# Microsecond Enhanced Indirect Model Predictive Control for Dynamic Power Management in MMC Units

^{*}

## Abstract

**:**

## 1. Introduction

## 2. MMC Modeling

#### 2.1. Mathematical Model of the MMC

#### 2.2. RSCAD Model of the MMC

#### 2.3. Comparison between Mathematical and RSCAD Model of the MMC

#### 2.3.1. Small Signal Analysis

#### Active Power Step Change

#### Reactive Power Step Change

#### Active Power Reversal

#### 2.3.2. Model Error Analysis

## 3. Indirect Implicit MPC with Laguerre’s Function

#### 3.1. Discrete Mathematical Model of MMC

#### 3.2. MPC Definition

Algorithm 1: MPC using Laguerre’s function |

Initialization: Augmented model (${A}_{m},{B}_{m},{C}_{m}\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}{D}_{m}$), ${A}_{l}$, N, a, A, ${N}_{p}$ and b.Step 1: Measure $\mathsf{\Delta}\overrightarrow{x}$ at k-th instance.Step 2: Compute optimal $\overrightarrow{\eta}$ by minimising the quadratic cost function (11a) using Hildreth’s Quadratic programming procedure considering the constraints.Step 3: Compute $\nabla \overrightarrow{u}\left(k\right)$ using with $\overrightarrow{\eta}$ from step 2 to calculate equations (10). Step 4: Using the receding horizon principle, apply $\nabla \overrightarrow{u}\left(k\right)$ corresponding to kth instance and neglecting the inputs at other sampling instances. Step 5: Go to step 1. |

#### 3.3. Indirect Implicit MPC Simulation Results in Matlab

#### 3.3.1. Constraint Satisfaction Problem

#### Rate Constraint

#### Amplitude Constraint

#### 3.3.2. Sensitivity Analysis

#### Sample Time (${T}_{s}$)

#### Weighting, Predictive Horizon and Laguerre’s Parameter

#### 3.4. Indirect Implicit MPC Simulations in RSCAD

#### 3.4.1. Simulations of Disturbance in Active and Reactive Power

#### 3.4.2. Constrained Satisfaction Problem in RSCAD

## 4. Comparison of PI and MPC Controls for the MMC in RTDS

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**An arm representation of MMC in the RSCAD [26].

**Figure 4.**Implemented controlling loops for: (

**a**) active and reactive power control; (

**b**) output current control; and (

**c**) circulating current control.

**Figure 5.**Comparison of: (

**a**) active power; (

**b**) reactive power; (

**c**) output current ${i}_{d}^{\mathsf{\Delta}}$; and (

**d**) output current ${i}_{q}^{\mathsf{\Delta}}$; in MATLAB (blue) and RSCAD (red) during step change of the active power from 1 p.u. to 0.3 p.u. at the time instance $t=0.6\phantom{\rule{4.pt}{0ex}}\mathrm{s}$.

**Figure 6.**Comparison of (

**a**) circulating current (${i}_{d}^{\mathsf{\Sigma}}$ and ${i}_{q}^{\mathsf{\Sigma}}$); (

**b**) Σ components of capacitor voltage in dqz reference frame; in MATLAB (blue) and RSCAD (red) during step change in active power from 1 p.u. to 0.3 p.u. at the time instance $t=0.6\phantom{\rule{4.pt}{0ex}}\mathrm{s}$.

**Figure 7.**Comparison of the zero sequence circulating current ${i}_{z}^{\mathsf{\Sigma}}$ MATLAB (blue) and RSCAD (red), during step change in active power from 1 p.u. to 0.3 p.u. at the time instance $t=0.6\phantom{\rule{4.pt}{0ex}}\mathrm{s}$.

**Figure 8.**Comparison of: (

**a**) active power; (

**b**) reactive power; (

**c**) ${i}_{d}^{\mathsf{\Delta}}$ current; and (

**d**) ${i}_{q}^{\mathsf{\Delta}}$ current; in MATLAB (blue) and RSCAD (red) during step change in reactive power from 0 p.u. to 0.25 p.u. at the time instance $t=0.6\phantom{\rule{4.pt}{0ex}}\mathrm{s}$.

**Figure 9.**Comparison of: (

**a**) circulating currents ${i}_{d}^{\mathsf{\Sigma}}$ and ${i}_{q}^{\mathsf{\Sigma}}$; (

**b**) $\mathsf{\Sigma}$ components of capacitor voltage in the $dqz$ reference frame; in MATLAB (blue) and RSCAD (red), during step change in Reactive power from 0 p.u. to 0.25 p.u. at the time instance $t=0.6\phantom{\rule{4.pt}{0ex}}\mathrm{s}$.

**Figure 10.**Comparison of: (

**a**) active power; (

**b**) reactive power; (

**c**) output current ${i}_{d}^{\mathsf{\Delta}}$; and (

**d**) output current ${i}_{q}^{\mathsf{\Delta}}$; in MATLAB (blue) and RSCAD (red) during active power reversal from 1 p.u. to -1 p.u. at the time instance $t=0.6\phantom{\rule{4.pt}{0ex}}\mathrm{s}$.

**Figure 11.**Comparison of: (

**a**) circulating currents ${i}_{d}^{\mathsf{\Sigma}}$ and ${i}_{q}^{\mathsf{\Sigma}}$; (

**b**) $\mathsf{\Sigma}$ components of arm capacitor voltage in the $dqz$ reference frame; in MATLAB (blue) and RSCAD (red) during active power reversal from 1 p.u. to -1 p.u. at the time instance $t=0.6\phantom{\rule{4.pt}{0ex}}\mathrm{s}$.

**Figure 12.**Comparison of zero sequence circulating current ${i}_{z}^{\mathsf{\Sigma}}$ in MATLAB (blue) and RSCAD (red), during active power reversal from 1 p.u. to -1 p.u. at the time instance $t=0.6\phantom{\rule{4.pt}{0ex}}\mathrm{s}$.

**Figure 13.**Case- study with unconstrained scenario: (

**a**) ${i}_{d}^{\mathsf{\Delta}}$ current; (

**b**) ${i}_{q}^{\mathsf{\Delta}}$ current; (

**c**) ${u}_{d}$ voltage; and (

**d**) ${u}_{q}$ voltage.

**Figure 14.**Case- studies with rate constraint, line 1: $|\mathsf{\Delta}{u}_{d}|=|\mathsf{\Delta}{u}_{q}|=30$, line 2: $|\mathsf{\Delta}{u}_{d}|=|\mathsf{\Delta}{u}_{q}|=40$, line 3: $|\mathsf{\Delta}{u}_{d}|=|\mathsf{\Delta}{u}_{q}|=50$, and line 4: unconstrained.

**Figure 15.**Case-studies with amplitude constraint: line 1: $|{u}_{d}\phantom{\rule{4pt}{0ex}}|=|{u}_{q}|\phantom{\rule{4pt}{0ex}}=30$, line 2: $|{u}_{d}|\phantom{\rule{4pt}{0ex}}=|{u}_{q}|\phantom{\rule{4pt}{0ex}}=60$, line 3: $|{u}_{d}|\phantom{\rule{4pt}{0ex}}=|{u}_{q}|\phantom{\rule{4pt}{0ex}}=80$ and line 4: Unconstrained.

**Figure 17.**Effect of R on the dynamic response and stability with: (

**a**) ${T}_{s}=2\times {10}^{-3}$; (

**b**) ${T}_{s}=4\times {10}^{-3}$; (

**c**) ${T}_{s}=6\times {10}^{-3}$; and (

**d**) ${T}_{s}=10\times {10}^{-3}$.

**Figure 18.**Effect of: (

**a**) ${N}_{p}$ with $a=[0.237,\phantom{\rule{0.166667em}{0ex}}0.237]$, $N=[4,\phantom{\rule{0.166667em}{0ex}}4]$; (

**b**) a with ${N}_{p}=4$, $N=[4,\phantom{\rule{0.166667em}{0ex}}4]$; and (

**c**) N with $a=[0.237,\phantom{\rule{0.166667em}{0ex}}0.237]$, ${N}_{p}=4$ on the dynamic response and stability.

**Figure 19.**Change at the time instance $t=0.5\phantom{\rule{4.pt}{0ex}}\mathrm{s}$ in: (

**a**) active power reference from 0 p.u. to 0.5 p.u.; (

**b**) reactive power reference from 0 p.u. to 0.2 p.u.

**Figure 20.**Change at the time instance $t=0.5\phantom{\rule{4.pt}{0ex}}\mathrm{s}$ in: (

**a**) active power reference from −0.5 p.u. to 0.5 p.u.; (

**b**) reactive power reference from −0.2 p.u. to 0.2 p.u.

**Figure 21.**Effect of constraints on (

**a**) ${i}_{d}^{\mathsf{\Delta}}$; (

**b**) ${v}_{Md}-{v}_{d}^{\mathsf{\Delta}}$; (

**c**) rate of change of ${v}_{Md}-{v}_{d}^{\mathsf{\Delta}}$, where line 1: Unconstrained, line 2: Constrained.

**Figure 22.**Comparison of PI and MPC controller during: (

**a**) active power reversal; (

**b**) circulating current suppression; and (

**c**) AC voltage support.

**Table 1.**Parameters of the test system from Figure 3.

Parameter | Symbol | Value | Unit |
---|---|---|---|

Base power | P | 800 | MVA |

Arm resistance | ${R}_{arm}$ | 0 | $\mathsf{\Omega}$ |

Equivalent transformer inductance | ${L}_{eq}^{ac}$ | 35 | mH |

Equivalent transformer resistance | ${R}_{eq}^{ac}$ | 0.363 | $\mathsf{\Omega}$ |

Submodule capacitance | C | 10 | mF |

Number of submodules | ${N}_{SM}$ | 400 | - |

Fundamental grid frequency | f | 50 | Hz |

Rated pole to pole DC voltage | ${V}_{dc}$ | 400 | kV |

Rated line to line primary voltage | ${V}_{ACprim}$ | 380 | kV |

Rated line to line secondary voltage | ${V}_{ACsec}$ | 220 | kV |

**Table 2.**Proportional and Integral gain parameter of the classical CCSC, OCC, and power PI controllers from Figure 4.

Parameter | Value |
---|---|

Active and reactive power controller proportional gain | 0.08 [pu] |

Active and reactive power controller integral gain | 4 [p.u.] |

Output current controller proportional gain | 0.8 [p.u.] |

Output current controller integral gain | 80 [p.u.] |

Circulating current suppression controller proportional gain | 0.8 [p.u.] |

Circulating current suppression controller integral gain | 80 [p.u.] |

**Table 3.**MAE in the mathematical model during the step change in Active Power from 1 p.u. to 0.3 p.u. at $t=0.6\phantom{\rule{4.pt}{0ex}}\mathrm{s}$.

Signals | Error in Percentage | ||
---|---|---|---|

Pre-Disturbance | During-Disturbance | Post-Disturbance | |

${P}_{MATLAB}$ | 0.05 | 0.48 | 0.02 |

${Q}_{MATLAB}$ | 0.04 | 0.91 | 0.04 |

${i}_{z}^{\mathsf{\Sigma}}$ | 1.37 | 0.92 | 0.43 |

${i}_{d}^{\mathsf{\Sigma}}$ | 1.05 | 1.78 | 0.42 |

${i}_{q}^{\mathsf{\Sigma}}$ | 1.00 | 1.55 | 0.39 |

${v}_{Cz}^{\mathsf{\Sigma}}$ | 0.15 | 0.74 | 0.57 |

**Table 4.**MAE in the mathematical model during the step change in Reactive Power from 0 p.u. to 0.24 p.u. at $t=0.6\phantom{\rule{4.pt}{0ex}}\mathrm{s}$.

Signals | Error in Percentage | ||
---|---|---|---|

Pre-Disturbance | During-Disturbance | Post-Disturbance | |

${P}_{MATLAB}$ | 0.04 | 0.16 | 0.03 |

${Q}_{MATLAB}$ | 0.03 | 0.22 | 0.02 |

${i}_{z}^{\mathsf{\Sigma}}$ | 1.50 | 1.52 | 1.49 |

${i}_{d}^{\mathsf{\Sigma}}$ | 0.98 | 1.85 | 0.95 |

${i}_{q}^{\mathsf{\Sigma}}$ | 0.98 | 0.79 | 0.97 |

${v}_{Cz}^{\mathsf{\Sigma}}$ | 0.25 | 0.32 | 0.24 |

**Table 5.**MAE in the mathematical model during active power reversal at $t=0.6\phantom{\rule{4.pt}{0ex}}\mathrm{s}$.

Signals | Error in Percentage | ||
---|---|---|---|

Pre-Disturbance | During-Disturbance | Post-Disturbance | |

${P}_{MATLAB}$ | 0.05 | 1.28 | 0.05 |

${Q}_{MATLAB}$ | 0.06 | 2.55 | 0.11 |

${i}_{z}^{\mathsf{\Sigma}}$ | 1.35 | 2.33 | 1.79 |

${i}_{d}^{\mathsf{\Sigma}}$ | 1.02 | 3.74 | 1.34 |

${i}_{q}^{\mathsf{\Sigma}}$ | 0.99 | 3.96 | 1.25 |

${v}_{Cz}^{\mathsf{\Sigma}}$ | 0.52 | 1.41 | 0.22 |

Name | Symbol | Value |
---|---|---|

Arm inductance | ${L}_{arm}$ | 0.15 [p.u.] |

Arm resistance | ${R}_{arm}$ | 0.0015 [p.u.] |

AC filter inductance | ${L}_{r}$ | 0.12 [p.u.] |

AC filter resistance | ${R}_{r}$ | 0.003 [p.u.] |

Sample time | ${T}_{s}$ | 2 [ms] |

Cases | Variables | Value (Sampling Instance) | Value (Sampling Instance) |
---|---|---|---|

Small Disturbance | Active power | 0.5 p.u. (10) | 1 p.u. (30) |

Reactive power | 0.2 p.u. (20) | 1 p.u. (30) | |

Large Disturbance | Active power reversal | −1 p.u. (40) | 1 p.u. (80) |

Reactive power reversal | −1 p.u. (50) | 0.5 p.u. (80) |

MPC | DLQR | Relative Error |
---|---|---|

$0.121\pm 0.223j$ | $0.121\pm 0.223j$ | $5.28\times {10}^{-4}-2.46\times {10}^{-4}j$ |

$0.121\pm 0.223j$ | $0.121\pm 0.223j$ | $-3.63\times {10}^{-4}+2.36\times {10}^{-4}j$ |

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**MDPI and ACS Style**

Shetgaonkar, A.; Lekić, A.; Rueda Torres, J.L.; Palensky, P.
Microsecond Enhanced Indirect Model Predictive Control for Dynamic Power Management in MMC Units. *Energies* **2021**, *14*, 3318.
https://doi.org/10.3390/en14113318

**AMA Style**

Shetgaonkar A, Lekić A, Rueda Torres JL, Palensky P.
Microsecond Enhanced Indirect Model Predictive Control for Dynamic Power Management in MMC Units. *Energies*. 2021; 14(11):3318.
https://doi.org/10.3390/en14113318

**Chicago/Turabian Style**

Shetgaonkar, Ajay, Aleksandra Lekić, José Luis Rueda Torres, and Peter Palensky.
2021. "Microsecond Enhanced Indirect Model Predictive Control for Dynamic Power Management in MMC Units" *Energies* 14, no. 11: 3318.
https://doi.org/10.3390/en14113318