*3.1. Large RES Production Power Plants*

As represented in Figure 3, the large RES production power plants are based on multiple small–medium generators, usually grouped as a medium voltage (MV) cluster interconnected to the single HV POC. This topology is common in practice, regardless of RES source (PV, wind, hydro, etc.) and generators technology (static, synchronous, asynchronous, etc.).

**Figure 3.** The topology of a renewable energy sources (RES) power plant composed of a medium voltage (MV) network and a unique point of connection (POC) to the high voltage (HV) network.

By adopting the well-known power flow equations, the MV distribution grid can be described by Equation (1), when considering active/reactive power for the node *k*:

$$\begin{cases} P\_k = \sum\_{i=1}^n V\_k \cdot V\_{i\cdot} \mathbf{Y}\_{ki\cdot} \cos(\theta\_k - \theta\_i - \gamma\_{ki}) \\\ Q\_k = \sum\_{i=1}^n V\_k \cdot V\_{i\cdot} \mathbf{Y}\_{ki\cdot} \sin(\theta\_k - \theta\_i - \gamma\_{ki}) \end{cases} \tag{1}$$

where *Pk* and *Qk* are the active/reactive power at the node *k*, while *Vk* and *Vi* are the RMS voltage, respectively, at the node *k* and node *i*. The magnitude of admittance coefficients in branch *ik* is represented by *Yki*, whereas θ*<sup>k</sup>* and θ*<sup>i</sup>* are the voltage phase angle at node *k* and node *i*. Finally, γ*ki* is the admittance phase of branch *ik*, whereas *n* the total number of nodes constituting the analyzed grid. By considering the per-unit notation (i.e., rated values are the basis) and linearizing (1) at a given operating point, the Jacobian matrix is defined as in Equation (2):

$$
\begin{bmatrix} \left[\Delta p\right] \\\\ \left[\Delta q\right] \end{bmatrix} = \begin{bmatrix} \left[\frac{dp}{d\theta}\right] & \left| \quad \left[\frac{dp}{d\upsilon}\right] \right| \\\\ \left[\frac{dq}{d\theta}\right] & \left[\frac{dq}{d\upsilon}\right] \end{bmatrix} \begin{bmatrix} \left[\Delta\theta\right] \\\\ \left[\Delta\upsilon\right] \end{bmatrix} \tag{2}
$$

By taking into account a behavior around the operating point, the partial derivatives matrices *dp dv* , *dp d*ϑ , *dq dv* , *dq d*ϑ are the link between active/reactive power and magnitude/phase angle of voltage at buses. As a matter of fact, such matrix coefficients embed the information about the characteristic parameters of the network lines. For the purposes of voltage control, the last equation is to be particularized as in Equation (3), neglecting the active power variations ([Δ*p*] = 0) and assuming the only reactive power sources as actuators [57]:

$$
\begin{bmatrix} \Delta q \end{bmatrix} = \left[ \left[ \frac{dq}{dv} \right] - \left[ \frac{dq}{d\theta} \right] \cdot \left[ \frac{dp}{d\theta} \right]^{-1} \cdot \left[ \frac{dp}{dv} \right] \right] [\Delta v] \tag{3}
$$

By setting a system-operating point, the power flow problem is solved, thus deducing the following equations for the linearized system:

$$\begin{bmatrix} \left[ \Delta q \right] \end{bmatrix} = \begin{bmatrix} A \end{bmatrix} \cdot \begin{bmatrix} \Delta v \end{bmatrix} \tag{4}$$

where [Δ*q*] and [Δ*v*] are the vectors (*n*,1) of reactive power/voltage variations, whilst the (*n*,*n*) matrix [*A*] models the electric coupling between reactive powers and voltage magnitudes. Hence, the generators are electrically coupled according to the coefficients (5):

$$
\left[\frac{dq}{dv}\right] - \left[\frac{dq}{d\theta}\right] \left[\frac{dp}{d\theta}\right]^{-1} \cdot \left[\frac{dp}{dv}\right] \tag{5}
$$

In other words, a voltage variation at every network node causes a reactive power variation in all the *n* nodes, according to the matrix [*A*] coefficients, as expressed in Equation (6):

$$
\begin{bmatrix}
\Delta q\_1\\\Delta q\_i\\\Delta q\_n
\end{bmatrix} = \begin{bmatrix}
a\_{11} & a\_{1i} & a\_{1n} \\
a\_{i1} & a\_{ii} & a\_{in} \\
a\_{n1} & a\_{ni} & a\_{nn}
\end{bmatrix} \cdot \begin{bmatrix}
\Delta v\_1\\\Delta v\_i\\\Delta v\_n
\end{bmatrix} \tag{6}
$$

It is remarkable to notice that [*A*] is considered full rank in the most practical applications, while the discussion of idiosyncratic cases (i.e., [*A*] singular) is beyond the study aims. Finally, Equation (7) is capable of modeling the voltage at the POC, where [*S*] is the vector (1,*n*) of the sensitivity coefficients *dv*/*dq* for combining the POC to the network nodes:

$$
\Delta v\_b = [S] \cdot [\Delta q] = \sum\_{i=1}^n s\_i \ast \Delta q\_i \tag{7}
$$

Depending on the relative coefficients *dv*/*dq*, the reactive power variation achieved at different grid nodes thus produces at the POC the voltage variation as in Equation (7). The matrix [*A*] (*n*,*n*) and the vector [*S*] (1,*n*) can also be determined by a numerical sensitivity analysis. Indeed, once all the network parameters are established and the reactive power of each DG plant is increased, the consequent voltage variation can be calculated as already discussed in [58]. On the other hand, by starting from the inverse of the electric coupling matrix, the dynamic decoupling matrix is found as in Equation (8). Such a matrix is then composed by the coefficients *dv*/*dq*; thus its definition is then given by Equation (9):

$$\left[\left[DD\right]\right] = \left[A\right]^{-1} \tag{8}$$

$$\begin{bmatrix} \Delta v \end{bmatrix} = \begin{bmatrix} DD \end{bmatrix} \cdot \begin{bmatrix} \Delta q \end{bmatrix} \tag{9}$$

To finally calculate the constants values for the two PI controls, a traditional synthesis is sufficient, once the cascade system is determined. By observing Figure 1, the latter is constituted by the capability matrix, the decoupling matrix, the reactive power regulators, and finally the AVRs or SFCs.

## *3.2. Traditional Power Plants*

The grid topology for a traditional power plant based on fossil fuel is offered in Figure 4, while Figure 5 shows the equivalent electrical model. By comparing the two topologies (i.e., Figure 3 RES versus Figure 4 traditional), the main difference is made manifest: the MV distribution grid in the RES power plant case. In the traditional power plant case, each generator is connected to the main busbar by the only generator transformer, which is characterized by a reactance *xti* [49]. This assumption is true even in the case of a generator directly connected to the main busbar, where the compound action provides *xti*, i.e., the equivalent reactance introduced by the control. Therefore, it is possible to categorize the traditional power plant as a subcase of the RES power plant case, where the internal distribution network is merely given by the reactance of generator transformers *xti*. In such a way, the electrical coupling of generators is only determined by the reactance of generator transformers *xti* and the equivalent reactance of upstream network *xcc*, as clarified in [49]. In this perspective, the network topology represented in Figure 3 and its mathematical model constitutes the general case, albeit it is introduced for the RES case. As a matter of fact, this representation can describe a production power plant either based on RES or on traditional fossil sources. Therefore, the application of control strategy on traditional power plant is also a particular case of the RES production power plant.

**Figure 4.** The typical topology of a traditional power plant: several generators parallel-connected to a busbar.

**Figure 5.** Equivalent electrical model.

For the traditional power plant case, the model can be attained by considering Figure 5. In such a case, the relation between generator voltage variations and reactive power variations is expressed with the algebraic relations (10):

$$\begin{cases} \Delta \mathbf{v}\_1 = \mathbf{x}\_{t1} \cdot \Delta q\_1 + \mathbf{x}\_{cc} \cdot \sum\_{i=1}^n \Delta q\_i \\\\ \vdots \\\\ \Delta \mathbf{v}\_i = \mathbf{x}\_{ti} \cdot \Delta q\_i + \mathbf{x}\_{cc} \cdot \sum\_{i=1}^n \Delta q\_i \\\\ \vdots \\\\ \Delta \mathbf{v}\_n = \mathbf{x}\_{ln} \cdot \Delta q\_n + \mathbf{x}\_{cc} \cdot \sum\_{i=1}^n \Delta q\_i \end{cases} \tag{10}$$

where the quantities are expressed in the per unit notation. Particularly, *xti* is the reactance of the *i*-th generator transformer, while *xcc* is the equivalent reactance of the upstream network. The symbol *vi* represents the voltage at the terminals of *i*-th generator and *qi* the reactive power of the *i*-th generator. Finally, *vb* is the voltage at POC to the transmission network. By expressing Equation (10) in matrix form, the important Equation (11) is determined. For a traditional electric plant, an additional important result is provided in Equation (12), where the elements of the dynamic decoupling matrix are clearly defined by the reactance *xti* and *xcc* [49].

$$\begin{bmatrix} \Delta v\_1 \\ \vdots \\ \Delta v\_i \\ \vdots \\ \Delta v\_n \end{bmatrix} = \begin{bmatrix} \mathbf{x}\_{t1} + \mathbf{x}\_{cc} & \mathbf{x}\_{cc} & \mathbf{x}\_{cc} \\ & \vdots \\ & \mathbf{x}\_{tc} & \mathbf{x}\_{tc} + \mathbf{x}\_{cc} & \mathbf{x}\_{cc} \\ & & \vdots \\ & & \vdots \\ & & \mathbf{x}\_{cc} & \mathbf{x}\_{tc} + \mathbf{x}\_{cc} \end{bmatrix} \begin{bmatrix} \Delta q\_1 \\ \vdots \\ \Delta q\_i \\ \vdots \\ \Delta q\_n \end{bmatrix} \tag{11}$$

$$DD\_{i,j} = \begin{cases} \begin{array}{ll} \mathbf{x\_{cc}} & \text{if} \quad i \neq j \\ \mathbf{x\_{li}} + \mathbf{x\_{cc}} & \text{if} \quad i = j \end{array} \tag{12}$$

## **4. Case Studies**

Currently, the proposed algorithm results are already implemented in Italian transmission systems (involving coal-fired and combined cycle gas power stations rated above 100 MVA) to achieve a coordinated production of reactive power [48]. In this section, the experimental data collected from the field for some traditional power plants equipped with a SART apparatus and the simulations for a large PV power plant are reported and compared. Three different power plant configurations are considered. In each of them, tests have been conducted applying at time *t* = 35 s a step in the *qliv* signal. Simulations have been carried out adopting the proposed control system, according to the mathematical model presented in the third chapter. The mathematical model of the control has been implemented in DOME (a Python based simulation tool) [59], together with the models of the networks for the three cases. Three additional files (Case A, Case B and Case C) are made available as supplementary materials. Datasets and experimental details are in these files.
