Open AccessArticle

Point of Common Connection Voltage Modulated Direct Power Control with Disturbance Observer to Increase in Renewable Energy Acceptance in Power System

Yong Woo Jeong

and

Woo Young Choi

Department of Control and Instrumentation Engineering, Pukyong National University, Busan 48513, Republic of Korea

Author to whom correspondence should be addressed.

Energies 2024, 17(21), 5319; https://doi.org/10.3390/en17215319

Submission received: 6 September 2024 / Revised: 3 October 2024 / Accepted: 22 October 2024 / Published: 25 October 2024

(This article belongs to the Special Issue Energy, Electrical and Power Engineering: 3rd Edition)

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Graphical abstract
"> Figure 1
Schematic of the grid-connected voltage source inverter with the step-up transformer and the proposed PCCVM-DPC and proposed DOB. "> Figure 2
Schematic of the Simulink/MATLAB simulation environments. "> Figure 3
Closed-loop bode plot of PCCVM-DOC with PI and PI + DOB. "> Figure 4
Fast Fourier Transform (FFT) result of <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>p</mi> <mi>c</mi> <mi>c</mi> <mo>,</mo> <mi>a</mi> </mrow> </msub> </semantics></math>. "> Figure 5
Validation results of the estimated disturbances of PI + DOB. "> Figure 6
Validation results of the active power for PCCVM-DPC with PI and PI + DOB. "> Figure 7
Validation results of the reactive power for PCCVM-DPC with PI and proposed PI+DOB. "> Figure 8
Validation results of the current harmonics for PCCVM-DPC with PI and proposed PI + DOB. "> Figure 9
Validation results of the power control performance during the transient region. "> Figure 10
Validation results of the current harmonics for PCCVM-DPC with PI and the proposed PI + DOB, considering the parameter uncertainties (THD of grid voltage: 2.9%). ">

Versions Notes

Abstract

In this paper, we present a disturbance observer-based point of common connection voltage-modulated direct power control (PCCVM-DPC) system, which increases the robustness of the PCCVM-DPC system. First, the mathematical analysis of the disturbances for the step-up transformer’s nonlinearity, the grid voltage harmonics, and the parameter uncertainties is presented. By analyzing the disturbance terms of the PCCVM-DPC system, we present the disturbance observer (DOB) for the PCCVM-DPC system. To assess the efficacy of our approach, we perform comparative studies of the PCCVM-DPC without DOB and PCCVM-DPC with DOB by constructing the simulation environment based on the commercial step-up transformer and ESS inverter datasheet. We have validated that the active and reactive power control performance of the PCCVM-DPC with DOB outperforms the PCCVM-DPC without DOB from the observation that the current total harmonic distortion reduced by more than 40% compared to the PCCVM-DPC without the DOB.

Keywords:

grid voltage modulated power control; point of common connection; energy storage system; voltage source inverter; disturbance observer; total harmonic distortion

Graphical Abstract

1. Introduction

At the end of 2023, global renewable power capacity had reached 3870 GW, and the photovoltaic power generation accounted for the largest share of the global total, with a capacity of 1419 GW [1]. As the penetration of renewable energies is increasing into the power grid system, irregular power generation from the renewable energy system is starting to raise grid instability. To remedy these issues, charging and discharging strategies of a grid-connected energy storage system (ESS) have been researched [2,3,4] coupled with the power grid [5]. To realize the charging/discharging strategy, an active and reactive power control performance of the grid-connected ESS inverter system is essential, and the performance of power control influences the power fluctuation of the point of common connection (PCC) side, which can cause inefficiency of the power grid system [6]. Specifically, an amount of active and reactive power flowing through the electrical substation can not exceed the installed transformer capacity, so the excessive power fluctuation caused by the grid-connected ESS inverter system can reduce the utilization efficiency of the installed electrical substation system. Therefore, research on the inverter control performance enhancement to reduce power fluctuation can provide a practical solution to increase the renewable energy acceptance of the grid system while not increasing the current electric substation capacity.

A total harmonic distortion (THD) of the phase currents and voltages at the PCC side directly impacts the electronic device’s lifetime [7,8,9], so in the case that the 3-phase voltages can be considered as the balanced conditions, the performance of the grid-connected ESS inverter control can be evaluated with the THD. The voltage harmonics are caused by the nonlinear loads and inadequately generated voltage from the generators [10], so it can be regarded as an uncontrollable factor to the inverter control system except for the weak grid conditions. Therefore, reducing the THD of the phase currents from the given voltage harmonic condition is important while performing active and reactive power control.

A vector current control (VCC) method is one of the control methods to stabilize a voltage source inverter (VSI). From the desired active and reactive power reference, the set point of the d and q-axis currents can be calculated, and a proportional and integral (PI) control structure of the VCC realizes the power control of the VSI [11,12,13,14]. However, the PI control of the VCC does not guarantee robustness for unknown uncertainties such as parameter variation, so it is not easy to perform current decoupling [15,16]. In order to improve the robust stability of the grid-connected inverter for wind power or photovoltaic power generation, the robust model of grid-connected inverter is established, and the influence mechanism of power-grid impedance and voltage distortion on the stability and current quality of grid-connected inverter is analyzed in [17]. To deal with the grid side current and voltage sensor failure, a robust control scheme for a three-phase grid-tied inverter system is developed by presenting the grid current and voltage estimation architecture [18].

In the case of AC–DC power converters, a DC voltage fluctuation can influence the robustness of the VSI system. To enhance the robustness by controlling the DC voltage of the grid-connected inverter, a second-order linear active disturbance rejection control is presented in [19], and an extended state observer-based sliding mode control for three-phase power converters is presented in [20]. In [21], a novel discrete-time robust adaptive super-twisting sliding mode controller is presented to enhance the robustness of the grid-tied power converter. To enhance the robustness of the system uncertainties and unknown perturbations, predictive control for the three-phase NPC converter has been presented by combining the reinforcement learning methodologies [22,23]. In [24], a disturbance observer-based inverter control is proposed by assuming that the disturbance of the grid system only has a sinusoidal waveform that oscillates at the fundamental frequency, particularly when the grid voltage does not include harmonics. In [25], parameter uncertainties of the inverter system have been compensated by using the disturbance observer. Since its application is for the three-phase PWM rectifier which flows the energy from the grid to the load, it does not consider the nonlinearity of the transformer and its distorted harmonics. However, in the case of the renewable power generator, which transfers the power to the grid, consideration of this nonlinearity is essential to control power fluctuation. Although all methodologies mentioned above contribute to enhancing the robustness of the VSI system, the three-phase current control is performed in the rotating reference frame, which indicates that it has a significant dependency on the phase angle estimation of the grid voltages. A phase-locked loop (PLL) is one of the frequently utilized tools to estimate the phase angle of the grid system [26,27], and it is reported that the estimated position contains a delayed phase during the transient regions [28,29]. Therefore, the non-uniform estimation result of the phase angle at the transient regions impacts the VSI control performance. Furthermore, the harmonics of the phase voltages directly impact the estimated result, so the enhancement of the phase angle estimator is needed [30,31,32].

Recently, a grid voltage modulation direct power control (GVM-DPC) [33] has been proposed to reduce the dependency on the phase-locked loop (PLL) position observer. From the comparative study between the VCC and the GVM-DPC, the outperformed power control performance in the transient region has been verified [34,35]. In [36], an improved DC-link voltage regulation strategy has been presented to enhance the DC voltage regulation performance by using the DC current observer, but the uncertainties of the inverter system are not considered such as resistance/inductance uncertainties and the distorted grid voltage harmonics. To enhance the robustness of the GVM-DPC, sliding mode control (SMC) with the zero-dimensional sliding surface is applied and shows enhanced performance [37]. Although the sliding mode-based control structure has an outperformed performance for the unknown and bounded disturbance of the system, to the best of our experience, a heuristic gain tuning process is needed for the control design step. In [38], the point of common connection voltage modulated direct power control (PCCVM-DPC) for the grid-tied photovoltaic inverter has been presented by constructing the conventional feedforward and feedback PI controllers, and shows the outperformance of the PCCVM-DPC from the comparative studies compared to the PLL based dq current controller.

In the case of the grid-connected inverter for the renewable energy generator and energy storage system, a step-up transformer is utilized to reduce the power loss of the generated renewable energy during the power transmission, and its nonlinearity can cause the power fluctuation at the point of common connection side, which can reduce the acceptance of the renewable energy to the conventional electrical substation. However, to the best of the author’s knowledge, research has not been performed that analyzes the disturbances caused by the step-up transformer’s nonlinearity and the distorted grid voltages and presents a disturbance observer-based PCCVM-DPC system. For that reason, in this paper, we present a point of common connection voltage-modulated direct power control (PCCVM-DPC) system by considering the nonlinearity of the transformer characteristics. By analyzing the disturbance terms of the PCCVM-DPC system, we newly present the disturbance observer and its compensation logic for the three-phase inverter control system. To assess the efficacy of our approach, we present comparative validation results of the conventional grid voltage modulated-direct power control (GVM-DPC) and the PCCVM-DPC by simulating the transformer based on the commercial transformer datasheet. The outperformed power quality at the PCC of the proposed method shows its effectiveness.

The main contributions of this paper are summarized as follows:

Modeling of the active and reactive power dynamics for the PCCVM-DPC, including the step-up transformer’s nonlinearity.
Analyzing the model disturbance including the parameter uncertainties of the inverter system, the distorted voltage harmonics at the PCC side and the transformer’s nonlinearity.
Presenting disturbance observer for the PCCVM-DPC system and analyzing the robustness of the closed-loop system.

The remainder of this paper consists of Section 2, which describes the system modeling of the ESS/VSI including the transformer nonlinear characteristics and the disturbance observer and the PCCVM-DPC, Section 3 for the validation results, and Section 4 for the conclusion.

2. Main Contents

To reduce the power fluctuation and the total harmonic distortion (THD) of the phase currents at the point of common connection (PCC), in this section, we newly derive the instantaneous active and reactive power dynamics at the PCC, including the step-up transformer characteristics. We analyze the disturbances that hinder the power tracking performance from the newly derived dynamics. Furthermore, we newly present a disturbance observer and a point of common connection voltage modulated direct power control (PCCVM-DPC) structure, including the disturbance rejection control. And, the closed-loop transfer functions are analyzed to expect the stability and effectiveness of the proposed method.

2.1. Active and Reactive Power Dynamics at the PCC Including Step-Up Transformer

Figure 1 shows the schematic of a voltage source inverter (VSI), connected to the grid system. Let us distinguish the three-phase system by defining j as

j \in {a, b, c}

. Let us define the output voltage of the VSI as

u_{j}, \forall j \in {a, b, c}

, the line filter inductance as

L_{f, j}

, and the resistance as

R_{f, j}

. Let us define the currents flowing from the IGBT of the inverter to the beginning of the low voltage bushing as

i_{i n v, j}

. Let us define the voltage at the end of the line filter as

v_{i n v, j}

. Then, the voltage drop equation from

u_{j}

to the end of

R_{f, j}

is represented as

\begin{matrix} u_{j} & = L_{f, j} \frac{d}{d t} i_{i n v, j} + R_{f, j} i_{i n v, j} + v_{i n v, j}, \forall j \in {a, b, c} . \end{matrix}

(1)

Let us define the line resistance including the transformer’s primary winding as

R_{p, j}

, the line inductance including the transformer’s primary winding as

L_{p, j}

, and the terminal voltage of the primary part as

v_{p, j}

\forall j \in {a, b, c}

. Then, the voltage drop equation from

v_{i n v, j}

to the end of

R_{p, j}

is represented as

\begin{matrix} v_{i n v, j} & = L_{p, j} \frac{d}{d t} i_{i n v, j} + R_{p, j} i_{i n v, j} + v_{p, j} . \end{matrix}

(2)

Let us define the primary winding current as

i_{p, j}

, the number of primary winding turns as

n_{p}

, the secondary winding current as

i_{s, j}

, the secondary voltage of the transformer as

v_{s, j}

, and the number of secondary winding turns as

n_{s}

. Then, the current and voltage relationship of the step-up transformer is represented as

\begin{matrix} i_{s, j} & = \frac{n_{p}}{n_{s}} i_{p, j}, v_{s, j} = \frac{n_{s}}{n_{p}} v_{p, j}, \forall j \in {a, b, c} . \end{matrix}

(3)

To consider the hysteresis characteristic of the transformer, let us define the excitation current as

i_{e x c, j}

, core loss resistance as

R_{e x c, j}

, and magnetization inductance as

L_{e x c, j}

\forall j \in {a, b, c}

. Then, the relationship between

i_{p, j}

and

i_{i n v, j}

is represented as

\begin{matrix} i_{p, j} & = i_{i n v, j} - i_{e x c, j}, \forall j \in {a, b, c} . \end{matrix}

(4)

Let us define the line resistance including the transformer’s secondary winding as

R_{s, j}

, and the line inductance including the transformer’s secondary winding as

L_{s, j}

. Let us define the flowing current at the PCC as

i_{p c c, j}

, and the measured voltage at the PCC as

v_{p c c, j}

\forall j \in {a, b, c}

. Then, we can represent the voltage drop equation from

v_{s, j}

to the end of

R_{s, j}

\begin{matrix} v_{s, j} & = L_{s, j} \frac{d}{d t} i_{p c c, j} + R_{s, j} i_{p c c, j} + v_{p c c, j}, \forall j \in {a, b, c} . \end{matrix}

(5)

By considering that

i_{s, j} = i_{p c c, j}

and plugging (1)–(4) into (5), we can represent the dynamics of

i_{p c c, j}

with respect to

u_{j}

\begin{matrix} u_{j} & = \{L_{f, j} \frac{n_{s}}{n_{p}} + L_{p, j} \frac{n_{s}}{n_{p}} + L_{s, j} \frac{n_{p}}{n_{s}}\} \frac{d}{d t} i_{p c c, j} \\ + \{R_{f, j} \frac{n_{s}}{n_{p}} + R_{p, j} \frac{n_{s}}{n_{p}} + R_{s, j} \frac{n_{p}}{n_{s}}\} i_{p c c, j} \\ + L_{f, j} \frac{d}{d t} i_{e x c, j} + L_{p, j} \frac{d}{d t} i_{e x c, j} + R_{f, j} i_{e x c, j} + R_{p, j} i_{e x c, j} \\ + \frac{n_{p}}{n_{s}} v_{p c c, j}, \forall j \in {a, b, c} . \end{matrix}

(6)

For the simplicity of the notation, let us define the equivalent inductance as

L_{e q, j}

, the resistance as

R_{e q, j}

, uncertainty caused by the leakage currents of the transformer as

Δ_{e x c, j}

, such that

\begin{matrix} L_{e q, j} = L_{f, j} \frac{n_{s}}{n_{p}} + L_{p, j} \frac{n_{s}}{n_{p}} + L_{s, j} \frac{n_{p}}{n_{s}}, \\ R_{e q, j} = R_{f, j} \frac{n_{s}}{n_{p}} + R_{p, j} \frac{n_{s}}{n_{p}} + R_{s, j} \frac{n_{p}}{n_{s}}, \\ Δ_{e x c, j} = L_{f, j} \frac{d}{d t} i_{e x c, j} + L_{p, j} \frac{d}{d t} i_{e x c, j} + R_{f, j} i_{e x c, j} + R_{p, j} i_{e x c, j}, \forall j \in {a, b, c} . \end{matrix}

(7)

By using the Clarke transformation, current dynamics in the stationary reference frame are represented as

\begin{matrix} L_{e q, k} \frac{d}{d t} i_{p c c, k} & = - R_{e q, k} i_{p c c, k} + u_{k} - Δ_{e x c, k} - \frac{n_{p}}{n_{s}} v_{p c c, k}, \forall k \in {α, β} . \end{matrix}

(8)

Since electrical parameters have an uncertainty according to the temperature, let us separate the nominal values (

R_{0}

L_{0}

) and parameter uncertainties (

Δ R_{k}

Δ L_{k}

), such that

\begin{matrix} L_{e q, k} = L_{0} + Δ L_{k}, R_{e q, k} = R_{0} + Δ L_{k}, \forall k \in {α, β} . \end{matrix}

(9)

By replacing

L_{e q, k}

and

R_{e q, k}

of (8) with (9), we obtain

\begin{matrix} \frac{d}{d t} i_{p c c, k} & = - \frac{R_{0}}{L_{0}} i_{p c c, k} + \frac{1}{L_{0}} u_{k} + \frac{1}{L_{0}} Δ V_{k} - \frac{1}{L_{0}} \frac{n_{p}}{n_{s}} v_{p c c, k}, \end{matrix}

(10)

where

Δ V_{k} = - Δ R_{k} i_{p c c, k} - Δ L_{k} \frac{d}{d t} i_{p c c, k} - Δ_{e x c, k}, \forall k \in {α, β} .

The PCC voltages

v_{p c c, k}

can be decomposed as the fundamental frequency component and the harmonics, so let us define

V_{g_{f}}

as the magnitude of the fundamental grid voltage, and

ω_{f}

is the fundamental frequency of the grid voltage. Then, we can represent

v_{p c c, k}

\begin{matrix} v_{p c c, α} & = V_{g_{f}} cos (ω_{f} t) + α (t), \\ v_{p c c, β} & = V_{g_{f}} sin (ω_{f} t) + β (t), \end{matrix}

(11)

where

α (t)

and

β (t)

indicate harmonic components. The time derivative of (11) is represented as

\begin{matrix} \frac{d}{d t} v_{p c c, α} & = - ω_{f} V_{g_{f}} sin (ω_{f} t) + \dot{α} (t) \\ = - ω_{f} v_{p c c, β} + Δ v_{p c c, α} \\ \frac{d}{d t} v_{p c c, β} & = ω_{f} V_{g_{f}} cos (ω_{f} t) + \dot{β} (t) \\ = ω_{f} v_{p c c, α} + Δ v_{p c c, β} \end{matrix}

(12)

where

Δ v_{p c c, α} = ω_{f} β (t) + \dot{α} (t), Δ v_{p c c, β} = - ω_{f} α (t) + \dot{β} (t) .

Let us define the instantaneous fundamental components of the real and reactive powers at the PCC as

P_{p c c}

and

Q_{p c c}

, where

\begin{matrix} P_{p c c} & = \frac{3}{2} (v_{p c c, α} i_{p c c, α} + v_{p c c, β} i_{p c c, β}), \\ Q_{p c c} & = \frac{3}{2} (v_{p c c, β} i_{p c c, α} - v_{p c c, α} i_{p c c, β}) . \end{matrix}

(13)

A time derivative of (13) is expressed as follows:

\begin{matrix} \frac{d}{d t} P_{p c c} = \frac{3}{2} (\frac{d v_{p c c, α}}{d t} i_{p c c, α} + v_{p c c, α} \frac{d i_{p c c, α}}{d t} + \frac{d v_{p c c, β}}{d t} i_{p c c, β} + v_{p c c, β} \frac{d i_{p c c, β}}{d t}) \\ \frac{d}{d t} Q_{p c c} = \frac{3}{2} (\frac{d v_{p c c, β}}{d t} i_{p c c, α} + v_{p c c, β} \frac{d i_{p c c, α}}{d t} - \frac{d v_{p c c, α}}{d t} i_{p c c, β} - v_{p c c, α} \frac{d i_{p c c, β}}{d t}) . \end{matrix}

(14)

By substituting (10) and (12) with

\frac{d}{d t} v_{p c c, k}, \frac{d}{d t} i_{p c c, k}

of (14), the dynamics of

P_{p c c}

and

Q_{p c c}

are represented as

\begin{matrix} \frac{d}{d t} P_{p c c} = - \frac{R_{0}}{L_{0}} P_{p c c} - ω_{f} Q_{p c c} + \frac{1}{L_{0}} d_{P} + \frac{3}{2 L_{0}} (v_{p c c, α} u_{α} + v_{p c c, β} u_{β}) \\ \frac{d}{d t} Q_{p c c} = - \frac{R_{0}}{L_{0}} Q_{p c c} + ω_{f} P_{p c c} + \frac{1}{L_{0}} d_{Q} + \frac{3}{2 L_{0}} (v_{p c c, β} u_{α} - v_{p c c, α} u_{β}) \end{matrix}

(15)

where

\begin{matrix} d_{P} = \frac{3}{2} {L_{0} Δ v_{p c c, α} i_{p c c, α} + L_{0} Δ v_{p c c, β} i_{p c c, β} \\ + v_{p c c, α} Δ V_{α} + v_{p c c, β} Δ V_{β} - \frac{n_{p}}{n_{s}} (v_{p c c, α}^{2} + v_{p c c, β}^{2})} \\ d_{Q} = \frac{3}{2} {L_{0} Δ v_{p c c, β} i_{p c c, α} - L_{0} Δ v_{p c c, α} i_{p c c, β} \\ + v_{p c c, β} Δ V_{α} - v_{p c c, α} Δ V_{β}} . \end{matrix}

(16)

Let us define PCCVM-DPC inputs as

u_{P}

and

u_{Q}

, which is similar to the GVM-DPC system [37], where

\begin{matrix} [\begin{matrix} u_{P} \\ u_{Q} \end{matrix}] & = [\begin{matrix} v_{p c c, α} u_{α} + v_{p c c, β} u_{β} \\ - v_{p c c, β} u_{α} + v_{p c c, α} u_{β} \end{matrix}] . \end{matrix}

(17)

By plugging (17) into (15), we can obtain concise active and reactive power dynamics as

\begin{matrix} \frac{d}{d t} P_{p c c} = - \frac{R_{0}}{L_{0}} P_{p c c} - ω_{f} Q_{p c c} + \frac{3}{2 L_{0}} u_{P} + \frac{1}{L_{0}} d_{P} \\ \frac{d}{d t} Q_{p c c} = - \frac{R_{0}}{L_{0}} Q_{p c c} + ω_{f} P_{p c c} - \frac{3}{2 L_{0}} u_{Q} + \frac{1}{L_{0}} d_{Q} . \end{matrix}

(18)

Therefore, the harmonics of the PCC voltages, uncertainties of electric parameters, and the nonlinearity of the transformer are represented as lumped disturbance terms of the active and reactive power dynamics. Based on these active and reactive power dynamics, the disturbance observer for the PCCVM-DPC system and disturbance rejection control structure are presented in the next subsections.

2.2. Disturbance Observer Design for PCCVM-DPC

Since multiple uncertainties are included in the disturbance terms

d_{P}

and

d_{Q}

of (18), it is expected that the proper disturbance estimation and compensation control logic can reduce the power fluctuation during the power control, which enhances the acceptance of the grid-connected ESS. In this section, we propose the structure of the disturbance observer for the PCCVM-DPC system and its stability analysis. To design the proposed DOB, let us define the estimated value of

d_{P}

d_{Q}

{\hat{d}}_{P}

{\hat{d}}_{Q}

. Let us define the estimated value of

P_{p c c}

Q_{p c c}

{\hat{P}}_{p c c}

{\hat{Q}}_{p c c}

, which are used for disturbance estimation. Let us define the estimation error of

d_{P}

d_{Q}

{\tilde{d}}_{P}

{\tilde{d}}_{Q}

, and the estimation error of

P_{p c c}

Q_{p c c}

{\tilde{P}}_{p c c}

{\tilde{Q}}_{p c c}

, such that

\begin{matrix} {\tilde{P}}_{p c c} & = P_{p c c} - {\hat{P}}_{p c c}, {\tilde{Q}}_{p c c} = Q_{p c c} - {\hat{Q}}_{p c c} \\ {\tilde{d}}_{P} & = d_{P} - {\hat{d}}_{P}, {\tilde{d}}_{Q} = d_{Q} - {\hat{d}}_{Q} . \end{matrix}

(19)

Let us define the observer gains as

l_{p}

and

l_{i}

. Then, from (18), the structure of DOB for the PCCVM-DPC system can be designed as

\begin{matrix} \frac{d}{d t} {\hat{P}}_{p c c} & = - \frac{R_{0}}{L_{0}} P_{p c c} - ω_{f} Q_{p c c} + \frac{3}{2 L_{0}} u_{P} + \frac{1}{L_{0}} {\hat{d}}_{P} \\ {\hat{d}}_{P} & = L_{0} (l_{p} {\tilde{P}}_{p c c} + l_{i} \int {\tilde{P}}_{p c c} d τ) \\ \frac{d}{d t} {\hat{Q}}_{p c c} & = - \frac{R_{0}}{L_{0}} Q_{p c c} + ω_{f} P_{p c c} - \frac{3}{2 L_{0}} u_{Q} + \frac{1}{L_{0}} {\hat{d}}_{Q} \\ {\hat{d}}_{Q} & = L_{0} (l_{p} {\tilde{Q}}_{p c c} + l_{i} \int {\tilde{Q}}_{p c c} d τ) . \end{matrix}

(20)

From now on, let us describe the stability analysis and the frequency response analysis of the proposed DOB structure. From (18) and (20), we obtain following equation:

\begin{matrix} \frac{d}{d t} {\tilde{P}}_{p c c} & = - l_{p} {\tilde{P}}_{p c c} - l_{i} \int {\tilde{P}}_{p c c} d τ + L_{0}^{- 1} d_{P}, \\ \frac{d}{d t} {\tilde{Q}}_{p c c} & = - l_{p} {\tilde{Q}}_{p c c} - l_{i} \int {\tilde{Q}}_{p c c} d τ + L_{0}^{- 1} d_{Q} . \end{matrix}

(21)

By defining the Laplace transform as

L (\cdot)

, we can represent each results of the Laplace transform for

{\tilde{P}}_{p c c}

{\tilde{d}}_{P}

d_{P}

{\tilde{Q}}_{p c c}

{\tilde{d}}_{Q}

, and

d_{Q}

{\tilde{P}}_{p c c} (s) = L (P_{p c c})

{\tilde{D}}_{P} (s) = L ({\tilde{d}}_{P})

and

D_{P} (s) = L (d_{P})

{\tilde{Q}}_{p c c} (s) = L (Q_{p c c})

{\tilde{D}}_{Q} (s) = L ({\tilde{d}}_{Q})

and

D_{Q} (s) = L (d_{Q})

. From (21), we obtain

{\tilde{P}}_{p c c} (s) = \frac{L_{0}^{- 1} s}{s^{2} + l_{p} s + l_{i}} D_{P} (s), {\tilde{Q}}_{p c c} (s) = \frac{L_{0}^{- 1} s}{s^{2} + l_{p} s + l_{i}} D_{Q} (s) .

(22)

By performing time derivative of

{\tilde{d}}_{P}

in (19), we obtain following equations:

\begin{matrix} {\dot{\tilde{d}}}_{P} & = {\dot{d}}_{P} - {\dot{\hat{d}}}_{P} \\ = {\dot{d}}_{P} - L_{0} (l_{p} {\dot{\tilde{P}}}_{p c c} + l_{i} {\tilde{P}}_{p c c}) \\ {\dot{\tilde{d}}}_{Q} & = {\dot{d}}_{Q} - L_{0} (l_{p} {\dot{\tilde{Q}}}_{p c c} + l_{i} {\tilde{Q}}_{p c c}) . \end{matrix}

(23)

From the Laplace transform of (23) and substituting

{\tilde{P}}_{p c c}

as (22), the transfer function between

d_{P}

and

{\tilde{d}}_{P}

are represented as

\begin{matrix} \frac{{\tilde{D}}_{P}}{D_{P}} & = 1 - \frac{l_{p} s + l_{i}}{s^{2} + l_{p} s + l_{i}} = \frac{s^{2}}{s^{2} + l_{p} s + l_{i}} = \frac{s^{2}}{s^{2} + 2 ζ_{o} ω_{o} s + ω_{o}^{2}}, \end{matrix}

(24)

where

ω_{o}

is the natural frequency and

ζ_{o}

is the damping ratio of the observer system such as

ω_{o} = \sqrt{l_{i}}

and

ζ_{o} = \frac{l_{p}}{2 \sqrt{l_{i}}}

. To discuss the gain tuning process of

l_{p}

and

l_{i}

, let us represent (24) as

\begin{matrix} \frac{{\tilde{D}}_{P} (s)}{D_{P} (s)} = K_{o} \frac{s^{2}}{(s / p_{o 1} + 1) (s / p_{o 2} + 1)}, \end{matrix}

(25)

where

p_{o 1} = ζ_{o} ω_{o} + j ω_{o} \sqrt{1 - ζ_{o}^{2}}

p_{o 2} = ζ_{o} ω_{o} - j ω_{o} \sqrt{1 - ζ_{o}^{2}}

and

K_{o} = {l_{i}}^{- 1}

. To avoid unnecessary oscillation of the estimated result while increasing the robustness, we set

ζ_{o}

to have a critical damping condition, and we set

ζ_{o} ω_{o}

to be 20 times more than the fundamental frequency of the grid voltages, such that

ζ_{o} = 1

ω_{o} = 20 \times 2 π 60

2.3. Disturbance Rejection Control of the PCCVM-DPC System

Let us define the desired active power at the PCC as

P_{p c c}^{d}

, and the reactive power command as

Q_{p c c}^{d}

. Then, we can define tracking errors of each power as

\begin{matrix} e_{P} & = P_{p c c}^{d} - P_{p c c}, \\ e_{Q} & = Q_{p c c}^{d} - Q_{p c c} . \end{matrix}

(26)

By differentiating (26) and plugging (18), we obtain

\begin{matrix} {\dot{e}}_{P} & = {\dot{P}}_{p c c}^{d} - {\dot{P}}_{p c c} \\ = {\dot{P}}_{p c c}^{d} + \frac{R_{0}}{L_{0}} P_{p c c} + ω_{f} Q_{p c c} - \frac{3}{2 L_{0}} u_{P} - \frac{1}{L_{0}} d_{P}, \\ {\dot{e}}_{Q} & = {\dot{Q}}_{p c c}^{d} - {\dot{Q}}_{p c c} \\ = {\dot{Q}}_{p c c}^{d} + \frac{R_{0}}{L_{0}} Q_{p c c} - ω_{f} P_{p c c} + \frac{3}{2 L_{0}} u_{Q} - \frac{1}{L_{0}} d_{Q} . \end{matrix}

(27)

Let us represent the control inputs

u_{P}

and

u_{Q}

\begin{matrix} u_{P} = u_{e q, P} - \frac{2 L_{0}}{3} {\tilde{u}}_{P}, \\ u_{Q} = u_{e q, Q} + \frac{2 L_{0}}{3} {\tilde{u}}_{Q} \end{matrix}

(28)

where

u_{e q, j}, \forall j \in {P, Q}

are equivalent feedback control law, and

{\tilde{u}}_{j}, \forall j \in {P, Q}

are the disturbance rejection control law, such that

\begin{matrix} u_{e q, P} = - \frac{2}{3} {\hat{d}}_{P} + \frac{2 L_{0}}{3} (\frac{R_{0}}{L_{0}} P_{p c c} + ω_{f} Q_{p c c} + {\dot{P}}_{p c c}^{d}), \\ u_{e q, Q} = \frac{2}{3} {\hat{d}}_{Q} - \frac{2 L_{0}}{3} (\frac{R_{0}}{L_{0}} Q_{p c c} - ω_{f} P_{p c c} + {\dot{Q}}_{p c c}^{d}), \end{matrix}

(29)

\begin{matrix} {\tilde{u}}_{P} = - k_{p} e_{P} - k_{i} \int e_{P} d τ, \\ {\tilde{u}}_{Q} = - k_{p} e_{Q} - k_{i} \int e_{Q} d τ \end{matrix}

(30)

where

k_{p}

and

k_{q}

are the control gains. By substituting

u_{P}

and

u_{Q}

of (27) with (28)–(30), the tracking error dynamics of PCCVM-DPC system with the proposed control method are represented as

\begin{matrix} {\dot{e}}_{P} = - k_{p} e_{P} - k_{i} \int e_{P} d τ - \frac{1}{L_{0}} {\tilde{d}}_{P}, \\ {\dot{e}}_{Q} = - k_{p} e_{Q} - k_{i} \int e_{Q} d τ - \frac{1}{L_{0}} {\tilde{d}}_{Q} . \end{matrix}

(31)

By defining Laplace transform as

L (\cdot)

, we can represent each results of Laplace transform for

e_{P}

and

e_{Q}

E_{P} = L (e_{P})

E_{Q} = L (e_{Q})

. Then, we can represent the transfer function from

{\tilde{d}}_{j}

e_{j}

\forall j \in {P, Q}

\begin{matrix} \frac{E_{j}}{{\tilde{D}}_{j}} & = - \frac{1}{L_{0}} \frac{s}{s^{2} + k_{p} s + k_{i}} = \frac{s^{2}}{s^{2} + 2 ζ_{c} ω_{c} s + ω_{c}^{2}} \end{matrix}

where

ω_{c}

is the natural frequency and

ζ_{c}

is the damping ratio of the control system such as

ω_{c} = \sqrt{k_{i}}

and

ζ_{c} = \frac{k_{p}}{2 \sqrt{k_{i}}}

. To discuss the gain tuning process of

k_{p}

and

k_{i}

, let us reformulate the transfer function as

\begin{matrix} \frac{E_{j}}{{\tilde{D}}_{j}} & = K_{c} \frac{s}{(s / p_{c 1} + 1) (s / p_{c 2} + 1)} \end{matrix}

(32)

where

p_{c 1} = ζ_{c} ω_{c} + j ω_{c} \sqrt{1 - ζ_{c}^{2}}

p_{c 2} = ζ_{c} ω_{c} - j ω_{o} \sqrt{1 - ζ_{c}^{2}}

and

K_{c} = {(L_{0} k_{i})}^{- 1}

. To avoid unnecessary oscillation of the control result while increasing the robustness, we set

ζ_{c}

to have a critical damping condition, and we set

ζ_{c} ω_{c}

to be seven times more than the fundamental frequency of the grid voltages, such that

ζ_{c} = 1

ω_{c} = 7 \times 2 π 60

Remark 1.

In the case where the disturbance observer is not utilized for the PCCVM-DPC system,

{\hat{d}}_{P}

and

{\hat{d}}_{Q}

are regarded as 0, which means

{\tilde{D}}_{j} = D_{j}

. Therefore, from (19) and (32), we can obtain the closed-loop transfer function between

d_{j}

e_{j}

\forall j \in {P, Q}

\frac{E_{j}}{D_{j}} = K_{c} \frac{s}{(s / p_{c 1} + 1) (s / p_{c 2} + 1)} .

(33)

From (25) and (32), we obtain the closed-loop transfer function of the proposed PCCVM-DPC control method as

\begin{matrix} \frac{E_{j}}{D_{j}} & = K_{o} K_{c} \frac{s^{3}}{(s / p_{o 1} + 1) (s / p_{o 2} + 1) (s / p_{c 1} + 1) (s / p_{c 2} + 1)} . \end{matrix}

(34)

From

K_{o}

of (25), we can expect that the robustness of the proposed PCCVM-DPC with DOB for the disturbance is improving as

l_{i}

is increasing. Compared to PCCVM-DPC without the DOB, which is discussed in Remark 1, we expect that the power control performance of the proposed PCCVM-DPC with the DOB outperforms the one without the DOB. The gain tuning of the second order transfer function can be performed by setting the natural frequency and the damping ratio.

3. Validation

3.1. Simulation Environment

To validate the proposed PCCVM-DPC with the DOB, we constructed the simulation environment with Simulink/Simscape specialized power systems library as shown in Figure 2. The fundamental frequency of the grid voltages is set as

f_{0} = 60

Hz. The pulse width modulated (PWM) frequency of the inverter is set as

f_{p w m} = 20

kHz, PWM resolution is set to be

2^{6}

bit. The PWM signals are center-aligned and the rising time delay for the complementary PWM signals is set as

t_{d} = 0.3 μ

s. The measurement sampling frequency of the

i_{p c c, j}, \forall j \in {a, b, c}

, the control sample time is set as

T_{c t r l} = 1 / f_{p w m}

, and forward Euler method is utilized for the controller and observer. To simulate the characteristics of the inverter, the simulation frequency is set to be 512 times more than the PWM frequency such as

T_{s} = 1 / (512 f_{p w m})

and the backward-Euler discrete method is utilized for the inverter and transformer of the simulator. We set the step-up transformer and the ESS inverter parameters by referring to the LS electronic product datasheets and the IEC 60076-11 (Transformer: 250 kVA 400 V/22 kV–60 Hz, ESS inverter: two modules of 125 kW inverter) [39,40,41]. Table 1 shows the simulation configuration parameters and the power grid system parameters illustrated in Figure 1. By considering the fundamental frequency of the grid voltages and the DC-offset of the closed loop bode plot, we selected the control gains to have duplicated poles, and its natural frequencies were set as 420 Hz, such as

p_{c 1} = p_{c 2} = 2 π \times 420

. Furthermore, observer gains were also selected to have duplicated poles, and their natural frequencies were set as 1200 Hz, such as

p_{o 1} = p_{o 2} = 2 π \times 1200

Figure 3 shows the comparative bode plot results between PI (33) and PI + DOB (34) based on the selected

k_{p}

k_{i}

l_{p}

and

l_{i}

. The blue line indicates the amplitude of (33), and the red line indicates the amplitude of (34). From (16), we can expect that the unknown disturbances consist of sinusoidal harmonics in which the fundamental frequency is 60 Hz, and from the bode plot, we can expect that the power tracking performance of the proposed PI + DOB is robust to the unknown disturbances compared to the conventional PI control structure.

3.2. Validation Results for Different Voltage Harmonic Conditions

The total harmonic distortion of the voltage is one of the important factors in operating power grid devices. We intentionally injected various harmonics into the grid voltages to validate the control performance for the disturbances. These harmonics are represented as

α (t), β (t)

of the (11) and, consequently, changing

d_{P}

and

d_{Q}

of (16).

Case 1-1: $α (t)$ ≃ 0 and $β (t) ≃ 0$ .
Case 2-1: $α (t) = 0.015 cos (5 \times 2 π f_{0} t) + 0.025 cos (7 \times 2 π f_{0} t)$ and
$β (t) = 0.015 sin (5 \times 2 π f_{0} t) + 0.025 sin (7 \times 2 π f_{0} t)$ .
Case 3-1: $α (t) = 0.03 cos (5 \times 2 π f_{0} t) + 0.05 cos (7 \times 2 π f_{0} t)$ and
$β (t) = 0.03 sin (5 \times 2 π f_{0} t) + 0.05 sin (7 \times 2 π f_{0} t)$ .

Figure 4 shows the total harmonic distortion (THD) of each case and fast Fourier transform (FFT) results of

V_{p c c, a}

. Active power command of the power conversion system (PCS) is applied up to 125 kW, which is the maximum value of the one power electronics building block of LS electronics PCS products. The blue colored line of this subsection indicates the validation results of Case 1-1, the red colored line indicates the results of Case 2-1, and the yellow colored line indicates the result of Case 3-1, respectively.

Figure 5 shows the estimated disturbance results. As the amplitude of the harmonics is increasing, we can see that the amplitude of estimated disturbance harmonics are increasing, as we discussed (11), (12) and (15). Figure 6 shows the comparative studies of the active power control performance between PCCVM-DPC with PI and PCCVM-DPC with PI + DOB. Figure 6a,b show the time plot of active power control performance for the PI and PI + DOB control structure, respectively. Figure 6c,d shows the fast Fourier transform (FFT) results of the active power control performance, for the PI and PI + DOB control structure, respectively. As expected from the bode plot, we confirmed that the dramatic performance enhancement for the harmonic disturbances is achieved. Figure 7 shows the comparative studies of the reactive power control performance between PCCVM-DPC with PI and PCCVM-DPC with PI + DOB. Figure 7a,b shows the time plot of reactive power control performance for the PI and PI + DOB control structure, respectively. Figure 7c,d show the FFT results of the reactive power control performance, for the PI and PI + DOB control structure, respectively. As expected from the bode plot, we confirmed that the dramatic performance enhancement for the harmonic disturbances is achieved. Figure 8 shows the FFT results of the phase currents for each case. From these results, we see that the proposed PCCVM-DPC with PI + DOB dramatically reduces the current THD by more than 40% of the THD of the PI control structure.

3.3. Validation Results for the Transient Regions

In this section, we present a comparative study between the conventional PCCVM-DPC and the proposed PCCVM-DPC with the DOB by considering the transformer’s nonlinearity and the distorted harmonic grid voltage conditions. Figure 9 shows the transient response when the active power reference is commanded from 62.5 kW to 125 kW, and the distorted grid voltages are the same as in Case 2-1 where the THD of the grid voltage is 2.9%. The blue color line indicates the active and reactive power control results of the conventional PCCVM-DPC, which does not contain the disturbance observer. The red color line indicates the active and reactive power control results of the proposed PCCVM-DPC with DOB. Since neither methodology utilizes the phase angle of the grid system, we can observe the outperformance of the PCCVM-DPC’s transient response. For the comparative studies during the transient region between the voltage-modulated DPC and the PLL-based controller, please refer to [34,38].

3.4. Validation Results for the Parameter Uncertainties

In this section, we present a comparative study between the conventional PCCVM-DPC and the proposed PCCVM-DPC with the DOB for the parameter uncertainties. Figure 10 shows the FFT results of the phase current at the PCC for both the conventional PCCVM-DPC and the proposed DOB-based PCCVM-DPC when circuit parameter uncertainty exists. The distorted grid voltages are the same as those in Case 2-1, where the THD of the grid voltage is 2.9Assuming that parameter estimation is performed irregularly, we scaled the actual plant parameters of the simulator (

R, L

) proportionally to the estimated values

R_{0}, L_{0}

. The blue line indicates the FFT results when the ratio of

R / R_{0}

and

L / L_{0}

is 0.8, the red line indicates when

R / R_{0}

and

L / L_{0}

are 1, and the yellow line indicates when

R / R_{0}

and

L / L_{0}

are 1.2. From these validations, we confirmed that the proposed PCCVM-DPC with DOB increases the robustness of the power control system to parameter uncertainties compared to the conventional PCCVM-DPC.

The blue line represents the active and reactive power control results of the conventional PCCVM-DPC, which does not include a disturbance observer. The red line shows the active and reactive power control results of the proposed PCCVM-DPC with a disturbance observer (DOB). Since neither methodology utilizes the phase angle of the grid system, we can observe that the PCCVM-DPC demonstrates superior transient response performance.

4. Conclusions

This paper proposed the point of common connection voltage modulated direct power control (PCCVM-DPC) for energy storage systems and power conversion system (ESS/PCS) with the disturbance observer and disturbance rejection structure to control the instantaneous active and reactive powers under the step-up transformer’s nonlinearity, parametric uncertainties, and the grid voltages harmonics. From the validation with the Simscape and the actual ESS inverter and transformer datasheet, we observed that PCCVM-DPC with the proposed control structure has robust power control performance although the measured voltages at the PCC contain inadequate harmonics. We validated the robustness of the proposed method from the observation that the phase current THD of the proposed DOB-based PCCVM-DPC has been reduced by more than 40% compared to the conventional PCCVM-DPC in various distorted grid voltage conditions. Furthermore, we validated the robustness of the proposed method for the parameter uncertainties. Although the power control performance of the proposed method outperforms the conventional PCCVM-DPC, the proposed DOB-based PCCVM-DPC structure is designed in the condition that the three-phase voltages of the grid system are in the balanced conditions, which only have the harmonic distortions. Therefore, an extension of research on the PCCVM-DPC for the unbalanced voltage conditions can be an interesting future research topic. Furthermore, the application of the proposed PCCVM-DPC and the disturbance observer to the electrified railway system can also be a meaningful research topic [42,43].

Author Contributions

Conceptualization, methodology, visualization, software and writing—original draft preparation, validation, supervision, writing—review and editing, Y.W.J.; funding acquisition, W.Y.C. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. RS-2023-00213640, Accurate Autonomous Tracking Control by Unmatched Disturbance Compensator with Reference Re-design Filter), the Industrial Source Technology Development Program (20018144, Development and PoC of Personal Mobility service robot platform) funded by the Ministry of Trade, Industry and Energy (MOTIE, Korea) and the Pukyong National University Industry-university Cooperation Foundation’s 2024 Post-Doc. support project.

Data Availability Statement

https://github.com/YongWooJeong/energies-17-05319 (accessed on 21 October 2024).

Acknowledgments

The author appreciates Chung Choo Chung for pre-reviewing the proposed formulation and the simulation environments of this paper before submission.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Schematic of the grid-connected voltage source inverter with the step-up transformer and the proposed PCCVM-DPC and proposed DOB.

Figure 2. Schematic of the Simulink/MATLAB simulation environments.

Figure 3. Closed-loop bode plot of PCCVM-DOC with PI and PI + DOB.

Figure 4. Fast Fourier Transform (FFT) result of

v_{p c c, a}

Figure 4. Fast Fourier Transform (FFT) result of

v_{p c c, a}

Figure 5. Validation results of the estimated disturbances of PI + DOB.

Figure 6. Validation results of the active power for PCCVM-DPC with PI and PI + DOB.

Figure 7. Validation results of the reactive power for PCCVM-DPC with PI and proposed PI+DOB.

Figure 8. Validation results of the current harmonics for PCCVM-DPC with PI and proposed PI + DOB.

Figure 9. Validation results of the power control performance during the transient region.

Figure 10. Validation results of the current harmonics for PCCVM-DPC with PI and the proposed PI + DOB, considering the parameter uncertainties (THD of grid voltage: 2.9%).

Table 1. Parameters used in simulation [39,40,41].

Param	Value	Unit	Param	Value	Unit
$f_{p w m}$	20	$k H z$	PWM resolution	$2^{5}$	-
$T_{s}$	$1 / (512 f_{p w m})$	s	$T_{c t r l}$	$1 / f_{p w m}$	s
$f_{0}$	60	Hz	$V_{d c}$	1100	V
$L_{f}$	6.00	mH	$R_{f}$	0.15	$Ω$
$L_{p}$	91.7	$μ$ H	$R_{p}$	2.7	m $Ω$
$L_{s}$	0.33	H	$R_{s}$	9.63	$Ω$
$L_{e x c}$	663.15	H	$R_{e x c}$	1.851	M $Ω$
$V_{p, j}$	380	V	$V_{s, j}$	22.9	kV
$k_{p}$	5277.9	-	$k_{i}$	6.940e6	-
$l_{p}$	1.508e4	-	$l_{i}$	5.685e7	-

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Jeong, Y.W.; Choi, W.Y. Point of Common Connection Voltage Modulated Direct Power Control with Disturbance Observer to Increase in Renewable Energy Acceptance in Power System. Energies 2024, 17, 5319. https://doi.org/10.3390/en17215319

AMA Style

Jeong YW, Choi WY. Point of Common Connection Voltage Modulated Direct Power Control with Disturbance Observer to Increase in Renewable Energy Acceptance in Power System. Energies. 2024; 17(21):5319. https://doi.org/10.3390/en17215319

Chicago/Turabian Style

Jeong, Yong Woo, and Woo Young Choi. 2024. "Point of Common Connection Voltage Modulated Direct Power Control with Disturbance Observer to Increase in Renewable Energy Acceptance in Power System" Energies 17, no. 21: 5319. https://doi.org/10.3390/en17215319

APA Style

Jeong, Y. W., & Choi, W. Y. (2024). Point of Common Connection Voltage Modulated Direct Power Control with Disturbance Observer to Increase in Renewable Energy Acceptance in Power System. Energies, 17(21), 5319. https://doi.org/10.3390/en17215319

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Point of Common Connection Voltage Modulated Direct Power Control with Disturbance Observer to Increase in Renewable Energy Acceptance in Power System

Abstract

1. Introduction

2. Main Contents

2.1. Active and Reactive Power Dynamics at the PCC Including Step-Up Transformer

2.2. Disturbance Observer Design for PCCVM-DPC

2.3. Disturbance Rejection Control of the PCCVM-DPC System

3. Validation

3.1. Simulation Environment

3.2. Validation Results for Different Voltage Harmonic Conditions

3.3. Validation Results for the Transient Regions

3.4. Validation Results for the Parameter Uncertainties

4. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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