and frequency are demonstrated in Figs. 6 and 7, respectively. Under all load change conditions, the stability of the microgrid system is ensured by the developed lower-level controller. From Fig. 6, one can notice that all bus voltage errors are maintained within $\pm$0.05 p.u. Moreover, the bus frequency errors are smaller than 0.005 p.u. as shown in Fig. 7.

In Figs. 8 and 9, the trajectories of load current sharing rrors of Subsystems 1 and 3 are depicted. From Fig. 8, one can see that even in presence of unbalanced $LC$-filter parameters, the largest load current sharing error is smaller than 0.7A.

B Test With Detailed Switch-Level Microgrid Model

In this test, the proposed control scheme is further testified with the detailed switch-level microgrid model, which is enough to imitate the real-world performance. All the power converters are implemented by their detailed modules included in the simpower systems toolbox of Simulink/MATLAB. The system parameters and control parameters remain the same as in TABLESⅠ and Ⅱ. The system responses are delivered in Figs. 10-14.

Figure  10.  Generation powers of the detailed switch-level model

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Figure  11.  Bus voltages of the detailed switch-level model

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Figure  12.  Bus frequencies of the detailed switch-level model

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Figure  13.  Load current sharing errors of Subsystem 1 of the detailed switch-level model

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Figure  14.  Load current sharing errors of Subsystem 3 of the detailed switch-level model

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From Fig. 10, it is noticed that the trajectories of the generation power are almost the same as those obtained from the mathematical model in Fig. 5. Fair power sharing of the total load demand is achieved successfully. Moreover, as shown in Figs. 11 and 12, the tracking objectives of the bus voltages and frequencies are achieved simultaneously with the proposed lower-level controller. The bus voltage errors and frequency errors are smaller than 0.05p.u. and 0.005p.u., respectively. Furthermore, in presence of the switching frequency based harmonics, the load current sharing errors are shown in Figs. 13 and 14 to substantiate the proposed approach.

Ⅴ. CONCLUSIONS

In this study, a distributed upper-level controller and a decentralized lower-level controller are proposed for multiple-bus microgrids with paralleled DGs. For the upper-level control, a distributed consensus-based power sharing algorithm is applied to obtain the desired power generation references for all subsystems. Then, a decentralized droop control is utilized to calculate the frequency references, which are further used to obtain the desired voltage trajectories. For the lower-level control, decentralized controllers are designed to force the subsystem bus voltage to track the desired references. In addition, the proportional load current sharing among DGs is achieved. Finally, simulation results with microgrid models of different levels of detail are presented. The effectiveness of the proposed microgrid control scheme is demonstrated by these results.