#### 2.1. Changes in Zero Sequence Component According to Transformer Structure

In the general magnetic circuit design of a three-phase transformer shown in

Figure 2a, there is a three-limb core type, in which the tops and bottoms of three vertical parallel limbs are connected by a horizontal yoke. Another type of three-phase transformer has a five-limb core structure that includes three limbs with coiled windings and two limbs with uncoiled windings, as shown in

Figure 2b. In a three-phase transformer made of three single-phase transformers, the impedance of the positive sequence component, negative sequence component, and zero sequence component are the same, the unbalanced component appears equally on the primary and secondary sides of the transformer like a balanced component. However, in the case of a three-phase transformer, the positive sequence component and zero sequence component react differently, depending on the connection type and the core structure.

The zero sequence current of the microgrid inverter is the sum of each phase’s current, as shown in (1):

where,

${i}_{abc}$ is the current of

a,

b and

c phase, and

${i}_{0}$ is the sum of the zero sequence currents.

${i}_{abc.0}$ is the zero sequence currents in each phase. The zero sequence current shows the third harmonic of the fundamental harmonic, and the magnitude and phase (

${i}_{abc.0}$) are the same in each phase. If the three phases are balanced, the zero sequence current is zero, but if they are unbalanced, the zero sequence current is not zero.

Figure 3a shows the zero-sequence-component magnetic flux of a three-limb core-type transformer. The magnitude and direction of the zero-sequence-component magnetic flux of each phase is the same, so it does not disappear, but appears as three times the zero sequence component, which is the sum of each component. This tripled zero-sequence-component magnetic flux flows along a path through air or oil outside the transformer. Since this path has a large amount of magnetic resistance, a large amount of magneto motive force occurs. As a result, the load-side zero sequence component is controlled by the transformer. Problems occur, such as an increased transformer temperature and an increased loss [

14]. On the primary side, which has capacitors that determine the output voltage, the zero sequence component is not transmitted, so a zero-sequence-component voltage does not occur on the primary side. In addition, when short circuits and ground faults occur and a three-limb core-type transformer is being used, the secondary-side zero sequence component is not transmitted to the primary side, so a problem occurs, in that the failure cannot be detected by the zero sequence current.

Figure 3b shows the zero-sequence-component magnetic flux of a five-limb core-type transformer. Five-limb core-type transformers provide a path through which the zero-sequence-component magnetic flux can pass. Therefore, the zero sequence component flows through the transformer’s core, which has low magnetic resistance, and the load-side zero sequence component is transmitted entirely to the primary side. The zero sequence current is transmitted to the primary-side capacitor owing to the load-side zero sequence component, and unbalance occurs in the output voltage.

#### 2.2. Zero-Sequence-Component Control Technique in a Five-Limb Core-Type Transformer

When a three-limb core-type transformer is used, the zero sequence component created by the unbalanced load is not transferred to the primary side. Therefore, it is impossible to perform zero sequence current control at the inverter. However, when a five-limb core-type transformer is used, the load-side zero sequence current is transmitted to the primary side, so zero sequence current control is possible, and an additional control technique can be applied to control the zero sequence current.

Equation (2) converts the stationary coordinate frame variables for each phase into a synchronous reference frame.

where

${f}_{abc}$ are functions of phase

a,

b, and

c.

$T(\theta )$ is transformation matrix of

abc frame to synchronous coordinate frame.

${f}_{dq0}^{\omega}$ represents the

d,

q,0 axis of the synchronous coordinate frame.

${f}_{0}^{\omega}$ is the 0-axis of the synchronous coordinate frame [

22,

23]. In (2), the zero sequence component

$({f}_{a}+{f}_{b}+{f}_{c})$ is shown on the synchronous reference frame’s 0-axis (

${f}_{0}^{\omega}$). Therefore, to remove the zero sequence component, we must use the

dq0 control, which controls up to the

0-axis in the existing

dq controller.

Figure 4 shows a three-phase, four-wire inverter with output transformer. It is a structure in which two switches that connect to the neutral terminal are added to the six existing switches, and 0-axis control commands generated by Equation (2) are used to control the switches connected to the neutral wire and to control the zero sequence component.

A simulation analysis of the transformer, which has zero-sequence-component magnetic transfer properties, was performed using the winding, linear core, and the air gap within the PSIM’s magnetic element [

24].

Figure 5 shows the PSIM’s magnetic element device.

Figure 5a shows the winding element, which converts electric energy to magnetic energy. The winding element is the same coil.

Figure 5b is a linear core, it can simulate magnetic flux according to magnetoresistance.

Figure 5c is airgap, and it represents the magnetic flux flowing through air.

Figure 6 shows an equivalent circuit of transformers implemented via PSIM software.

Figure 6a shows a three-limb core-type transformer in which the tops and bottoms of three vertical parallel limbs are connected by a horizontal yoke. The zero sequence component has the same magnitude and phase in each phase. In a three-limb core-type transformer, since there is no path connecting the top and bottom of the transformer for the zero sequence component to flow, zero sequence component flux flows through the airgap outside the transformer.

Figure 6b shows a five-limb core-type transformer. In a five-limb core-type transformer, zero sequence component flux flows through the extra two linear cores.

The three-phase inverter circuit in

Figure 7 was designed, and a simulation analysis was performed for the changes in the output voltage according to the transformer structure under unbalanced load conditions (

R_{a} = 20 Ω,

R_{b} = 12 Ω, and

R_{c} = 200 Ω). Labels a1, a2, b1, b2, c1, c2 of simulation are connected to the equivalent model of the transformer shown in

Figure 6a or

Figure 6b.

Figure 8 shows the simulation results of the output voltage and output current in which a three-limb core-type transformer is used, and an unbalanced load is connected. Output voltage (

${V}_{load\_abc}$) has a small unbalance. If a three-limb core-type transformer is used, the zero sequence component that occurs from the load flows through the air gap, which has a high magnetic resistance. Therefore, the zero sequence component is not transmitted smoothly to the primary side, and only the positive sequence component and negative sequence component appear on the primary side. In the simulation results, a 20-V voltage unbalance occurred owing to the negative sequence component. Even if this configuration has a small unbalance voltage, the phase short and ground faults cannot be detected by the high impedance for zero-sequence magnetic flux, which can be a big problem for the micro-grid system reliability.

Figure 9 shows the simulation results of a simulation using a normal proportional-integral (PI) controller without zero-sequence-component compensation when a five-limb core-type transformer is used to detect the load’s short circuits and ground faults. The five-limb core-type transformer has a magnetic flux path that the zero sequence component can pass through, so the load’s zero sequence component is entirely transmitted to the primary side. Therefore, the positive sequence component, negative sequence component, and zero sequence component are all transmitted to the capacitor, and the load side’s voltage unbalance is transmitted to the primary side. Thus, a large voltage unbalance occurs. Based on the parameters used, a voltage unbalance of around 220 V occurred.

In the case of a three-limb core-type transformer, the zero sequence component can be removed without additional controls, but it flows through the transformer, which affects the transformer’s lifespan. In addition, grid faults such as short circuits and ground faults cannot be detected. On the other hand, the five-limb core-type transformer can detect grid faults, but the zero sequence component is transmitted to the primary side and a large unbalance occurs, so a zero-axis controller must be added.

Figure 10 shows the

dq0 controller block diagram. Where,

${V}_{dq0}$ is the output voltage in the synchronous coordinate frame.

${V}_{dq0}^{ref}$ is the reference of the output voltage in synchronous coordinate frame. A voltage controller consisting of a PI controller generates an inductor current reference using voltage error.

${i}_{L.dq0}$ is the inductor current in the synchronous coordinate frame, and

${i}_{dq{0}_{ref}}$ is the reference of the inductor current in the synchronous coordinate frame. A current controller consisting of a PI controller generates the reference of duty.

${D}_{dq.fb}$ is the reference of feedback duty.

$-\omega L{i}_{q}/{V}_{dc}$ is the feedforward component of 3 phase PWM inverter [

11,

12,

13].

${D}_{dq0\_ref}$ is the duty reference in each axis of the PWM inverter. A zero sequence controller that controls the zero sequence component is added to the

dq controller that controls the positive sequence component, and the zero sequence component is controlled. To provide a balanced voltage, the zero-sequence-component voltage command must be 0. The zero-sequence voltage error in the

0-axis voltage controller is PI controlled, and zero-sequence current commands are created. The voltage controller’s output becomes a zero-sequence current command proportional to the magnitude of the current unbalance, and controls the zero sequence current.

#### 2.3. Negative-Sequence-Component Control Technique

If a three-limb core-type transformer is used or a three-phase four-wire inverter with a five-limb core-type transformer is built, and a

dq0 controller is used, the zero sequence component can be removed. However, the negative sequence component still remains and causes an output voltage unbalance. The negative sequence component reverses the order of the phases as in (3):

where,

${V}_{abc\_positive}$ is the positive sequence component for the

abc frame and

${V}_{abc\_negative}$ is the negative sequence component for the

abc frame. The Equations (4) and (5) for converting the negative sequence component to a synchronous reference frame are as follows:

where

${f}_{negative\_dq}^{\omega}$ is the negative sequence component for the synchronous coordinates frame. As shown in (5), the negative sequence component appears as a second harmonic in the synchronous reference frame.

Figure 11 shows a controller block diagram for an inverter with an added resonant controller for removing the negative sequence component.

Equation (6) is the transfer function of the PI + R controller, and the PI + R controller is the sum of the PI controller and the resonant controller.

where,

${k}_{pv}$ is the proportional gain of the voltage controller and

${k}_{iv}$ is the integral gain of the voltage controller, and

${k}_{pr}$ is the resonant gain of the voltage controller. In the resonant controller transfer function of Equation (2), at a frequency of

${\omega}_{2\mathrm{th}}$, the controller has an infinite gain. Therefore, by using the resonance controller, it is possible to remove the second harmonic present in the negative sequence component.

The proposed controller adds a resonant controller for the second harmonic in parallel, and can remove the negative sequence component.

Figure 12 shows a Bode plot of a proportional-integral + resonant (PI + R) controller; it can be seen that there is a large gain in the second harmonic frequency [

20,

21].

Figure 13 shows the results of a PSIM simulation in which a PI + R controller is used to control the negative sequence component. Under the same parameter conditions as

Figure 7, a resonant controller was added in parallel to remove the secondary harmonic. The resonant controller removes the negative sequence component, so it can be seen that even when an unbalanced load is connected, a balanced voltage is produced.

Figure 14 shows a fast Fourier transform (FFT) analysis. If a conventional PI controller (

Figure 14a) is used, the negative sequence component appears as a second harmonic in the synchronous reference frame. If the secondary-side harmonic is controlled by the resonant controller (

Figure 14b), the negative sequence component is removed, and balanced voltage can be produced.

#### 2.4. Nonlinear Compensation Using PI + MR Controller

To use a DC load, the electrical energy user changes the AC voltage into DC through a rectifier, such as a diode rectifier [

25,

26,

27]. This kind of rectifier load creates a 6

n harmonic on the load side, and 6

n − 1, 6

n + 1 harmonics on the phase side, owing to a nonlinear load. If the three-phase diode rectifier’s input phase current is described as a Fourier series, in can be written as

where

${i}_{L,abc}$ is the input current of the diode rectifier. Equation (7) is converted to a synchronous reference frame.

where

${i}_{L,dq}$ is the input current of the diode rectifier in the synchronous frame. As shown in (8), if the 6

n harmonic is removed from the synchronous reference frame, this has the benefit of removing both the 6

n − 1 and 6

n + 1 harmonics that appear in the phase. The existing PI + R controller compensates for both the 6

n − 1 and 6

n + 1 harmonics in the stationary reference frame, so the frequencies that must be removed become twice as numerous.

Equation (9) is the transfer function of PI + MR controller for nonlinear load, which is the sum of PI controller and multi-resonant controller.

where

${k}_{{\mathit{pr}}_{6}}$ is the resonant gain of 6th harmonic,

${\omega}_{6\mathrm{th}}$ is the radian of the 6th harmonic,

${k}_{{\mathit{pr}}_{12}}$ is the resonant gain of the 12th harmonic,

${\omega}_{12\mathrm{th}}$ is the radian of the 12th harmonic.

Figure 15 shows a block diagram of the resonant controller that controls the nonlinearity. The controller is designed to eliminate the 6th- and 12th-order components, which are the largest components of harmonics that make nonlinearity.

Figure 16 shows a Bode plot of the resonant controller.