# A New Real Time Lyapunov Based Controller for Power Quality Improvement in Unified Power Flow Controllers Using Direct Matrix Converters

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## Abstract

**:**

## 1. Introduction

## 2. Modeling of the UPFC Power System

_{S}and V

_{R}are the sending-end and receiving-end voltages, respectively, of G

_{S}and G

_{R}generators feeding load Z

_{load}. The DMC is connected to the transmission line 2 through shunt and series coupling transformers, respectively T

_{1}and T

_{2}. Also, as shown in Figure 1, Z

_{L}

_{1}and Z

_{L}

_{2}are the impedances of the transmission lines 1 and 2, respectively, represented as a series inductance and resistance.

#### 2.1. UPFC Power System Dynamic Model

_{a}, T

_{b}, T

_{c}), a three-phase series output transformer (T

_{A}, T

_{B}, T

_{C}) and a three-phase DMC are also included.

_{S}, in dq coordinates [21].

_{Sd}, V

_{Sq}are the mains voltage components, and I

_{d}, I

_{q}the line currents, both in dq coordinates (Figure 1).

_{Sq}) is equal to zero, P and Q will be given by Equation (2).

_{Sd}.

_{2}and R

_{2}) are estimated by L

_{2}= X

_{2}/ω, R

_{2}= R

_{L}

_{2}+ R

_{L}

_{1}||R

_{Zload}.

_{Cd}and V

_{Cq}, being V

_{R}

_{0d}and V

_{R}

_{0q}system disturbances.

#### 2.2. Direct Matrix Converter Model

_{kj}. These switches have turn-on and turn-off capability, allowing the connection of each one of three output phases directly to a given three-phase input voltage.

_{kj}(k, j ∈ {1, 2, 3}) can assume two possible states: “S

_{kj}= 1” (switch closed) or “S

_{kj}= 0” (switch opened), being the nine DMC switches represented as a 3 × 3 matrix (7). Taking into account circuit topology restrictions, no input phase short-circuits and no open output phases, there are only 27 possible switching combinations [14].

**S**, being ${\left[\begin{array}{ccc}{v}_{A}& {v}_{B}& {v}_{C}\end{array}\right]}^{T}=$ $S$${\left[\begin{array}{ccc}{v}_{a}& {v}_{b}& {v}_{c}\end{array}\right]}^{T}$. Then, it is necessary to represent the output voltages and the input currents yielding 27 switching patterns, as time variant vectors in αβ coordinates [14]. The relationship between input phase currents and output phase currents depends on the transposition of matrix

**S**: ${\left[\begin{array}{ccc}{i}_{a}& {i}_{b}& {i}_{c}\end{array}\right]}^{T}=$ ${\mathit{S}}^{T}$${\left[\begin{array}{ccc}{i}_{A}& {i}_{B}& {i}_{C}\end{array}\right]}^{T}$.

## 3. DMC-UPFC Lyapunov-Based Control

_{X}= e

_{X}

^{2}/2, where e

_{X}is the tracking error of the X variable, e

_{X}= X

_{Ref}− X. To ensure a globally asymptotically-stable control solution from Lyapunov’s direct method of stability [22], the time derivative ${\dot{V}}_{X}$ of the positive-definite Lyapunov function V

_{X}must be globally negative-definite ${\dot{V}}_{X}<0$, or ${e}_{X}{\dot{e}}_{X}<0$. This can be achieved by imposing an error dynamic such as:

_{X}is a positive definite constant, resulting in ${e}_{X}{\dot{e}}_{X}=-{k}_{X}{e}_{X}^{2}<0$. Equation (8) ensures asymptotic stability, as the e

_{X}error tends to zero with time constant 1/k

_{X}, and robustness as the error dynamic in (8) is independent of the system parameters.

#### 3.1. Active and Reactive Power Control

_{Ref}, Q

_{Ref}, so that e

_{X}= [e

_{P}, e

_{Q}]

^{T}, the tracking errors (or deviations) being e

_{P}= P

_{Ref}− P and e

_{Q}= Q

_{Ref}− Q:

_{P}= e

_{P}

^{2}/2, V

_{Q}= e

_{Q}

^{2}/2. Therefore, the controller must enforce:

_{P}, k

_{Q}are positive definite constants. Replacing the e

_{P}, e

_{Q}time derivates using (6), Equation (12) is received:

_{Cd}

_{Ref}and V

_{Cq}

_{Ref}the DMC controller must impose the output voltages V

_{Cd}

_{Ref}and V

_{Cq}

_{Ref}.

_{Cd}

_{Ref}, V

_{Cq}

_{Ref}that must be injected in the line by the series transformer of the space-vector PWM controlled DMC.

#### 3.2. Input Reactive Power Control

_{id}, V

_{iq}, i

_{id}, i

_{iq}represent input voltages and input currents in dq components (at the shunt transformer secondary), and V

_{dM}, V

_{qM}, i

_{dM}, i

_{qM}are DMC voltages and input currents in dq components, respectively,

_{i}= Q

_{i}

_{Ref}= 0. The control input is ${i}_{qM}$ and the output is ${i}_{iq}$, as ${Q}_{i}=-{V}_{id}{i}_{iq}$ at constant ${V}_{id}$. Defining the tracking error ${e}_{Qi}={Q}_{i\mathrm{Ref}}-{Q}_{i}=-{Q}_{i}$, to guarantee Lyapunov asymptotic stability in this 2nd order system, the controller must enforce an exponentially stable error dynamics tending to zero such as ${\ddot{e}}_{Qi}+{k}_{1}{\dot{e}}_{Qi}+{k}_{2}{e}_{Qi}=0$, where k

_{1}and k

_{2}are positive definite constants. Therefore:

_{qM}

_{Ref}is the value of i

_{qM}(control input) that satisfies (15). Solving (17) for the reference value i

_{qM}

_{Ref}of the q component of the matrix current i

_{qM}, Equation (18) is obtained:

_{Cd}

_{Ref}, V

_{Cq}

_{Ref}and i

_{qM}

_{Ref}, as discussed in Section 2.2. The optimum space-vector is computed in the next section by minimizing a cost functional of the reference voltages and current errors.

#### 3.3. Optimum Space-Vector

_{s}, the real-time modulation control method first calculates the reference components, V

_{Cd}

_{Ref}(t

_{s}), V

_{Cq}

_{Ref}(t

_{s}) and i

_{qM}

_{Ref}(t

_{s}) given in (13) and (18). Then, the weighted cost functional (19) is evaluated for each one of the 27 matrix vectors to select the one presenting the minimum value of the cost functional.

## 4. Implementation of DMC-UPFC Controller

_{Cd}

_{Ref}(t

_{s}), V

_{Cq}

_{Ref}(t

_{s}) and i

_{qM}

_{Ref}(t

_{s}) once, while the cost functional, C(t

_{s}), is evaluated 27 times to find the vector with components V

_{Cd}(t

_{s}), V

_{Cq}(t

_{s}) and i

_{qM}(t

_{s}) that minimizes the weighted cost functional. This selects the space-vector which minimizes the cost functional, and is applied to the DMC-UPFC in the following sampling time (t

_{s}+ ΔT).

## 5. Simulation and Experimental Evaluation

_{1}, T

_{2}(2 kVA transformers with voltage ratio 220/115 V and 66.5/66.5 V, respectively). Voltage sensors were used to measure the voltages (LEM LV 25-P), while current sensors were used to measure the DMC currents (LEM LA25NP).

_{s}) of 18 μs. This sampling time is suited to switching the DMC at frequencies around 5 kHz, but is not enough to guarantee safe commutation between the DMC bidirectional switches, as this requires much faster changing signals. Therefore, to ensure safe commutation, the four-step output current commutation strategy was used [18], implemented in a field-programmable gate array (FPGA) using a Xilinx (Virtex 5) board, as represented in Figure 4.

_{L}

_{1}= 12 mH, L

_{L}

_{2}= 15 mH and series resistances R

_{L}

_{1}= R

_{L}

_{2}= 0.2 Ω, respectively.

_{Ref}and Q

_{Ref}references (ΔP

_{Ref}and ΔQ

_{Ref}, respectively). Figure 6a,b show, respectively, simulation and experimental results for the proposed controller, considering P and Q power step responses to ΔP

_{Ref}= +0.4 pu and ΔQ

_{Ref}= +0.2 pu, with initial reference values P

_{Ref}= 0.4 pu, Q

_{Ref}= 0.2 pu.

_{Ref}= +0.4 pu and ΔQ

_{Ref}= −0.2 pu, when initial reference values are P

_{Ref}= 0.4 pu, Q

_{Ref}= 0.2 pu. These results show no cross-coupling, steady-state errors below ripple errors, and fast response times for different changes of power references. Line currents are almost sinusoidal with small ripple content.

_{Ref}= 0.4 pu, Q

_{Ref}= 0.2 pu. Line currents are almost sinusoidal with small ripple content, while matrix currents shows some ripple content as usual in DMC [15].

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 5.**Flowchart of the control method proposed to compute V

_{Cd}

_{Ref}, V

_{Cq}

_{Ref}(13) and i

_{qM}

_{Ref}(18).

**Figure 6.**Active and reactive series power response for P and Q steps (first ΔP

_{Ref}= +0.4 pu and after ∆Q

_{Ref}= +0.2 pu). (

**a**) Simulation results; (

**b**) Experimental results (Ch1 = Ch2 = 0.5 pu/div).

**Figure 7.**Active and reactive series power response for P and Q steps (first ΔQ

_{Ref}= +0.2 pu and after ΔP

_{Ref}= +0.4 pu). (

**a**) Simulation results; (

**b**) Experimental results (Ch1 = Ch2 = 0.5 pu/div).

**Figure 8.**Active and reactive power response and line currents for a P and Q step ΔP

_{Ref}= +0.4 pu and ΔQ

_{Ref}= +0.2 pu). (

**a**) Simulation Results; (

**b**) Experimental results, Ch1 = Ch2 = 0.5 pu/div (P Series and Q Series) and Ch3 = Ch4 = 1.0 pu/div (iB and iC).

**Figure 9.**Line currents (iB, iC) and input matrix converter currents (ib, ic) for ΔP

_{Ref}= +0.4 pu and ΔQ

_{Ref}= +0.2 pu: (

**a**) Simulation results; (

**b**) Experimental results (Ch1 = Ch2 = Ch3 = Ch4 =1.0 pu/div).

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**MDPI and ACS Style**

Monteiro, J.; Pinto, S.; Delgado Martin, A.; Silva, J.F.
A New Real Time Lyapunov Based Controller for Power Quality Improvement in Unified Power Flow Controllers Using Direct Matrix Converters. *Energies* **2017**, *10*, 779.
https://doi.org/10.3390/en10060779

**AMA Style**

Monteiro J, Pinto S, Delgado Martin A, Silva JF.
A New Real Time Lyapunov Based Controller for Power Quality Improvement in Unified Power Flow Controllers Using Direct Matrix Converters. *Energies*. 2017; 10(6):779.
https://doi.org/10.3390/en10060779

**Chicago/Turabian Style**

Monteiro, Joaquim, Sónia Pinto, Aranzazu Delgado Martin, and José Fernando Silva.
2017. "A New Real Time Lyapunov Based Controller for Power Quality Improvement in Unified Power Flow Controllers Using Direct Matrix Converters" *Energies* 10, no. 6: 779.
https://doi.org/10.3390/en10060779