A Unified Control of Super-capacitor System Based on Bi-directional DC-DC Converter for Power Smoothing in DC Microgrid

Xialin Li; Pengfei Li; Leijiao Ge; Xunyang Wang; Zhiwang Li; Lin Zhu; Li Guo; Chengshan Wang

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A Unified Control of Super-capacitor System Based on Bi-directional DC-DC Converter for Power Smoothing in DC Microgrid PDF

- ORCID：
Xialin Li
✉
- ORCID：
Pengfei Li
✉
- ORCID：
Leijiao Ge
✉
- ORCID：
Xunyang Wang
✉
- ORCID：
Zhiwang Li
✉
- ORCID：
Lin Zhu
✉
- ORCID：
Li Guo
✉
- ORCID：
Chengshan Wang
✉

the Key Laboratory of Smart Grid of Ministry of Education, Tianjin University, Tianjin 300072, China； the School of Information and Electronic Engineering, Hebei University of Engineering, Handan 056038, China； the Binhai Power Supply Bureau of State Grid Tianjin Electric Power Company, Tianjin 300480, China

Updated：2023-05-22

DOI：10.35833/MPCE.2021.000549

OUTLINE

Abstract

To improve the equivalent inertia of DC microgrids (DCMGs), a unified control is proposed for the first time for a bi-directional DC-DC converter based super-capacitor (SC) system, whereby power smoothing and SC terminal voltage regulation can be achieved in a DCMG simultaneously. The proposed control displays good plug-and-play features using only local measurements. For quantitative analysis and effective design of the critical parameter of unified control, two indices, equivalent power supporting time and inertia contributed by the unified controlled SC system, are introduced firstly. Then, with a simple but effective reduced-order model of a DCMG, analytical solutions are obtained for the two indices. In addition, a systematic design method is presented for the proposed unified control. Finally, to verify the proposed unified control, a switching model is developed for a typical DCMG in PSCAD/EMTDC, and theoretical analyses are conducted for different operating conditions.

Keywords

DC microgrid; DC-DC converter; super-capacitor; unified control; equivalent power supporting time and inertia; power smoothing

I. Introduction

IN remote areas and islands, AC/DC microgrids (MGs) integrated with conventional diesel generator set (DGS), energy storage systems (ESSs), and highly penetrated renewable power generations (RPGs) are emerging as promising solutions to power supply problems. However, when subjected to uncertain power disturbances, dynamic stability and power smoothing issues are the main challenges presented by the limited dynamic response and power creep rate of the DGS [

1]-[4]. In weak-grid-connected DCMG, no appropriate power smoothing methods are found to deal with the instability of DC-AC converters [5]. Therefore, it is critical to smooth the transient power and enhance the equivalent inertia of DCMGs.

In low-voltage DCMGs, system inertia can be improved by using the energy stored in DC bus capacitors via adaptive virtual inertia control with variable droop gain [

6] or DC virtual synchronous control [7], [8]. As the energy stored in DC bus capacitors is limited, the control methods in [6]-[8] cannot achieve the desired power smoothing performance when large power disturbances occur. Thus, power-type ESSs such as super-capacitors (SCs) can be used to improve the immunity to disturbances and to enhance the transient stability of DCMGs. To this end, this paper focuses on how to use SC systems to improve the equivalent inertia of DCMGs.

The main publications on control methods can be categorized into direct methods based on power perturbation by divider filtering [

9]-[11] and indirect methods based on the equivalent output impedance of hybrid ESSs with different dynamics [12]-[15]. In direct methods, the high-frequency and low-frequency components of power disturbances can be obtained by filtering the measured power disturbance [9], the output current reference from a centralized DC voltage controller [10], or the locally measured DC bus voltage [11]. These components can be set as the power or current references of the SC and DC voltage control units, respectively. In indirect methods, the equivalent output impedances of the SC and DC voltage control units are designed with different dynamics to realize multiscale decoupling of power smoothing. In [12]-[16], the methods such as virtual capacitors, integral droop control, and virtual resistive-capacitive (RC) droop control have been proposed to make the equivalent output impedance of the SC system capacitive, thereby realizing an effective power smoothing performance in case of a large power disturbance.

To the best of our knowledge, the methods in [

9]-[16] still have the following disadvantages.

1) Poor plug-and-play feature. Additional sensors, high bandwidth, reliable communication [

9], or a centralized controller [10] are necessary in direct methods, which share the risk of a single point of failure. Additional communication can also affect the plug-and-play features of an SC system. In [12]-[16], although both the DC voltage control unit and the SC system adopt droop control, the power smoothing performance is strongly dependent on the equivalent output impedance. Therefore, the plug-and-play capacity of an SC system is limited.

2) Ignoring the impact of SC terminal voltage regulation on the power smoothing performance. The power smoothing performance of the SC system is mainly related to the terminal voltage. Most existing studies suggest that the dynamics of terminal voltage regulation and power smoothing control of SC systems be decoupled. When the power smoothing transient is over, the steady terminal voltage of the SC system is designed to recover to its rated value [

14], [15] or to within a certain range such as 30%-70% of the rated value [16], or 50% of the rated value [17]. If restored to the rated value, the SC system loses the ability to balance disturbances in the case of a power increase of RPGs. Although the SC system can deal with uncertain power disturbances when its terminal voltage is recovered to 50% or to within a certain range, there is little work on determining whether these voltage settings are optimal for power smoothing control over a wide operating range of the DCMG.

To overcome these issues, the main contributions of this paper can be summarized as follows.

1) First, a unified control has been proposed for a bidirectional DC-DC converter based SC system, which can improve the equivalent inertia of the DCMG, smooth the active power of grid-forming units, and control the SC terminal voltage simultaneously. It should be noted that the proposed control has good plug-and-play features by only using local measurements.

2) Second, to represent and quantify the power smoothing capability more clearly, two indices, i.e., equivalent power supporting time and equivalent inertia contributed by the unified controlled SC system, are introduced. The analytical solutions for these two indices can be obtained using a simple but effective reduced-order model of the DCMG.

3) Third, based on the two analytical indices, a systematic design method is presented for the critical parameters of the proposed unified control, which is further verified by detailed simulations based on a typical DCMG switching model constructed in PSCAD/EMTDC.

The remainder of this paper is organized as follows. Section II presents a typical DCMG structure. The unified control for the SC system is proposed in Section III. An analytical solution of equivalent power support time T_s is presented in Section IV by a simple but effective reduced-order model of DCMG. The systematic analysis, design of the proposed unified control, and simulation verification are provided in Section V. Finally, Section VI concludes the paper.

II. Typical DCMG Structure

A typical DCMG structure is shown in Fig. 1, including grid-forming units, grid-following units, and grid-supporting units [

18]. For a grid-connected DCMG, the DC bus voltage is typically controlled by a bi-directional DC-AC converter. In the islanded mode, an ESS such as a battery system with a bi-directional DC-DC converter is generally used to control the DC bus voltage. These units can be considered as grid-forming units of the DCMG. Grid-following units can consist of RPGs such as wind power (WP) and photovoltaics (PV) in the maximum power point track (MPPT) mode, controllable DGs in power dispatching mode, and loads.

Fig. 1 Typical DCMG structure.

In remote areas or islands, DC-AC converters powered by a weak grid or DGS can be considered as the grid-forming units of the DCMG. Since these converters do not have the ability to respond quickly, a large power disturbance from the grid-following units may have a significant impact on the dynamic stability of the DC bus voltage. Thus, the SC system was adopted as a grid-supporting unit to improve the equivalent inertia of the DCMG, suppress power disturbance, and smooth the active power of the grid-forming unit. In this paper, a simple but effective control method is proposed for the SC system, referred to as unified control. u_dc is the DC bus voltage, and P_s, P_p, and P_sc are the power outputs of grid-forming units, grid-following units, and grid-supporting units, respectively.

III. Unified Control for SC System

In this section, the requirements of the power smoothing control in the DCMG are initially introduced. Then, a simple unified control is proposed for the SC system, to improve the equivalent inertia of the DCMG, smooth the active power of the grid-forming unit, and control the SC terminal voltage simultaneously.

A. Requirements of Power Smoothing Control Performance in DCMG

In this paper, the expected power smoothing control performances of the DCMG are illustrated, as shown in Fig. $2 (a)$ and (b). Considering a large power disturbance $Δ P_{p}$ (e.g., a power variation from RPGs or loads) at time t₀, the SC system is expected to compensate for the unbalanced power immediately, with a dynamic response $Δ P_{s c}$ . To this end, the power of the grid-forming unit $Δ P_{s}$ , DC bus voltage u_dc, and terminal voltage of the SC system u_sc smoothly reach an acceptable equilibrium point with u_sc₁ and u_sc₂ denoting the terminal voltages of the SC system in the pre- and post-disturbance states, respectively. $Δ E_{s m o o t h}$ is the released energy, and T_s is the equivalent power supporting time.

Fig. 2 Ideal dynamic responses of power and voltage. (a) Ideal power dynamic response. (b) Ideal voltage dynamic response.

In some published studies, the DC bus voltage u_dc was regulated to a constant reference value. In contrast, the DC voltage droop control strategy of DC-AC converter [

19], [20] is adopted for the grid-forming unit of the DCMG in this paper, as shown in Fig. 3. All values of the variables are per-unit, i.e., normalized by the base values of power and DC voltage (denoted as P_B and U_dcB);

u_{d c, s e t}

and

u_{d c, p u}

are the reference value and measurement of the DC voltage, respectively;

P_{s, p u}

is the active power of DC-AC injected into DCMG; R_d is the droop gain;

k_{p u, s}

and

k_{i u, s}

are the proportional and integral gains of the DC voltage controller, respectively;

i_{d r e f, p u}

and

i_{q r e f, p u}

are the reference values of the currents projected onto the d- and q-axis, respectively; and d_abc is the the duty ratio.

Fig. 3 DC voltage droop control of DC-AC converter.

With droop control, the relationship between the power of grid-forming or grid-following units (referred to $P_{s, p u}$ and $P_{p, p u}$ , respectively) and DC bus voltage $u_{d c, p u}$ in steady state can be expressed as:

u_{d c, p u} = u_{d c, s e t} - P_{s, p u} R_{d} = u_{d c, s e t} - P_{p, p u} R_{d}

(1)

Hence, by measuring $u_{d c, p u}$ , the SC system can obtain the power of grid-forming or grid-following units readily, without any additional sensors or communication, which is critical for the plug-and-play feature. In this paper, $u_{d c, s e t} = 1$ , and given that the variation range of the total power of the grid-following units in DCMG is $- 1 \leq P_{p, p u} \leq 1$ , the operation range of the DC bus voltage will be $1 - R_{d} \leq u_{d c, p u} \leq 1 + R_{d}$ .

As depicted by the gray shaded area in Fig. 2(a), the released energy $Δ E_{s m o o t h}$ can be represented by the power smoothing ability of the SC system, whereby the detailed expression is given by:

Δ E_{s m o o t h} = \int_{t_{0}}^{t_{\infty}} Δ P_{s c} d t = \frac{1}{2} C_{s c} (u_{s c 2}^{2} - u_{s c 1}^{2})

(2)

where C_sc is the capacitance value of SC; and u_sc₁ and u_sc₂ are the terminal voltages of the SC system in the pre- and post-disturbance states, respectively.

Regarding the DGS in the isolated DCMGs, which is limited by the response or power climbing rate, a certain duration of time is required by the SC system to support the DCMG. In other emergency scenarios, a certain duration of time is also critical for the power-coordinated control response of the DCMG such as load shedding or power curtailment of RPGs. Thus, to evaluate the power smoothing ability of the SC system more intuitively, the concept of equivalent power supporting time T_s is proposed and defined as:

T_{s} = Δ E_{s m o o t h} / Δ P_{p}

(3)

The main requirements of power smoothing control can be summarized as follows.

1) Control performance, as shown in Fig. 2(a), can be realized using only local measurements. Moreover, a given set value of T_s can be satisfied via the specific design of the critical control parameters.

2) The existing works in [

14]-[17] do not discuss how to set the steady terminal voltage value of the SC system from the perspective of being more effectively to cope with uncertain power disturbances more effectively. If the SC terminal voltage control can be coupled dynamically with power smoothing control on the same timescale, the design of the SC control system can be simplified.

Motivated by these considerations, a novel unified control of an SC system is proposed for power smoothing applications.

B. Proposed Unified Control of SC System

1)　Unified Control Structure

The proposed unified control of the SC system unifies the SC terminal voltage control and power smoothing control, as shown in Fig. 4, in two parts consisting of DC voltage consensus control and base value regulation of SC terminal voltage. It should be noted that only local measurements of the DC voltages and currents are used here, indicating good plug-and-play feature.

The easiest way to smooth the power disturbance and improve the inertia of the DCMG is to connect the SC directly to the DC bus as part of the DC bus capacitor. However, as the terminal voltage level of the SC is usually different from that of the DC bus, the SC is always connected to the DC bus via a bidirectional DC-DC converter. Thus, to make the SC system an equivalent DC bus capacitor, a simple unified control is proposed, as shown in Fig. 4, where i_lsc is the inductor current; and i_hsc and C_hsc are the output current and DC capacitor on the DC bus side, respectively.

Fig. 4 Proposed unified control of SC system with plug-and-play feature.

With this simple consensus control, the per-unit value of the SC terminal voltage can be dynamically tracked to that of the DC bus voltage, such that:

u_{s c} \approx U_{s c B} u_{d c} / U_{d c B} = U_{s c B} u_{d c, p u}

(4)

where u_sc and U_scB are the terminal voltage and base value of the SC, respectively; and u_dc and U_dcB are the actual and base values of the DC bus voltage, respectively. To improve the response of the consensus control in the current control loop, the current feedforward (F is the feedforward gain) is adopted.

It is known that when subjected to power disturbance, the SC system can improve the equivalent inertia of DCMG and smooth the active power of DC voltage control unit without any delay, by simply tracking the DC bus voltage.

The function of the control loop of the base value regulation of SC terminal voltage in Fig. 4 is to provide an adaptive base value U_scB to realize SC terminal voltage control and satisfy a given set value of equivalent power supporting time T_s simultaneously, which is described in detail in the following subsection.

2)　Base Value Regulation of SC Terminal Voltage

Adjusting U_scB online is the key to realizing the dynamic coupling of the SC terminal voltage control in power smoothing applications, which is also critical for DCMG to suppress uncertain power disturbances. To this end, the following method is proposed to set U_scB online:

U_{s c B} = U_{r a t e d} [u_{s c, s e t} - \frac{u_{w o r k}}{2 R_{d}} (1 + R_{d} - u_{d c, p u})]

(5)

where U_rated is the rated value of the SC terminal voltage; and $u_{s c, s e t}$ and u_work (in per-unit) are the setting value and allowable working area of the SC terminal voltage, respectively. In this paper, the setting value $u_{s c, s e t}$ is selected to be $1 / (1 + R_{d})$ in order to prevent u_sc from exceeding the rated value U_rated.

In (5), there is a droop feature between U_scB and the per-unit value of the DC bus voltage $u_{d c, p u}$ . Specifically, the higher the DC bus voltage $u_{d c, p u}$ is, the greater U_scB will be. In this scenario, the operation state is mainly related to the low power level of the grid-following units $P_{p, p u}$ (e.g., the output power of the RPGs is greater than the load consumption). Then, the main cause of power perturbation is from a reduction in the output of RPGs or an increase in the load. Therefore, with a larger base value of U_scB, the SC system has more stored energy to cope with such transient power disturbances. In contrast, a lower DC bus voltage value $u_{d c, p u}$ indicates that the DCMG is in a state of lower RPG output or that the load is heavier. Obviously, a lower value of U_scB can ensure that the SC system has sufficient charging capacity after being subjected to an RPG power rush or a sudden load decrease. By substituting (5) into (4), the final relationship between the SC terminal voltage and DC bus voltage can be derived as:

u_{s c} = U_{r a t e d} [\frac{u_{w o r k} u_{d c, p u}^{2}}{2 R_{d}} + (u_{s c, s e t} - \frac{1 + R_{d}}{2 R_{d}} u_{w o r k}) u_{d c, p u}]

(6)

It should be noted that the working area u_work in Fig. 4 and (6) play a key role in the real power smoothing application of the SC system, especially to satisfy a given set value of T_s. To this end, the value of u_work should be designed systematically, which is introduced in the following section.

IV. Analytical Solution of T_s by a Simple but Effective Reduced-order Model of DCMG

In this section, a simple but effective reduced-order model of DCMG based on the DC bus voltage response is derived and verified. Then, the analytical solutions to the equivalent inertia and equivalent power supporting time T_s contributed by the unified controlled SC system are obtained, which provide a theoretical foundation for the systematic design of the unified control of the SC system in Section V.

A. Reduced-order Model of DCMG

The dynamics of the DC bus voltage can be expressed as:

\frac{C_{b u s} U_{d c B}^{2}}{P_{B}} \frac{d Δ u_{d c, p u}}{d t} = Δ P_{s, p u} + Δ P_{s c, p u} - Δ P_{p, p u}

(7)

where C_bus is the lumped capacitance; and $C_{b u s} U_{d c B}^{2} / P_{B}$ is regarded as the original inertia of the DCMG denoted as H_dc₀.

With the droop control in Fig. 3, and by ignoring the dynamics of the inner current control loop, the dynamic response of the DC-AC converter can be obtained as:

Δ P_{s, p u} = - \frac{G_{u d c} (s)}{1 + R_{d} G_{u d c} (s)} Δ u_{d c, p u}

(8)

where G_udc(s) is the proportional-integral (PI) controller of the DC voltage control loop.

To obtain the response of $Δ u_{d c, p u}$ to $Δ P_{s, p u}$ , (9) is defined based on the conservation of energy.

\frac{1}{2} C_{s c, 0} (u_{s c 2}^{2} - u_{s c 1}^{2}) = \frac{1}{2} C_{s c, e q} (u_{d c 2}^{2} - u_{d c 1}^{2})

(9)

where C_sc_,0 is the actual capacitance of SC; and $C_{s c, e q}$ is the equivalent capacitance of SC from the DC bus side. Assuming that the DC bus voltage changes from u_dc₁ to u_dc₂ after the disturbance, the SC terminal voltage is changed from u_sc₁ to u_sc₂ accordingly.

Based on (4), (6), and (9), the detailed expression of $C_{s c, e q}$ can be obtained as:

C_{s c, e q} = A_{r a t i o} C_{s c, 0}

(10)

where A_ratio is a transformation coefficient expressed as:

\begin{array}{l} A_{r a t i o} = \\ \frac{U_{r a t e d}^{2}}{U_{d c B}^{2}} [{(\frac{u_{w o r k}}{2 R_{d}})}^{2} (u_{d c 1, p u}^{2} + u_{d c 2, p u}^{2}) + {(u_{s c, s e t} - \frac{1 + R_{d}}{2 R_{d}} u_{w o r k})}^{2} + \\ \frac{u_{w o r k}}{R_{d}} (u_{s c, s e t} - \frac{1 + R_{d}}{2 R_{d}} u_{w o r k}) \frac{u_{d c 1, p u}^{2} + u_{d c 1, p u} u_{d c 2, p u} + u_{d c 2, p u}^{2}}{u_{d c 1, p u} + u_{d c 2, p u}}] \end{array}

(11)

Then, according to (9)-(11), the dynamic performance of the power smoothing control of the SC system can be described in a very concise form as:

Δ P_{s c, p u} = - \frac{C_{s c, e q} U_{d c B}^{2}}{P_{B}} \frac{d Δ u_{d c, p u}}{d t} = - H_{s c} \frac{d Δ u_{d c, p u}}{d t}

(12)

where $C_{s c, e q} U_{d c B}^{2} / P_{B}$ is defined as the equivalent inertia contributed by the unified controlled SC system denoted as H_sc.

Combining (7), (8), and (12), the equivalent DC voltage response model of DCMG subjected to a power disturbance can be expressed as:

Δ u_{d c, p u} = - G \frac{s + z}{s^{2} + 2 ξ_{d} ω_{n} s + ω_{n}^{2}} Δ P_{p, p u}

(13)

The related parameters in (13) are expressed as:

\{\begin{array}{l} G = \frac{R_{d} k_{p u, s} + 1}{H_{e q} R_{d} k_{p u, s} + H_{e q}} \\ z = \frac{R_{d} k_{i u, s}}{R_{d} k_{p u, s} + 1} \\ ξ_{d} = (\frac{H_{e q} R_{d} k_{i u, s} + k_{p u, s}}{H_{e q} R_{d} k_{p u, s} + H_{e q}}) / (2 \sqrt[]{\frac{k_{i u, s}}{H_{e q} R_{d} k_{p u, s} + H_{e q}}}) \\ ω_{n} = \sqrt[]{\frac{k_{i u, s}}{H_{e q} R_{d} k_{p u, s} + H_{e q}}} \\ H_{e q} = H_{d c 0} + H_{s c} \end{array}

(14)

Using model (13), the dynamic performance index of the power smoothing control of the SC system can be calculated analytically.

B. Verification of Second-order Model

To validate the second-order model of the DCMG in (13), a detailed switching model of the DCMG is developed in PSCAD/EMTDC, with the main circuit and control parameters listed in Table I, where C_vsc is the DC capacitor of grid-forming unit at DC bus side; and C_hl and C_hs are the DC capacitors of grid-following unit and grid-supporting unit at DC bus side, respectively. For the switching model in PSCAD/EMTDC, the control period of the converters and simulation time step are selected as 100 $μ s$ and 1 $μ s$ , respectively. The comparison results of reduced-order model (13) and the switching model under different conditions and parameters are shown in Fig. 5.

TABLE I Main Circuit and Control Parameters of DCMG System

Unit	Item	Value
AC-side circuit	AC RMS line voltage V_s	380 V
	Transformer	380 V/220 V
	Rated frequency	50 Hz
DC-AC (grid-forming unit)	LCL filter	2 mH/10 μF, 0.5 Ω/0.12 mH
	Rated power P_B	10 kW
	Rated RMS line voltage V_inv	220 V
	$u_{d c, s e t}$ / $i_{q r e f, p u}$	1/0
	$k_{p u, s}$ / $k_{i u, s}$ / $k_{p i, s}$ / $k_{i i, s}$	16/160/2/145
	Phase locked loop k_pp/k_ip	10/100
	C_vsc	4000 μF
DC-side circuit	Rated DC voltage U_dcB	400 V
SC DC-DC	$k_{p u, s c}$ / $k_{i u, s c}$ / $k_{p i, s c}$ / $k_{i i, s c}$	4/30/0.05/50
SC DC-DC	L_sc/C_hs	2 mH/2000 μF
Grid-following unit	Rated power	10 kW
	$k_{p u, l}$ / $k_{i u, l}$ / $k_{p i, l}$ / $k_{i i, l}$	1/20/0.05/50
	U_ref	300 V
	L₁/C_hl/C_ll	2 mH/2000 μF/2500 μF

Fig. 5 Comparison results of reduced-order model (13) and switching model under different conditions and parameters. (a) With different values of $Δ P_{p, p u}$ . (b) With different values of $C_{s c, 0}$ . (c) With different values of $u_{w o r k}$ . (d) With different values of $R_{d}$ .

Given the basic parameters $U_{r a t e d} = 200$ V, $C_{s c, 0} = 3$ F, $u_{w o r k} = 0.3$ , the simulation results under different power disturbances are presented in Fig. 5(b), (c), and (d) which compare the results for different values of C_sc_,0, u_work, and R_d, respectively. It is observed that for a wide range of variations in the main parameters, the step responses of model (13) are somewhat consistent with the simulation results based on the detailed switching model, which verifies the effectiveness of model (13).

C. Further Reduction of Second-order Model

Assuming that the dynamics of second-order model in (13) are designed to be overdamped, the characteristic equation of (13) will have two negative real poles (denoted by $- 1 / T_{1}$ and $- 1 / T_{2}$ ) and derived as:

\{\begin{array}{l} \frac{1}{T_{1}} = \frac{q}{2} + \frac{1}{2} \sqrt[]{q^{2} - 4 ω_{n}^{2}} \\ \frac{1}{T_{2}} = \frac{q}{2} - \frac{1}{2} \sqrt[]{q^{2} - 4 ω_{n}^{2}} \end{array}

(15)

where the variable q is expressed as:

q = \frac{H_{e q} R_{d} k_{i u, s} + k_{p u, s}}{H_{e q} R_{d} k_{p u, s} + H_{e q}}

(16)

The limits of q and $ω_{n}$ as H_eq goes to infinity can be readily obtained as:

\{\begin{array}{l} \underset{H_{e q} \to + \infty}{l i m} q = \frac{R_{d} k_{i u, s} + \frac{k_{p u, s}}{H_{e q}}}{R_{d} k_{p u, s} + 1} = \frac{R_{d} k_{i u, s}}{R_{d} k_{p u, s} + 1} = z \\ \underset{H_{e q} \to + \infty}{l i m} ω_{n} = \sqrt[]{\frac{k_{i u, s}}{H_{e q} R_{d} k_{p u, s} + H_{e q}}} = 0 \end{array}

(17)

With increasing H_eq, pole $- 1 / T_{1}$ gets closer to zero z while another pole $- 1 / T_{2}$ moves toward the origin, which is verified by the changing trajectory of the zero and poles of model (13), as shown in Fig. 6(a).

Fig. 6 Influence of system parameters. (a) Changing trajectory of zeros and poles when H_sc changes. (b) DC voltage response with different values of C_sc_,0 when $u_{w o r k} = 0.3$ and $R_{d} = 0.05$ . (c) DC voltage response with different values of u_work when $C_{s c, 0} = 5$ F and $R_{d} = 0.05$ . (b) DC voltage response with different values of R_d when $C_{s c, 0} = 5$ F and $u_{w o r k} = 0.2$ .

Since the pole $- 1 / T_{1}$ and zero $- z$ are very close to each other when DCMG is overdamped, model (13) can be further reduced to a simple first-order model:

Δ u_{d c, p u} = - \frac{z G T_{1}}{s + 1 / T_{2}} Δ P_{p, p u} = - R_{d} \frac{1 / T_{2}}{s + 1 / T_{2}} Δ P_{p, p u}

(18)

Figure 6(b), (c), and (d) compares the results of the step responses of (13) and (18) for different values of C_sc_,0, u_work, and R_d, respectively. It is observed that for a wide range of variations in the main parameters, the step responses of (18) are very consistent with those of (13), which verifies the first-order model (18). To this end, the simple model (18) can be used for the systematic analysis and design of the proposed unified control of the SC system.

According to (18), the analytical expression for the dynamic response of the DC bus voltage can be derived as:

Δ u_{d c, p u} = - R_{d} (1 - e^{- t / T_{2}}) Δ P_{p, p u}

(19)

Substituting (19) into (12), the power response of the SC system is obtained as:

Δ P_{s c, p u} = (H_{s c} R_{d} / T_{2}) e^{- t / T_{2}} Δ P_{p, p u}

(20)

The energy released or absorbed by the SC system to support DCMG can be calculated as:

Δ E_{s m o o t h, p u} = \int_{0}^{\infty} Δ P_{s c, p u} d t = H_{s c} R_{d} Δ P_{p, p u}

(21)

Then, using (2) and (21), the analytical solution of the equivalent supporting time T_s can be derived as:

T_{s} = H_{s c} R_{d}

(22)

Finally, with the simple models of DCMG derived in this section, the analytical solutions to equivalent inertia H_sc and equivalent supporting time T_s to the SC system are obtained.

V. Systematic Analysis, Design of Proposed Unified Control, and Simulation Verification

In this section, with a detailed DCMG system and associated parameters, a systematic analysis and design method for the proposed unified power smoothing control is presented and further verified by detailed simulations.

A. DCMG Configuration

To verify the proposed unified control and the related theoretical analysis, a detailed DCMG was built using PSCAD/EMTDC, as shown in Fig. 7.

Fig. 7 DCMG system for PSCAD/EMTDC simulation.

The DC bus voltage is controlled via the grid-connected DC-AC converter with the droop method described in Fig. 3, which can be regarded as the grid-forming unit of the DCMG. The grid-following unit is simulated using a DC-DC converter and an equivalent load resistance. The load terminal voltage is controlled to a constant value, and the power disturbance is simulated by changing the equivalent resistance. The SC system is connected to the DC bus by a bidirectional DC-DC converter with the proposed unified control shown in Fig. 4.

B. Systematic Analysis and Design of Proposed Control

This subsection initially presents the influence of physical and control parameters on the equivalent inertia H_sc. Then, the method of determining the feasible region of the equivalent supporting time T_s of the SC system is presented. Finally, according to specific power smoothing requirements within the feasible region of T_s, a simple method is developed to design an appropriate working area u_work for the SC system.

1)　Equivalent Inertia H_sc

In (10)-(12), it is observed that the equivalent inertia H_sc contributed by the SC system is mainly related to: ① physical parameters, including the capacitance C_sc_,0 of SC and its rated voltage U_rated, power of grid-following unit P_p_0,_pu in the pre-disturbance state, and the variation $Δ P_{p, p u}$ ; ② control parameters, including droop gain R_d and the working area u_work of the SC system.

1)　Influence of physical parameters on H_sc

Let $U_{d c B} = 400$ V, $R_{d} = 0.05$ , and $u_{w o r k} = 0.3$ . Figure 8(a) shows that SC can contribute a greater equivalent inertia H_sc, with increasing U_rated and C_sc_,0. In contrast, increasing P_B leads to a decrease in H_sc.The power supporting ability of the SC system decreases with increasing capacity of the DCMG.

Let $R_{d} = 0.05$ , $u_{w o r k} = 0.3$ , $C_{s c, 0} = 10$ F, $U_{r a t e d} = 200$ V, and $P_{B} = 10$ kW. The influence of the DC bus voltages u_dc_1,_pu and u_dc_2,_pu (specifically associated with P_p_0,_pu and $Δ P_{p, p u}$ , respectively) on H_sc is shown in Fig. 8(b). As the droop gain R_d is 0.05, and the per-unit value of P_p_,_pu is assumed to be $- 1 \leq P_{p, p u} \leq 1$ , the variation in the DC voltages $u_{d c 1, p u}$ and $u_{d c 2, p u}$ is between 0.95 and 1.05, as shown in Fig. 8(b). It is observed that a high level of $u_{d c 1, p u}$ and $u_{d c 2, p u}$ will increase the value of H_sc. This conclusion will help in the specific design of the equivalent supporting time T_s and working area u_work of the SC system.

Fig. 8 Influence of physical parameters on H_sc. (a) Relationship between H_sc and U_rated, P_B, and C_cs_,0. (b) Relationship between H_sc and u_dc_1,_pu and u_dc_2,_pu.

2)　Influence of Control Parameters on H_sc

Let $C_{s c, 0} = 10$ F, $U_{r a t e d} = 200$ V, $U_{d c B} = 400$ V, and $P_{B} = 10$ kW. The influence of droop gain R_d and working area u_work on H_sc is shown in Fig. 9. In Fig. 9(a) and (b), when u_dc_1,_pu and u_dc_2,_pu are both selected to be 1, the value of H_sc increases as u_work increases from 0 to 0.9.

However, within the same variation range of u_work, there is an optimized value of u_work to make the SC system provide the largest equivalent inertia H_sc when the DC voltage is at a low level (related to $u_{d c 1, p u} = u_{d c 2, p u} = 0.95$ p.u.). As observed in Fig. 9(b), the main influence of R_d on H_sc is that H_sc decreases with a larger value of R_d. However, when $u_{d c 1, p u} = u_{d c 2, p u} = 0.95$ p.u., and u_work is close to 0.9, a larger value of R_d may increase the value of equivalent inertia H_sc.

2)　Feasible Region of Equivalent Supporting Time T_s

In Section III-A, the concept of the equivalent power supporting time T_s is providedto evaluate the power smoothing ability of the SC system, as shown in (3). Then, using the proposed unified control and reduced-order model, an analytical solution of T_s is derived, as depicted in (22). For a given DCMG with a configured SC system, assuming that the main parameters U_dcB, P_B, R_d, C_sc_,0, and U_rated are well-designed, appropriate control parameters (specifically referred to as the working area u_work of the SC system) are chosen for the proposed unified control, so that a given set value of T_s is satisfied is an issue. In this paper, the feasible region of T_s is first determined. According to (22), the critical point for determining the feasible region of T_s is to find the available region of H_sc.

Fig. 9 Influence of control parameters on H_sc.

Based on (11) and (12), H_sc can be represented as a function constrained by:

H_{s c} ≜ f (u_{d c 1, p u}, u_{d c 2, p u}, u_{w o r k})

(23)

As shown in Fig. 8(b), H_sc increases with increasing $u_{d c 1, p u}$ and $u_{d c 2, p u}$ . To this end, if u_dc_1,_pu is selected as the minimum value (referred to as $1 - R_{d}$ ), H_sc satisfies:

H_{s c} \geq f (1 - R_{d}, u_{d c 2, p u}, u_{w o r k})

(24)

If the equivalent smoothing time T_s₀ is calculated using (24), the real power smoothing performance can ensure that $T_{s} \geq T_{s 0}$ .

In addition, with $u_{d c 1, p u} = 1 - R_{d}$ , the variation range of $| Δ P_{p, p u} |$ , u_dc_2,_pu, and u_work can be obtained as:

0 \leq | Δ P_{p, p u} | \leq 2 p . u .

(25)

1 - R_{d} \leq u_{d c 2, p u} = u_{d c 1, p u} + | Δ P_{p, p u} | R_{d} \leq 1 + R_{d}

(26)

0 \leq u_{w o r k} \leq 1 / (1 + R_{d})

(27)

Finally, Algorithm 1 is provided to calculate the feasible region of T_s.

Algorithm 1 : calculation of feasible region of T_s
Step 1: give the known parameters U_dcB, P_B, R_d, C_sc_,0, and U_rated.
Step 2: set $u_{d c 1, p u} = 1 - R_{d}$ , and get the variation range of $\| Δ P_{p, p u} \|$ , u_dc_2,_pu, and u_work using (25)-(27).
Step 3: calculate the feasible region of T_s using (11), (12), and (22) according to the different values of $\| Δ P_{p, p u} \|$ (or u_dc_2,_pu) and u_workwithin the applicable variation range.

For example, for a given DCMG with $U_{d c B} = 400$ V, $P_{B} = 10$ kW, $R_{d} = 0.05$ , $C_{s c, 0} = 33$ F, and $U_{r a t e d} = 200$ V, the feasible region of T_s is obtained using Algorithm 1, as shown in Fig. 10.

Fig. 10 Feasible region of T_s with different values of $| Δ P_{p, p u} |$ and u_work.

As shown in Fig. 10, for a given working area u_work of the SC system, the value of T_s increases as the power disturbance | $Δ P_{p, p u}$ | increases. When $Δ P_{p, p u}$ remains unchanged, there is an optimized value of u_work to ensure that the SC system has the largest value of T_s.

In the real operation of a DCMG, the power disturbance $| Δ P_{p, p u} |$ is always uncertain, but can be known within a certain range. Thus, the working area u_work becomes a critical control parameter for the SC system to satisfy a given set value of T_s.

3)　Design of Working Area u_work to Satisfy Power Smoothing Performance

Within the feasible region of T_s, the focus is on the following issue: when the value of an uncertain disturbance $| Δ P_{p, p u} |$ is greater than $Δ P_{p, s e t}$ (referred to a threshold value of power disturbance in the DCMG to be smoothed), how is an appropriate value of u_work designed to make the SC system to support the DCMG with the expected requirement denoted as $T_{s} \geq T_{m i n}$ ? Here, T_min is the minimum equivalent supporting time requirement, which should be within the feasible region of T_s.

The design of an appropriate value of u_work is summarized in Algorithm 2. Here, two cases with different power smoothing performance requirements are considered for the application of Algorithms1 and 2, with the basic parameters listed in Table I.

Algorithm 2 : design of an appropriate value of u_work
Step 1: set $\| Δ P_{p, p u} \| = Δ P_{p, s e t}$ , and obtain the relationship between u_work and T_s based on Algorithm 1.
Step 2: set the supporting time constraint T_min, then obtain the feasible range of u_work (which can make T_s greater than the T_min).
Step 3: select the largest value within the derived feasible range of u_work in Step 2 as the final design value.

1) Case 1: set $R_{d} = 0.05$ . And T_s satisfies $T_{s} \geq T_{m i n} = 15$ s when the uncertain power disturbance $| Δ P_{p, p u} | \geq Δ P_{p, s e t} = 0.25$ p.u..

2) Case 2: set $R_{d} = 0.075$ . And T_s satisfies $T_{s} \geq T_{m i n} = 20$ s when the uncertain power disturbance $| Δ P_{p, p u} | \geq Δ P_{p, s e t} = 1.0$ p.u..

From Step 1 in Algorithm 2, the feasible ranges of u_work satisfying the requirements of Case 1 and Case 2 are obtained as (0.2622, 0.6430) and (0.396, 0.696), respectively, as shown in Fig. 11(a). As long as the value of u_work is within the derived range, the preset power supporting time in the two cases is satisfied. Here, a method is proposed for selecting an optimal value.

Fig. 11 Influence of u_work. (a) Relationship between u_work and T_s. (b) Variation of ${\bar{A}}_{r a t i o}$ associated with u_work.

In (10)-(12), H_sc is mainly associated with the transformation coefficient A_ratio, which is related to the operation states u_dc_1,_pu and u_dc_2,_pu. In this paper, an average index of the transformation coefficient ${\bar{A}}_{r a t i o}$ is defined as:

{\bar{A}}_{r a t i o} = \int_{1 - R_{d}}^{1 + R_{d}} \int_{1 - R_{d}}^{1 + R_{d}} \frac{A_{r a t i o}}{4 R_{d}^{2}} d u_{d c 1, p u} d u_{d c 2, p u}

(28)

The parameter ${\bar{A}}_{r a t i o}$ can be used to represent the average equivalent inertia and average power smoothing ability of the SC system under all possible operating conditions of the DCMG. Figure 11(b) shows the relationship between ${\bar{A}}_{r a t i o}$ and u_work in Case 1. It is observed that the value of ${\bar{A}}_{r a t i o}$ is positively proportional to that of u_work. Therefore, after determining the feasible range of u_work, the largest value of u_work within the feasible range can be selected for a more effective utilization of the SC system, which corresponds to Step 3 in Algorithm 2.

4)　Actual Equivalent Power Supporting Time T_s with Designed Parameter of Working Area u_work

With the designed value of u_work, the actual equivalent power supporting time T_s contributed by the SC system can be determined according to (10)-(12) and (22) under all operating conditions of the DCMG. Taking Case 1 as an example, the feasible range of u_work is (0.2622, 0.6430). Then, $u_{w o r k} = 0.6430$ is considered according to Algorithm 2. To verify the effectiveness of this selection, the actual power smoothing performance represented by T_s is compared with $u_{w o r k} = 0.6430$ and $u_{w o r k} = 0.2622$ , as shown in Fig. 12.

Fig. 12 Actual equivalent supporting time T_s with $u_{w o r k} = 0.6430$ and $u_{w o r k} = 0.2622$ in Case 1.

In Fig. 12, for most operating states of the DCMG, the value of the actual equivalent power supporting time T_s of the SC system for $u_{w o r k} = 0.6430$ is larger than that of $u_{w o r k} = 0.2622$ , which means that the SC system can provide more equivalent inertia, verifying the effectiveness of Step 3 in Algorithm 2.

C. Simulation Verification

Based on the switching model in PSCAD/EMTDC, the proposed unified control of the SC system and previous theoretical analysis results are verified in this subsection.

1)　Case 1

In this case, the expected power smoothing performance and equivalent supporting time T_s are the same as those in Section V-B, which is summarized as: setting $R_{d} = 0.05$ , T_s satisfies $T_{s} \geq T_{m i n} = 15$ s when the uncertain power disturbance $| Δ P_{p, p u} | \geq Δ P_{p, s e t} = 0.25$ p.u..

Based on the previous analysis, the feasible range of u_work satisfying the requirement of Case 1 is (0.2622, 0.6430), as shown in Fig. 11(a). Figure 13 shows the simulation results of Case 1, for $u_{w o r k} = 0.6430$ and $u_{w o r k} = 0.2622$ . Figure 13(a)-(e) shows the simulation results of $P_{s, p u}$ (active power of DC-AC converter), $u_{d c, p u}$ (DC bus voltage), $P_{s c, p u}$ (power of SC system), $u_{s c, p u}$ (SC terminal voltage), and T_s, respectively.

When $t < 200$ s, the DC bus voltage $u_{d c, p u}$ is close to 1 p.u.. When u_work is selected to be 0.6430 and 0.2622, the SC terminal voltage value $u_{s c, p u}$ is approximately 0.631 and 0.821 p.u. (the rated value of the SC terminal voltage U_rated is 200 V), respectively, as shown in Fig. 13(d).

At $t = 200$ s, the power of grid-following unit P_p_,_pu is changed to 0.5 p.u., with $Δ P_{p, p u}$ of 0.5 p.u.. As shown in Fig. 13(c), the SC system responds immediately to support the DCMG and balance the related power disturbance, which means that the expected power smoothing performance shown in Fig. 2(a) is realized with the proposed unified control in Fig. 4. Moreover, as shown in Fig. 13(a), (b), and (d), the power of the grid-forming units $P_{s, p u}$ , DC bus voltage $u_{d c, p u}$ , and the terminal voltage of the SC system $u_{s c, p u}$ reach a new steady state smoothly, which indicates that with the proposed unified control, active power smoothing of the DC voltage control unit and SC terminal voltage control are realized simultaneously. In Fig. 13(c), when $u_{w o r k 1} = 0.6430$ and $u_{w o r k 2} = 0.2622$ , the actual equivalent power supporting time during this transient is $T_{s 1} = 24.88$ s and $T_{s 2} = 17.36$ s, respectively. Obviously, T_s satisfies the given requirement $T_{s} \geq T_{m i n} = 15$ s when the power disturbance $| Δ P_{p, p u} | \geq Δ P_{p, s e t} = 0.25$ p.u., verifying the effectiveness of Algorithm 2 in Section V. Moreover, from the result $T_{s 1} > T_{s 2}$ , it is observed that with a larger value of working area u_work, the SC system can operate for a larger value of equivalent supporting time, which is consistent with the theoretical analysis shown in Fig. 12.

Fig. 13 Simulation results of Case 1. (a) Active power of DC-AC converter. (b) DC bus voltage. (c) Power of SC system. (d) SC terminal voltage.

At $t = 350$ s, $P_{p, p u}$ is increased from 0.5 p.u. to 0.75 p.u., with $Δ P_{p, p u} = 0.25$ p.u.. From the results shown in Fig. 13(c), it can be observed that the actual T_s of the SC system is approximately $T_{s 1} = 18.31$ s and $T_{s 2} = 15.49$ s, respectively. The results also indicate that the designed value of u_work can achieve the expected power smoothing performance in this scenario.

Then, $P_{p, p u}$ is increased from 0.75 p.u. to 1 p.u. at $t = 450$ s and recovered back to 0.75 p.u. at $t = 550$ s. In Fig. 13(c), it is observed that the actual T_s is approximately 14.3 s, which is slightly less than the set time of 15 s. During $t = 450$ -550 s, the DC bus voltage u_dc,pu reaches the lowest value of 0.95 p.u., and the actual power smoothing performance is affected, which is consistent with the theoretical analysis in Fig. 12.

2)　Case 2

This case has the same conditions as that of Case 2 in Section V-B, which is re-emphasized as follows.

Setting $R_{d} = 0.075$ , T_s satisfies $T_{s} \geq T_{m i n} = 20$ s when the uncertain power disturbance $| Δ P_{p, p u} | \geq Δ P_{p, s e t} = 1$ p.u..

From the theoretical analysis shown in Fig. 11(a), the feasible range of u_work that satisfies the requirement is between 0.396 and 0.696. In particular, the equivalent supporting time T_s reaches the maximum value of 20.96 s, when u_work is selected to be 0.552. Figure 14 shows the simulation results of Case 2, considering the three conditions of $u_{w o r k 1} = 0.696$ , $u_{w o r k 2} = 0.396$ , and $u_{w o r k 3} = 0.522$ . Figure 14(a)-(d) shows the simulation results of $P_{s, p u}$ (active power of DC-AC), $u_{d c, p u}$ (DC bus voltage), $P_{s c, p u}$ (power of SC system), $u_{s c, p u}$ (SC terminal voltage), and $T_{s}$ , respectively.

Fig. 14 Simulation results of Case 2. (a) Active power of DC-AC converter. (b) DC bus voltage. (c) Power of SC system. (d) SC terminal voltage.

When $t < 200$ s, the DC bus voltage $u_{d c, p u}$ is approximately 1 p.u.. When u_work is selected to be 0.696, 0.396, and 0.522, the SC terminal voltage values are $u_{s c 1, p u} = 0.5822$ p.u., $u_{s c 2, p u} = 0.7322$ p.u., and $u_{s c 3, p u} = 0.6542$ p.u., respectively (the rated value of the SC terminal voltage U_rated is 200 V), as shown in Fig. 14(d).

At $t = 200$ s, $P_{p, p u}$ is increased to 1 p.u. with $Δ P_{p, p u}$ of 1 p.u.. As illustrated in Fig. 14(c), with the proposed unified control, the SC system discharges rapidly to support the DCMG at the moment of power disturbance, without any delay. In Fig. 14(a) and (b), the power output of the DC-AC converter (referred to as the grid-forming unit) and the DC bus voltage are smoothed as expected. According to the simulation results in Fig. 14(c), when u_work is selected to be 0.696, 0.396, and 0.522, the actual power supporting time T_s is $T_{s 1} = 19.34$ s, $T_{s 2} = 19.27$ s, and $T_{s 3} = 20.2$ s, respectively, which is very close to the designed value of $T_{m i n} = 20$ s. Thus, the simulation results are consistent with the theoretical analysis demonstrated in Fig. 11(a), which verifies the effectiveness of the systematic analysis and design method for the proposed unified control in Section V-B.

At $t = 350$ s, $P_{p, p u}$ is reduced to be 0.5 p.u. from 1 p.u., with a power disturbance $| Δ P_{p, p u} |$ of 0.5 p.u.. As the preset uncertain power disturbance is $Δ P_{p, s e t} = 1.0$ p.u., the actual T_s under a relatively small disturbance $| Δ P_{p, p u} |$ may not satisfy the requirement $T_{s} \geq T_{m i n} = 20$ s. The simulation results in Fig. $14 (c)$ show that the actual T_s associated with the control parameter u_work is about $T_{s 1} = 14.18$ s, $T_{s 2} = 16.8$ s, and $T_{s 3} = 16.43$ s, respectively, which is consistent with the previous theoretical analysis.

D. Discussion About Control

Two cases with different parameters and conditions are applied to verify the proposed unified control of the SC system and the theoretical analysis in this paper. From the simulation results, the following conclusions can be drawn.

1) The proposed unified control realizes power smoothing, and SC terminal voltage control, simultaneously on the same timescale. Compared with the methods in [

14]-[17], the unified control has a more simple but effective structure. Moreover, it should be noted that only local DC voltages and currents are used for power smoothing control, indicating good plug-and-play features.

2) The proposed simple reduced model of DCMG in Section IV and theoretical analysis in Section V are effective since the presented Algorithms1 and 2 derived from the analytical solutions of equivalent inertia and power supporting time have been verified by detailed simulation results.

In addition, it should be noted that the analytical solutions to equivalent inertia and power supporting time can be used not only for the design of the working area u_work of the SC system, but also for the optimal configuration of the SC system. For example, to satisfy the power smoothing performance, the questions such as how much the capacitance of the SC system is needed and how to determine the rated terminal voltage of the SC system can also be addressed with the proposed control, simple model, and theoretical results.

VI. Conclusion

In this paper, a novel unified control of the SC system is proposed for power smoothing in DCMG. Without any communication, it can simultaneously realize the smoothing of large power disturbance and terminal voltage control of the SC system, indicating satisfactory plug-and-play characteristics. Moreover, using the defined equivalent power supporting time and equivalent inertia contributed by the unified controlled SC system, a systematic design method for critical control parameters is presented and further verified by detailed simulation results. With the proposed unified control, reduced model, and design method, the expected power smoothing performance of the SC system represented by the equivalent power supporting time can be achieved in the feasible region, which is very useful for the practical operation of DCMGs.

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Home

Introduction

Editorial Board

For Author

Call For Papers

APC

Sponsor & Publisher

A Unified Control of Super-capacitor System Based on Bi-directional DC-DC Converter for Power Smoothing in DC Microgrid PDF

Abstract

Keywords

I. Introduction

II. Typical DCMG Structure

III. Unified Control for SC System

A. Requirements of Power Smoothing Control Performance in DCMG

B. Proposed Unified Control of SC System

IV. Analytical Solution of T_s by a Simple but Effective Reduced-order Model of DCMG

A. Reduced-order Model of DCMG

B. Verification of Second-order Model

C. Further Reduction of Second-order Model

V. Systematic Analysis, Design of Proposed Unified Control, and Simulation Verification

A. DCMG Configuration

B. Systematic Analysis and Design of Proposed Control

C. Simulation Verification

D. Discussion About Control

VI. Conclusion

References

Home

Introduction

Editorial Board

For Author

Call For Papers

APC

Sponsor & Publisher

A Unified Control of Super-capacitor System Based on Bi-directional DC-DC Converter for Power Smoothing in DC Microgrid PDF

Abstract

Keywords

I. Introduction

II. Typical DCMG Structure

III. Unified Control for SC System

A. Requirements of Power Smoothing Control Performance in DCMG

B. Proposed Unified Control of SC System

IV. Analytical Solution of Ts by a Simple but Effective Reduced-order Model of DCMG

A. Reduced-order Model of DCMG

B. Verification of Second-order Model

C. Further Reduction of Second-order Model

V. Systematic Analysis, Design of Proposed Unified Control, and Simulation Verification

A. DCMG Configuration

B. Systematic Analysis and Design of Proposed Control

C. Simulation Verification

D. Discussion About Control

VI. Conclusion

References

IV. Analytical Solution of T_s by a Simple but Effective Reduced-order Model of DCMG