# Impact of Reactive Current and Phase-Locked Loop on Converters in Grid Faults

^{*}

## Abstract

**:**

## 1. Introduction

- Integrated the pseudo-trajectory analysis method and phase portrait methods to achieve a concise evaluation of the dynamic performance of PLL-based converters, during and after a fault.
- Developed a relationship for deriving reactive current references which ensured stable LVRT behavior.
- Determined the impact of the PLL’s loss of synchronization on active power at the converter terminals, and thus on the DC link voltage.
- Proposed a comprehensive LVRT strategy, in alignment with the aforementioned contributions, in order to facilitate stable operation during faults, and also to prompt recovery following faults.

## 2. System Modeling

_{cov}is converted into its dq-components utilizing Park’s transformation [23]. The PLL uses the q-axis component of the grid voltage V

_{q}as the set point. During steady-state, when the Park’s transformation’s output voltage space vector rotates synchronously with the grid voltage space vector, if V

_{cov}is aligned with the d-component, V

_{q}vanishes.

_{PLL}-q

_{PLL}(dashed line in Figure 3) and of the grid d

_{g}-q

_{g}(solid line in Figure 3) rotate at different angular velocities, ω

_{g}and ω

_{PLL}.

_{PLL}; the grid voltage’s phase V

_{cov}is θ

_{g}. The PLL’s deviation is θ

_{ε}= θ

_{g}− θ

_{PLL}. θ

_{I}is the grid current’s phase and θ

_{C}is the phase of the current reference for the converter’s current control loop.

## 3. Dynamic Behavior Investigation during and after Fault with Reactive Current Reference

_{PLL}and input V

_{q}of the PLL, respectively. The PLL’s operation point moves along the trajectory (blue, green or red curves in Figure 4). The intersections of the trajectories and the abscissa are the equilibrium points. In Figure 4, two equilibrium points on the blue and green curves exist, with the left one being a stable equilibrium point (SEP

_{0}) and the one at the right representing an unstable equilibrium point (USEP

_{0}) [14]. No equilibrium point exists on the red curve, which implies that the operation point moving along the red curve diverges.

_{0}to the red solid line, and then moves continuously to the right in the direction of increasing angle θ

_{PLL}along the red curve.

_{f}exists, as shown in the green curve in Figure 4. The operating point jumps from the original SEP

_{0}to the green curve, and then moves during the fault to the left along the green curve to the temporary stable equilibrium point SEP

_{f}. After fault clearing, the operating point jumps from the SEP

_{f}back to the blue curve, and then moves along the blue curve to the original SEP

_{0}.

_{c}|:

_{Z}is close to π/2, which is the usual condition for transmission grids. If the converter operates at a unity power factor, the current reference’ phase θ

_{C}is 0, so |sin(θ

_{C}+ θ

_{Z})| ≈ 1. In this case, (8) can be further simplified to the following:

_{c}| in (3) constantly equals zero. Therefore, criterion (4) is satisfied for any combination of voltage and grid impedance. In some fault-ride-through strategies [30], (5) can be satisfied by adjusting the phase of the current reference θ

_{C}, which is equivalent to adjusting the ratio of current reference ${I}_{\mathrm{d}}^{\ast}$ and ${I}_{\mathrm{q}}^{\ast}$.

_{Z}can be obtained by an online estimation algorithm [27,28,29]. In addition, the current reference’s phase can also be controlled adaptively to satisfy (7). For example, V

_{q}or ω

_{PLL}is used as a set point to adjust the θ

_{C}= arctan(${I}_{\mathrm{q}}^{\ast}$/${I}_{\mathrm{d}}^{\ast}$) [5,31]. That is, if ${I}_{\mathrm{q}}^{\ast}$ = −1.0 p.u. and ${I}_{\mathrm{d}}^{\ast}$ = 0 p.u., then θ

_{C}= −π/2, which means that the converter delivers the maximum possible reactive power and zero active power into the grid.

_{C}and |m

_{c}| and m

_{g}. The abscissa represents θ

_{C}. The blue and red curves represent |m

_{c}| and m

_{g}, respectively. According to criterion (4), |m

_{c}| ≤ m

_{g}, if the operating point lies below the red curve, an SEP exists, e.g., the operating point located at the green point θ

_{C}= −θ

_{Z}or the yellow point θ

_{C}= −π/2. If the operating point moves to the right in the direction of increasing phase, this implies an absorption of reactive power from the grid during the fault. Such behavior of the converter is forbidden during a fault according to grid codes. Therefore, this research only investigated the case in which the operating point moves to the left.

_{C}= 0. When fault ride-through control [27,28,29,30,31,32,33] is activated, the operation point moves to the left along the blue trajectory with magenta arrows in Figure 5. Neither an online grid impedance estimation algorithm nor an adaptive controller loop can immediately place the operation point in the region m

_{g}, in order to meet criterion (4). Therefore, the SEP’s existence cannot be ensured until θ

_{C}is stabilized, and may even deteriorate the stability performance during this process.

_{g}is the change in grid voltage and k is the slope of the reactive current, which is often chosen as 2.

_{C}should be −π/2. However, when adaptive fault ride-through control is activated, the converter injects less reactive current into the grid than required by the grid code in most cases, even if the operating point is at the green point, i.e., θ

_{C}= −θ

_{Z}, due to the resistive part of the grid impedance, θ

_{Z}< π/2.

_{Z}approaches π/2, and sin(θ

_{C}+ θ

_{Z}) becomes negative and close to zero. This makes |m

_{c}| in the criterion (6) smaller, which will make the system more robust against a loss of stability.

_{K}is close to −π/2 during a fault. If the converter has an SEP due to the injection of reactive currents, the SEP

_{f}is located in the interval (−π/2,0), and lies to the left of the original SEP, as shown in Figure 5.

## 4. Investigation of Reactive Current Exit Behavior after Fault Clearing

#### 4.1. Reactive Current Exit Behavior

_{g}| < 0.4 p.u.), the reactive current reference ${I}_{\mathrm{q}}^{\ast}$ was set to −I

_{max}, and the active current reference ${I}_{\mathrm{d}}^{\ast}$ was set to 0, in order to meet requirements of the grid codes, as explained in the previous section. After fault clearing, different strategies exist for returning the reactive current to its pre-fault set point. Upon summarizing our test results from existing commercial converters, we identified two reactive current exit strategies after fault clearing, as shown in Figure 6.

_{max}to 0, while the active current reference ${I}_{\mathrm{d}}^{\ast}$ gradually increases to its pre-fault value. The current references should meet the constraint of (12) to avoid overcurrent [31].

_{max}to 0. The second stage starts after ${I}_{\mathrm{q}}^{\ast}$ reaches 0, and then the active current reference ${I}_{\mathrm{d}}^{\ast}$ increases to its pre-fault value.

_{ε}is given by the following:

_{ε}≠ 0, the actual output current does not match the calculated output current in the controller; therefore, the actual output power does not match the controller’s setting.

_{max}and ${I}_{\mathrm{d}}^{\ast}$ = 0 when the fault is cleared and the phase deviation at this time instant is denoted by θ

_{ε}, then the actual output active power is given by the following:

_{ε}< 0), then the actual output active power is negative. The converter draws power from the grid and transfers it into its DC link. If the phase deviation θ

_{ε}is π/2, then the actual output power of the converter is −|V

_{g}||I

_{max}|, i.e., the maximum output power that the converter will backfill from the grid to the converter’s DC Link.

_{ε}due to the stabilization of the PLL allows the actual output power to be finally stabilized at a value that is consistent with the DC link’s power balance. However, the PLL’s dynamics and the different strategies of ${I}_{\mathrm{d}}^{\ast}$ reduction lead to a complex dynamic process of the converter at the post-fault stage. The power may oscillate between positive and negative output, thus jeopardizing the stability of the converter, and even the grid’s operation.

- Reduces negative active power magnitude;
- Reduces power oscillations;
- Increases active power as soon as possible while satisfying 1 and 2.

_{pPLL}= 0.32, K

_{iPLL}= 0.32 1/s. The rate of reactive current exit used in the simulation was 5 p.u./s.

_{ε}for a reactive current reference. The abscissa represents the phase deviation between the converter frame and grid frame; the ordinate is the reactive current reference, and the colors represent the actual output active power. As the active power approached 1.0 p.u., the area’s color turned closer to dark red. As the active power approached −1.0 p.u., there existed an active power flow from the grid into the converter, and the area’s color turned closer to dark blue. The black dashed lines denote the boundaries between positive and negative active power. The following example serves as further explanation.

_{0}) and B(t

_{0}), were located to the left and right of the SEP, respectively, in Figure 7. The two trajectories depicted in green and magenta show the course of the phase deviation with increasing time as a function of the reference value of I

^{*}. It was noted that the initial point was defined by the fault configuration as well as the fault time, and could only slightly be influenced by converter control during a no SEP fault. After fault clearing, the PLL rapidly reduced the phase deviation, while A and B moved closer to the SEP. A and B experienced an overshoot after passing the SEP. At t

_{1}, the phase deviations of A and B reached their respective maxima after passing the SEP.

_{1}, whereas B moved from the dark red area to the blue area, i.e., the actual output power changed from positive to negative.

_{ε}for the reactive current reference of strategy 2. Figure 8b illustrates the heat map of strategy 2’s first stage (t

_{0}< t < t

_{1}), with ${I}_{\mathrm{d}}^{\ast}$ = 0 and the active power varying with ${I}_{\mathrm{q}}^{\ast}$ and the PLL’s phase deviation. Figure 8a shows the heat map of strategy 2’s second stage (t > t

_{1}), with ${I}_{\mathrm{q}}^{\ast}$ = 0 and the active power varying with the phase deviation and ${I}_{\mathrm{d}}^{\ast}$.

_{0}< t < t

_{1}) and second stage (t > t

_{1}) in Figure 8 are as follows:

_{1}, the |${I}_{\mathrm{q}}^{\ast}$| corresponding to points A and B were zero, and then entered the second stage: the ${I}_{\mathrm{d}}^{\ast}$ gradually increased, as shown in Figure 8a.

#### 4.2. Adjustment Strategy for the Current Reference after Fault Clearing

_{q}(θ

_{ε}), as illustrated in Figure 9.

_{ε}, i.e., the position of the initial point after a fault in Figure 7 or Figure 8, this research used V

_{q}and dV

_{q}/dt to determine the area where its phase deviation θε is located, as presented in Figure 9a.

_{q}’s maximum. When the initial phase deviation was located in area I, with strategy 1 (Figure 9d), during the approach to the SEP, the operating point passed through the dark blue region, i.e., the negative maximum active power, regardless of how the curve of ${I}_{\mathrm{q}}^{\ast}$ was changed. If strategy 2 (Figure 9c) was used, at strategy 2’s first stage, it was possible to keep the operating point in the light green or light blue region most of the time as it approached the SEP, i.e., the active power being close to 0. At strategy 2’s second stage, after ${I}_{\mathrm{q}}^{\ast}$ went to zero, raising ${I}_{\mathrm{d}}^{\ast}$, as provided in Figure 9b, allowed the output active power to remain at a small negative value, or always positive, as the operating point approached the SEP.

_{q}’s maximum and the SEP. According to the investigations in Section 3, the SEP

_{f}during the fault was in this area. Therefore, this area was the most likely area for initial phase deviation after fault clearing. Using either strategy 1 (Figure 9d) or strategy 2 (Figure 9c) caused the operating point to pass through the dark blue region, regardless of how the ${I}_{\mathrm{q}}^{\ast}$ was changed. However, a rapid reduction in ${I}_{\mathrm{q}}^{\ast}$ minimized the duration of the blue region.

_{q}’s minimum. With strategy 1 or strategy 2, the operating point was always in the red area, i.e., the converter always delivered active power into the grid. This allowed the DC link voltage to stabilize quickly after fault clearing.

_{q}’s minimum and the right USEP. With strategy 1, the operating point could pass through the blue and red areas as it approached the SEP, i.e., a rapid oscillation of the active power from positive to negative. With strategy 2, the operating point was always in the red or light green area as it approached the SEP, i.e., the active power was always positive, or to a lesser negative value.

_{q}and d V

_{q}/dt to determine the area where the operating point was located, as shown in Table 1. This information could be used to adjust the current reference dynamically.

## 5. Experimental Verification

#### 5.1. Experimental Setup

_{pc}= 5, K

_{ic}= 300 1/s, DC control parameter K

_{pdc}= 0.1, K

_{idc}= 100 1/s, and phase-locked loop control parameters K

_{pPLL}= 0.32, K

_{iPLL}= 0.32 1/s. The upper limit of current amplitude was 1.2 p.u. The grid side voltage was 400 V, SCR was 5, and the X/R ratio was 7. The reactive current exit strategy 1 in Figure 6 was used to improve the recovery rate of output active power.

#### 5.2. Inject Reactive Current to Improve the Stability Performance of the PLL

_{q}. At the moment of fault clearing, the operating point lay outside of the domain of attraction, so the system diverged and V

_{q}oscillated continuously. The active power (blue curve) oscillated due to the loss of synchronization, causing the converter to inject power from the grid into the DC link, resulting in a peak in the DC link voltage. A converter will stop operation after a rapidly increasing DC link voltage, since the hardware protection is triggered.

_{q}converged to 0 after the oscillation, and the active power (blue curve) oscillations were properly damped and stabilized at 1.0 p.u. At the moment of fault clearing, the initial phase deviation and the reactive current reference led to a negative active power peak, which caused the DC link voltage to rise rapidly.

#### 5.3. Fast Reactive Current Exit Strategy

#### 5.4. Fast Reactive Current Exit Strategy but Not Fast System Stabilization

#### 5.5. Variable PLL Parameters Will Speed up System Stabilization

_{q}were very small, and then immediately stabilized at 0. Comparing Figure 13 with Figure 15, the active and reactive power curves (red and blue curves) are smooth, and do not show pronounced oscillations, and are able to decrease or increase at the desired rate. Thus, there were no obvious peaks in the DC link voltage either.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**Two different strategies for reactive current exit. (

**a**) q-axis current reference strategy, (

**b**) d-axis current reference strategy.

**Figure 7.**Heat map of the reactive current reference of strategy 1: ${I}_{\mathrm{d}}^{\ast}$ increases as ${I}_{\mathrm{q}}^{\ast}$ decreases. The trajectories of the two different post-fault initial points (A(t

_{0}) and B(t

_{0})) are indicated by the green and magenta arrows, respectively. Blue regions denote active power injected from the grid into the DC link of the converter, while red regions denote the regular power direction from the DC link into the grid.

**Figure 8.**Heat map of the reactive current reference of strategy 2: ${I}_{\mathrm{d}}^{\ast}$ increases after ${I}_{\mathrm{q}}^{\ast}$ decreases to 0.

**Figure 9.**(

**a**) V

_{q}versus phase deviation, (

**b**) heat map of 2nd stage of strategy 2, (

**c**) heat map of 1st stage of strategy 2, (

**d**) heat map of strategy 1.

**Figure 11.**No reactive current injection; the SEP was not present when the fault persisted, resulting in the operating point not being located within the DOA after the fault, and therefore the system being unstable. This led to a reversal of the actual active and reactive power flow (PQ curves are below the black dashed line), resulting in an uncontrolled increase in DC voltage, with the DC voltage curve far above the upper limit (black dashed line).

**Figure 12.**With reactive current injection, the SEP was located in area II during the fault. Then, the system was stable after oscillation. The active power flow temporarily reversed after the fault (P curve is below the black dashed line), resulting in the DC voltage temporarily exceeding the upper limit (black dashed line).

**Figure 13.**With reactive current injection, a reduction rate of ${I}_{\mathrm{q}}^{\ast}$ = 5 p.u./s, the active power reversed for a long time due to the slower reactive current exit rate (P curve below the black dashed line), eventually causing the DC voltage to exceed the upper limit (black dashed line).

**Figure 14.**With reactive current injection, a reduction rate of ${I}_{\mathrm{q}}^{\ast}$ = 80 p.u./s, the faster reactive current exit rate caused a short-term reversal of active power (P curve below the black dashed line). Therefore, the DC voltage quickly returned to its nominal value after a fault.

**Figure 15.**With reactive current injection, variable PLL parameters, a reduction rate of ${I}_{\mathrm{q}}^{\ast}$ = 5 p.u./s.

Area | V_{q} | dV_{q}/dt | ${\mathit{I}}_{\mathbf{q}}^{\mathbf{*}}$ | ${\mathit{I}}_{\mathbf{d}}^{\mathbf{*}}$ |
---|---|---|---|---|

I | ≥0 | ≥0 | Rapidly decreasing | Increase after V_{q} ≈ 0 |

II | ≥0 | <0 | Rapidly decreasing | ${I}_{\mathrm{d}}^{\ast 2}={\left|{I}_{\mathrm{max}}\right|}^{2}-{I}_{\mathrm{q}}^{\ast 2}$ |

III | <0 | ≥0 | Rapidly decreasing | ${I}_{\mathrm{d}}^{\ast 2}={\left|{I}_{\mathrm{max}}\right|}^{2}-{I}_{\mathrm{q}}^{\ast 2}$ |

IV | <0 | <0 | Rapidly decreasing | Increase after V_{q} ≈ 0 |

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**MDPI and ACS Style**

Zhang, Z.; Schuerhuber, R. Impact of Reactive Current and Phase-Locked Loop on Converters in Grid Faults. *Energies* **2023**, *16*, 3122.
https://doi.org/10.3390/en16073122

**AMA Style**

Zhang Z, Schuerhuber R. Impact of Reactive Current and Phase-Locked Loop on Converters in Grid Faults. *Energies*. 2023; 16(7):3122.
https://doi.org/10.3390/en16073122

**Chicago/Turabian Style**

Zhang, Ziqian, and Robert Schuerhuber. 2023. "Impact of Reactive Current and Phase-Locked Loop on Converters in Grid Faults" *Energies* 16, no. 7: 3122.
https://doi.org/10.3390/en16073122