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# Operating principles

The control of single-phase systems often significantly differs from that of three-phase systems. Notably, numerous well-known concepts such as the separation of direct and quadrature axes - linked to active and reactive power flows - cannot be easily derived from coordinate transformations, simply because only one phase is available! This difference often justifies a preference for specific control techniques such as the use of Proportional Resonant (PR) controllers (TN110), rather than conventional vector control (TN106) in a rotating (also called synchronous) reference frame.

That said, vector control techniques can also be used in single-phase applications, provided that a second axis β is emulated in order to support all vector computations (even though β-related references and quantities may be disregarded at the end of the control process). The benefit of such an approach is that, since coordinate transformations can be used, active and reactive power flows can be manipulated as usual (i.e. within a rotating reference frame).

## Emulating the β axis

The most obvious technique for emulating the β axis is to delay the available α axis by a quarter period. This results in a simple, yet robust implementation, but which unavoidably impacts the overall control chain due to the said delay. An alternative approach was first proposed by [1], essentially consisting of using an estimator for emulating the current that would flow on the β axis. In practice, this estimator can be easily derived from the plant model, which is needed anyway for the purposes of the control design.

This approach can be easily understood by observing the following figure, taken from [1], which highlights:

• The Physical Axis (α axis), which can be modeled as a first-order system where the current can be physically measured as $//$.

• The Emulated Axis (β axis), which can re-use the same model for estimating the current that would flow into the system, such as $//$.

where $//$ is the utility grid voltage and $//$ is the converter voltage produced between the two phase-legs. In practice, $//$ and $//$ are often directly made available by the PLL block, which is implemented using a Second-Order Generalized Integrator (SOGI) TN104.

[1] B. Bahrani, A. Rufer, S. Kenzelmann and L. Lopes, “Vector control of single-phase voltage-source
converters based on fictive-axis emulation,” in IEEE Trans. Ind. Appl., Vol. 47, N° 2, Apr. 2011.

# B-Box / B-Board implementation

The provided Simulink file implements the grid current control as shown below.

The implementation of the FAE estimator itself is:

With $//$ and $//$

## C/C++ code

The imperix IDE provides numerous pre-written and pre-optimized functions. Dedicated routines for fictive axis emulation exist as such within the controllers.h/.cpp files.

As for controllers, FAE-related routines are based on:

• A pseudo-object FAEparameters, which contains pre-computed parameters as well as state variables.

• A configuration function, meant to be called during UserInit(), named ConfigFAE().

• A run-time function, meant to be called during the user-level ISR, such as UserInterrupt(), named RunFAE().

The necessary parameters are documented within the controller.h header file. They are namely:

• L and R, the parameter values of the grid inductor (plant).

• tsample, the sampling (interrupt) period.

The source code of the related routines is given below:

void ConfigFAE(FAEParameters* me, float R, float L, float tsample){
// Precompute the parameters offline:
me->a = tsample/(L + R * tsample);
me->b = L/(L + R * tsample);

// Initialize the state quantities:
me->state = 0.0;
}

float RunFAE(FAEParameters *me, float delta){
// Apply the first-order transfer function:
me->state = me->a * delta + me->b * me->state;

return me->state;
}

The code below gives a use case example of both routines:

#include "../API/controllers.h"             // Discrete-time controllers
FAEParameters FAE;                          // Pseudo-object containing the FAE parameters

float R = 0.015;                            // Grid inductor ESR (Ohm)
float L = 0.0025;                           // Grid inductor value (H)

tUserSafe UserInit(void){
ConfigFAE(&FAE, R, L, SAMPLING_PERIOD);
}

tUserSafe UserInterrupt(void){
// ... some code
Ig.imaginary = RunFAE(&FAE, Eg.imaginary - Ug.imaginary);
// ... some code
}
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