Intelligent integration strategies of wind farms in a super grid

Abstract

In this article, we have an interested implementing new intelligent strategies based on fuzzy controllers, for a better integration of renewable energy in the powerful electrical networks, commonly called super grids. A preliminary investigation has allowed us to learn more about the impact of this integration on algebraic variables of an electrical network on one hand and on the modes of operation of power plants on the other. Indeed, if necessary to ensure the strategic balance production and consumption, especially during peak hours, power operators shall ensure that the integration of one or more wind farms does not affect any aspect of the quality Energy distributed to subscribers. Perfect stability of large power grids at times of integration of renewable energy sources is ensured through the use of new strategies constituting a potential support which acts to overcome the hazards may hinder the operation of the components basic of these grids.

Keywords

Super grid wind farm thevenin equivalent generator fuzzy logic sensitivity

1 Introduction

Wind energy already represents a significant part in international energy supply concepts. More than ever, it is our duty to meet the challenges of a stable energy production, with a large proportion of wind energy. In the coming years, wind energy is expected to cover an increasing share of global needs. These prerequisites depend mainly among others capacity of wind power technology to be integrated into existing network topologies. To do this, an intelligent and flexible technology is required for the challenges facing network managers on wind turbines and wind farms with the power plant behavior.

The increasing demand for electric power in a global economy characterized by an alarming recession of global fossil fuel reserves, imperatively induced a collapse in energy generation balance.

The procedures for renewable energy survey made by the research teams around the world are only partially successful. Given these circumstances, it is imperative to explore new operational and technical management of these grids are generally subject to adverse events likely to affect the quality of the services requested by customers, even partially[10].

Various procedures for controlling the stability of super networks, with the guarantee of balance of energy balance were proposed. In this context, some works [1] were based on the medium-term dynamic simulation to see the impact of the action to be taken and selecting appropriate control systems. This is based on the application of a modal analysis of the voltage combined with the assessment of the sensitivity of different power plants with respect to the wind farms.

Indeed, controlling the behaviour of a conventional power plant could be affected by a dysfunction in one or more wind farms requires instantaneous acquisition of active and reactive powers of the power plants and the magnitude of this impact in the order to take an appropriate decision ensuring a strengthening of the stability of these plants through intelligent control strategy acting so as to restore the level of voltage and powers required in any node of the electrical network studied.

Other authors [2 –4] became interested in the study of dynamical models of the frequency to simulate the impact of the most important system parameters on the frequency response due to disturbances and consequently determine the optimal number of steps in the procedure of grid stabilisation.

Some done studies [5] are interested in developing techniques to control the vulnerability of a power system through a combined approach incorporating fuzzy logic and neural to avoid the collapse of the voltage.

In this context and in order to ensure a perfect stability of voltage and frequency throughout the network, we are interested to the implementation of a new smart strategy to achieve optimal integration of wind farms to a large network. Indeed, we have paid particular attention to assessing the sensitivity of the different power plants especially at moments of connection or disconnection of one or more wind farms wf^k ; k ={ 1, …, n }.

The sensitivity of different power plants induced by the connection of wind farms in the studied network will be processed through a new smart strategy to guarantee a perfect stability in frequency and voltage of this network. Indeed, we evaluated the quality of active and reactive powers developed by each production plant according to those injected by the various wind farms, specially at the instants of connection and disconnection of these wind farms. Specifically, the based strategy takes into account voltage fluctuations at different nodes and fluctuations of active and reactive powers induced by the connection or disconnection of wind farms. Based on the sensitivity of each production plant to the impact of wind farms connected or disconnected, we proceeded by appropriate adjustment of powers and voltages applied in each network node.

2 Study design of a large network

The matrix of nodal admittances of a n network nodes is constructed as follows: ${\bar{Y}}_{n} = [\begin{matrix} \bar{Y} (n_{1}, n_{1}) & \bar{Y} (n_{1}, n_{2}) & \dots & \bar{Y} (n_{1}, n_{n}) \\ \bar{Y} (n_{2}, n_{1}) & \bar{Y} (n_{2}, n_{2}) & \dots & \bar{Y} (n_{2}, n_{n}) \\ . & . & \dots & . \\ \bar{Y} (n_{n}, n_{1}) & \bar{Y} (n_{n}, n_{2}) & \dots & \bar{Y} (n_{n}, n_{n}) \end{matrix}]$ (1)

The injected apparent power in a node n_k is expressed as follows [17]: $\begin{matrix} {\bar{S}}_{n_{k}} = Y (n_{k}, n_{k}) . V_{n_{k}}^{2} . e^{- j . θ_{n_{k} n_{k}}} \\ + \sum_{j = 1; j \neq k}^{j = N_{\to k}} Y (n_{k}, n_{j}) . V_{n_{k}} . V_{n_{j}} . e^{j . (α_{n_{k}} - α_{n_{j}} - θ_{n_{k} n_{j}})} \end{matrix}$ (2)

N_→k is the number of nodes connected to the node n_k.

Therefore: ${\begin{matrix} P_{inj - n_{k}} = Y (n_{k}, n_{k}) . V_{n_{k}}^{2} . cos (θ_{n_{k} n_{k}}) \\ + \sum_{j = 1; j \neq k}^{j = N_{\to k}} Y (n_{k}, n_{j}) . V_{n_{k}} . V_{n_{j}} . cos (α_{n_{k}} - α_{n_{j}} - θ_{n_{k} n_{j}}) \\ Q_{inj - n_{k}} = - Y (n_{k}, n_{k}) . V_{n_{k}}^{2} . sin (θ_{n_{k} n_{k}}) \\ + \sum_{j = 1; j \neq k}^{j = N_{\to k}} Y (n_{k}, n_{j}) . V_{n_{k}} . V_{n_{j}} . sin (α_{n_{k}} - α_{n_{j}} - θ_{n_{k} n_{j}}) \end{matrix}$ (3)

We define the specific active and reactive powers P_{spec-n_k}, Q_{spec-n_k} at any network node n_k as follows: ${\begin{matrix} P_{spec - n_{k}} = P_{{Gn}_{k}} - P_{{Cn}_{k}} \\ Q_{spec - nk} = Q_{{Gn}_{k}} - Q_{{Cn}_{k}} \end{matrix}$ (4)

The active and reactive powers P_{Gn
_k}, Q_{Gn
_k} will only be considered in case of connection of a power plant in this node n_k. P_{Cn
_k}, Q_{Cn
_k} are the active and reactive powers consumed in this node. Similarly, we define the functions of active and reactive power at any node in the network to be studied:

The Jacobian matrix [J (P_Vα, Q_Vα)] is constructed as follows: $[J^{P_{V}}] = [\begin{matrix} \frac{\partial {FP}_{n_{1}}}{\partial V_{n_{1}}} & \frac{\partial {FP}_{n_{1}}}{\partial V_{n_{2}}} & \dots & \frac{\partial {FP}_{n_{1}}}{\partial V_{n_{n - 1}}} & \frac{\partial {FP}_{n_{1}}}{\partial V_{n_{n}}} \\ \frac{\partial {FP}_{n_{2}}}{\partial V_{n_{1}}} & \frac{\partial {FP}_{n_{2}}}{\partial V_{n_{2}}} & \dots & \frac{\partial {FP}_{n_{2}}}{\partial V_{n_{n - 1}}} & \frac{\partial {FP}_{n_{2}}}{\partial V_{n_{n}}} \\ \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} \\ \frac{\partial {FP}_{n_{n - 1}}}{\partial V_{n_{1}}} & \frac{\partial {FP}_{n_{n - 1}}}{\partial V_{n_{2}}} & \dots & \frac{\partial {FP}_{n_{n - 1}}}{\partial V_{n_{n - 1}}} & \frac{\partial {FP}_{n_{n - 1}}}{\partial V_{n_{n}}} \\ \frac{\partial {FP}_{n_{n}}}{\partial V_{n_{n 1}}} & \frac{\partial {FP}_{n_{n}}}{\partial V_{n_{n 2}}} & \dots & \frac{\partial {FP}_{n_{n}}}{\partial V_{n_{n - 1}}} & \frac{\partial {FP}_{n_{n}}}{\partial V_{n_{n}}} \end{matrix}]$ (5) $[J^{P_{α}}] = [\begin{matrix} \frac{\partial {FP}_{n_{1}}}{\partial α_{n_{1}}} & \frac{\partial {FP}_{n_{1}}}{\partial α_{n_{2}}} & \dots & \frac{\partial {FP}_{n_{1}}}{\partial α_{n_{n - 1}}} & \frac{\partial {FP}_{n_{1}}}{\partial α_{n_{n}}} \\ \frac{\partial {FP}_{n_{2}}}{\partial α_{n_{1}}} & \frac{\partial {FP}_{n_{2}}}{\partial α_{n_{2}}} & \dots & \frac{\partial {FP}_{n_{2}}}{\partial α_{n_{n - 1}}} & \frac{\partial {FP}_{n_{2}}}{\partial α_{n_{n}}} \\ \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} \\ \frac{\partial {FP}_{n_{n - 1}}}{\partial α_{n_{1}}} & \frac{\partial {FP}_{n_{n - 1}}}{\partial α_{n_{2}}} & \dots & \frac{\partial {FP}_{n_{n - 1}}}{\partial α_{n_{n - 1}}} & \frac{\partial {FP}_{n_{n - 1}}}{\partial α_{n_{n}}} \\ \frac{\partial {FP}_{n_{n}}}{\partial α_{n_{1}}} & \frac{\partial {FP}_{n_{n}}}{\partial α_{n_{2}}} & \dots & \frac{\partial {FP}_{n_{n}}}{\partial α_{n_{n - 1}}} & \frac{\partial {FP}_{n_{n}}}{\partial α_{n_{n}}} \end{matrix}]$ (6) $[J^{Q_{V}}] = [\begin{matrix} \frac{\partial {FQ}_{n_{1}}}{\partial V_{n_{1}}} & \frac{\partial {FQ}_{n_{1}}}{\partial V_{n_{2}}} & \dots & \frac{\partial {FQ}_{n_{1}}}{\partial V_{n_{n - 1}}} & \frac{\partial {FQ}_{n_{1}}}{\partial V_{n_{n}}} \\ \frac{\partial {FQ}_{n_{2}}}{\partial V_{n_{1}}} & \frac{\partial {FQ}_{n_{2}}}{\partial V_{n_{2}}} & \dots & \frac{\partial {FQ}_{n_{2}}}{\partial V_{n_{n - 1}}} & \frac{\partial {FQ}_{n_{2}}}{\partial V_{n_{n}}} \\ \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} \\ \frac{\partial {FQ}_{n_{n - 1}}}{\partial V_{n_{1}}} & \frac{\partial {FQ}_{n_{n - 1}}}{\partial V_{n_{2}}} & \dots & \frac{\partial {FQ}_{n_{n - 1}}}{\partial V_{n_{n - 1}}} & \frac{\partial {FQ}_{n_{n - 1}}}{\partial V_{n_{n}}} \\ \frac{\partial {FQ}_{n_{n}}}{\partial V_{n_{1}}} & \frac{\partial {FQ}_{n_{n}}}{\partial V_{n_{2}}} & \dots & \frac{\partial {FQ}_{n_{n}}}{\partial V_{n_{n - 1}}} & \frac{\partial {FQ}_{n_{n}}}{\partial V_{n_{n}}} \end{matrix}]$ (7) $[J^{Q_{α}}] = [\begin{matrix} \frac{\partial {FQ}_{n_{1}}}{\partial α_{n_{1}}} & \frac{\partial {FQ}_{n_{1}}}{\partial α_{n_{2}}} & \dots & \frac{\partial {FQ}_{n_{1}}}{\partial α_{n_{n - 1}}} & \frac{\partial {FQ}_{n_{1}}}{\partial α_{n_{n}}} \\ \frac{\partial {FQ}_{n_{2}}}{\partial α_{n_{1}}} & \frac{\partial {FQ}_{n_{2}}}{\partial α_{n_{2}}} & \dots & \frac{\partial {FQ}_{n_{2}}}{\partial α_{n_{n - 1}}} & \frac{\partial {FQ}_{n_{2}}}{\partial α_{n_{n}}} \\ \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} & \begin{matrix} . \\ . \\ . \end{matrix} \\ \frac{\partial {FQ}_{n_{n - 1}}}{\partial α_{n_{1}}} & \frac{\partial {FQ}_{n_{n - 1}}}{\partial α_{n_{2}}} & \dots & \frac{\partial {FQ}_{n_{n - 1}}}{\partial α_{n_{n - 1}}} & \frac{\partial {FQ}_{n_{n - 1}}}{\partial α_{n_{n}}} \\ \frac{\partial {FQ}_{n_{n}}}{\partial α_{n_{1}}} & \frac{\partial {FQ}_{n_{n}}}{\partial α_{n_{2}}} & \dots & \frac{\partial {FQ}_{n_{n}}}{\partial α_{n_{n - 1}}} & \frac{\partial {FQ}_{n_{n}}}{\partial α_{n_{n}}} \end{matrix}]$ (8)

The partial derivatives $\frac{\partial P_{inj - n_{k}}}{\partial V_{n_{k}}}, \frac{\partial P_{inj - n_{k}}}{\partial V_{n_{j}}}$ of the active power injected into a node n_k relative to voltages (V_nk, V_nj) j = { 1, …, n }, and the partial derivatives $\frac{\partial P_{inj - n_{k}}}{\partial α_{n_{k}}}, \frac{\partial P_{inj - n_{k}}}{\partial α_{n_{j}}}$ relative to the angle (α_{n
_k}, α_{n
_j}) are calculated from the following relationships [17]: ${\begin{matrix} \frac{\partial P_{inj - n_{k}}}{\partial V_{n_{k}}} = Y (n_{k}, n_{k}) . V_{n_{k}} . cos (θ_{n_{k} n_{k}}) \\ + \sum_{j = 1}^{j = n} Y (n_{k}, n_{j}) . V_{n_{j}} . cos (α_{n_{k}} - α_{n_{j}} - θ_{n_{k} n_{j}}) \\ \frac{\partial P_{inj - n_{k}}}{\partial α_{n_{j}}} = Y (n_{k}, n_{j}) . V_{n_{k}} . cos (α_{n_{k}} - α_{n_{j}} - θ_{n_{k} n_{j}}) \end{matrix}$ (9) ${\begin{matrix} \frac{\partial P_{inj - n_{k}}}{\partial α_{n_{k}}} = - Y (n_{k}, n_{k}) . V_{n_{k}}^{2} . sin (θ_{n_{k} n_{k}}) \\ - \sum_{j = 1}^{j = n} Y (n_{k}, n_{j}) . V_{n_{k}} . V_{n_{j}} . sin (α_{n_{k}} - α_{n_{j}} - θ_{n_{k} n_{j}}) \\ \frac{\partial P_{inj - n_{k}}}{\partial α_{n_{j}}} = Y (n_{k}, n_{j}) . V_{n_{k}} . V_{n_{j}} . sin (α_{n_{k}} - α_{n_{j}} - θ_{n_{k} n_{j}}) \end{matrix}$ (10)

Likewise, $\frac{\partial Q_{inj - n_{k}}}{\partial V_{n_{k}}}, \frac{\partial Q_{inj - n_{k}}}{\partial V_{n_{j}}}$ and $\frac{\partial Q_{inj - n_{k}}}{\partial α_{n_{k}}}, \frac{\partial Q_{inj - n_{k}}}{\partial α_{n_{j}}}$ are given by the following relationships [17]: ${\begin{matrix} \frac{\partial Q_{inj - n_{k}}}{\partial V_{n_{j}}} = - Y (n_{k}, n_{k}) . V_{n_{j}} . sin (θ_{n_{k} n_{k}}) \\ + \sum_{j = 1}^{j = n} Y (n_{k}, n_{j}) . V_{n_{k}} . sin (α_{n_{k}} - α_{n_{j}} - θ_{n_{k} n_{j}}) \\ \frac{\partial Q_{inj - n_{k}}}{\partial V_{n_{j}}} = Y (n_{k}, n_{j}) . V_{n_{k}} . sin (α_{n_{k}} - α_{n_{j}} - θ_{n_{k} n_{j}}) \end{matrix}$ (11) ${\begin{matrix} \frac{\partial Q_{inj - n_{k}}}{\partial α_{n_{k}}} = - Y (n_{k}, n_{k}) . V_{n_{k}}^{2} . cos (θ_{n_{k} n_{k}}) \\ + \sum_{j = 1}^{j = n} Y (n_{k}, n_{j}) . V_{n_{k}} . V_{n_{j}} . cos (α_{n_{k}} - α_{n_{j}} - θ_{n_{k} n_{j}}) \\ \frac{\partial Q_{inj - n_{k}}}{\partial α_{n_{j}}} = - Y (n_{k}, n_{j}) . V_{n_{k}} . V_{n_{k}} . cos (α_{n_{k}} - α_{n_{j}} - θ_{n_{k} n_{j}}) \end{matrix}$ (12)

Formulas 9, 10, 11 and 12 are exclusively news set by us after extremely laborious tests and with those, we are able to simulate the behavior of a network of what size it is by the mean of an outstanding software under Matlab environment done by our self.

3 Wind generator equivalent model

The equivalent circuit diagram of a single wind generator connected to an electrical grid as indicated by Fig. 3: ${\bar{I}}_{{wg}_{k}} = ({\bar{Y}}_{{wg}_{k}} + {\bar{Y}}_{tr} + {\bar{Y}}_{1}) . ({\bar{E}}_{{wg}_{k}} - {\bar{V}}_{n_{i}})$ (13)

Given that the studied wind farm is made up of r wind generator, we obtain the following equivalent circuit:

Referring to Fig. 4, we write:

${\bar{I}}_{wg} = ({\bar{Y}}_{tr} + {\bar{Y}}_{l}) . ({\bar{V}}_{wg} - {\bar{V}}_{n_{i}})$ (14)

On the other hand, the admittance ${\bar{Y}}_{{wg}_{k}}$ represents a proportion K_{wg
_k} of the total equivalent admittance ${\bar{Y}}_{{wg}_{eq}}$ , therefore: ${\bar{Y}}_{{wg}_{k}} = K_{{wg}_{k}} . {\bar{Y}}_{{wg}_{eq}}$ (15)

So: ${\bar{I}}_{wg} = {\bar{Y}}_{{wg}_{eq}} . \sum_{k = 1}^{k = r} K_{{wg}_{k}} . {\bar{E}}_{{wg}_{k}} - {\bar{V}}_{wg} . \sum_{k = 1}^{k = r} {\bar{Y}}_{{wg}_{k}}$ (16)

Finally: ${\bar{I}}_{wg} = {\bar{Y}}_{{wg}_{eq}} . [{\bar{E}}_{{wg}_{eq}} - {\bar{V}}_{wg}]$ (17)

Where: ${\begin{matrix} {\bar{Y}}_{{wg}_{eq}} = \sum_{k = 1}^{k = r} {\bar{Y}}_{{wg}_{k}} \\ {\bar{E}}_{{wg}_{eq}} = \sum_{k = 1}^{k = r} K_{{wg}_{k}} . {\bar{E}}_{{wg}_{k}} \end{matrix}$ (18)

Knowing that ${\bar{E}}_{{wg}_{eq}}$ is the total equivalent electromotive force (e.m.f) and ${\bar{Y}}_{{wg}_{eq}}$ the total equivalent admittance of the wind farm. The current through the bus k becomes:

where: ${\bar{Y}}_{{wg}_{eqt}} = {\bar{Y}}_{{wg}_{eq}} + {\bar{Y}}_{tr} + {\bar{Y}}_{l}$ (19)

The active and reactive powers generated by the equivalent wind farm at the bus n_i: ${\begin{matrix} P_{{wg}_{eq}} = Y_{{wg}_{eqt}} . E_{{wg}_{eq}}^{2} . cos (θ_{{wg}_{eqt}}) \\ - Y_{{wg}_{eqt}} . E_{{wg}_{eq}} . V_{n_{i}} . cos (α_{{wg}_{eq}} - α_{n_{i}} - θ_{{wg}_{eqt}}) \\ Q_{{wg}_{eq}} = - Y_{{wg}_{eqt}} . E_{{wg}_{eq}}^{2} . sin (θ_{{wg}_{eqt}}) \\ - Y_{{wg}_{eqt}} . E_{{wg}_{eq}} . V_{n_{i}} . sin (α_{{wg}_{eq}} - α_{n_{i}} - θ_{{wg}_{eqt}}) \end{matrix}$ (20)

knowing that: ${\begin{matrix} {\bar{E}}_{{wg}_{eq}} = E_{{wg}_{eq}} . e^{j . α_{{wg}_{eq}}} \\ {\bar{Y}}_{{wg}_{eqt}} = Y_{{wg}_{eqt}} . e^{j . θ_{{wg}_{eqt}}} \\ {\bar{V}}_{n_{i}} = V_{n_{i}} . e^{j . α_{n_{i}}} \end{matrix}$

Moreover, the active and reactive powers injected into the bus n_i are expressed as follows [17, 18]: ${\begin{matrix} P_{i n j - n_{i}} = V_{n_{i}} . \sum_{j = 1}^{j = N_{\to i}} Y (n_{i}, n_{j}) . V_{n_{j}} . \cos (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \\ Q_{i n j - n_{i}} = V_{n_{i}} . \sum_{j = 1}^{j = N_{\to i}} Y (n_{i}, n_{j}) . V_{n_{j}} . \sin (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \end{matrix}$ (21)

N_→i is the number of nodes connected to the node n_i.

4 Intelligent integration strategies

We proposed in this paper to implement an intelligent integration strategy of wind farms in a super grid. This strategy is mainly based on the calculation of the sensitivities of active and reactive powers P_{M
_i} and Q_{M
_i} generated by each power plant, at a bus n_i, versus the active and reactive powers P_{inj-n_x} and Q_{inj-n_x} injected at a bus n_x. ${\begin{matrix} P_{M_{i}} = \sum_{j = 1}^{j = N_{\to i}} Y (n_{i}, n_{j}) . V_{n_{i}} . V_{n_{j}} . cos (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \\ Q_{M_{i}} = \sum_{j = 1}^{j = N_{\to i}} Y (n_{i}, n_{j}) . V_{n_{i}} . V_{n_{j}} . sin (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \end{matrix}$ (22)

N_→i is the number of nodes connected to the node n_i.

And the voltage level V_{n
_x} at bus n_x is written as follows [17]: ${\begin{matrix} V_{n_{x}} = \frac{P_{inj - n_{x}}}{\sum_{j = 1}^{j = N_{\to x}} Y (n_{x}, n_{j}) . V_{n_{j}} . cos (α_{n_{x}} - α_{n_{j}} - θ_{n_{x} n_{j}})} \\ V_{n_{x}} = \frac{Q_{inj - n_{x}}}{\sum_{j = 1}^{j = N_{\to x}} Y (n_{x}, n_{j}) . V_{n_{j}} . sin (α_{n_{x}} - α_{n_{j}} - θ_{n_{x} n_{j}})} \end{matrix}$ (23)

Relationships (22) and (23) give: ${\begin{matrix} P_{M_{i}} = Y (n_{i}, n_{x}) . V_{n_{i}} . V_{n_{x}} . cos (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}}) \\ + \sum_{j = 1; j \neq x}^{j = N_{\to i}} Y (n_{i}, n_{j}) . V_{n_{i}} . V_{n_{j}} . cos (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \\ Q_{M_{i}} = Y (n_{i}, n_{x}) . V_{n_{i}} . V_{n_{x}} . sin (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}}) \\ + \sum_{j = 1; j \neq x}^{j = N_{\to i}} Y (n_{i}, n_{j}) . V_{n_{i}} . V_{n_{j}} . sin (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \end{matrix}$ (24) Even yet: ${\begin{matrix} P_{M_{i}} = Y (n_{i}, n_{x}) . V_{n_{i}} . V_{n_{x}} . cos (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}}) \\ + Γ_{n_{i} \to n_{j} (j \neq x)}^{P_{inj}} \\ Q_{M_{i}} = Y (n_{i}, n_{x}) . V_{n_{i}} . V_{n_{x}} . sin (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}}) \\ + Γ_{n_{i} \to n_{j} (j \neq x)}^{Q_{inj}} \end{matrix}$ (25)

Where:

$\begin{matrix} Γ_{n_{i} \to n_{j} (j \neq x)}^{P_{inj}} \\ = \sum_{j = 1; j \neq x}^{j = N_{\to i}} Y (n_{i}, n_{j}) . V_{n_{i}} . V_{n_{j}} . cos (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \\ Γ_{n_{i} \to n_{j} (j \neq x)}^{Q_{inj}} \\ = \sum_{j = 1; j \neq x}^{j = N_{\to i}} Y (n_{i}, n_{j}) . V_{n_{i}} . V_{n_{j}} . sin (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \end{matrix}$ (26)

Hence: ${\begin{matrix} P_{M_{i}} = Y (n_{i}, n_{x}) . V_{n_{i}} . \\ \frac{P_{inj - n_{x}}}{\sum_{j = 1}^{j = N_{\to x}} Y (n_{x}, n_{j}) . V_{n_{j}} . cos (α_{n_{x}} - α_{n_{j}} - θ_{n_{x} n_{j}})} \\ . cos (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}}) + Γ_{n_{i} \to n_{j} (j \neq x)}^{P_{inj}} \\ P_{M_{i}} = Y (n_{i}, n_{x}) . V_{n_{i}} . \\ \frac{Q_{inj - n_{x}}}{\sum_{j = 1}^{j = N_{\to x}} Y (n_{x}, n_{j}) . V_{n_{j}} . sin (α_{n_{x}} - α_{n_{j}} - θ_{n_{x} n_{j}})} \\ . cos (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}}) + Γ_{n_{i} \to n_{j} (j \neq x)}^{P_{inj}} \\ Q_{M_{i}} = Y (n_{i}, n_{x}) . V_{n_{i}} . \\ \frac{Q_{inj - n_{x}}}{\sum_{j = 1}^{j = N_{\to x}} Y (n_{x}, n_{j}) . V_{n_{j}} . sin (α_{n_{x}} - α_{n_{j}} - θ_{n_{x} n_{j}})} \\ . sin (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}}) + Γ_{n_{i} \to n_{j} (j \neq x)}^{Q_{inj}} \\ Q_{M_{i}} = Y (n_{i}, n_{x}) . V_{n_{i}} . \\ \frac{P_{inj - n_{x}}}{\sum_{j = 1}^{j = N_{\to x}} Y (n_{x}, n_{j}) . V_{n_{j}} . cos (α_{n_{x}} - α_{n_{j}} - θ_{n_{x} n_{j}})} \\ . sin (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}}) + Γ_{n_{i} \to n_{j} (j \neq x)}^{Q_{inj}} \end{matrix}$ (27)

We focus only on the sensitivity of the of active and reactive powers P_{M
_i} and Q_{M
_i} generated by each power plant, at a bus n_i, versus the active and reactive powers P_{wg
_x} and Q_{wg
_x} generated at a bus n_x. ${\begin{matrix} P_{M_{i}} = Y (n_{i}, n_{x}) . V_{n_{i}} . \\ \frac{P_{inj - n_{x}}}{\sum_{j = 1}^{j = N_{\to x}} Y (n_{x}, n_{j}) . V_{n_{j}} . cos (α_{n_{x}} - α_{n_{j}} - θ_{n_{x} n_{j}})} \\ . cos (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}}) + Γ_{n_{i} \to n_{j} (j \neq x)}^{P_{inj}} \\ Q_{M_{i}} = Y (n_{i}, n_{x}) . V_{n_{i}} . \\ \frac{Q_{inj - n_{x}}}{\sum_{j = 1}^{j = N_{\to x}} Y (n_{x}, n_{j}) . V_{n_{j}} . sin (α_{n_{x}} - α_{n_{j}} - θ_{n_{x} n_{j}})} \\ . sin (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}}) + Γ_{n_{i} \to n_{j} (j \neq x)}^{Q_{inj}} \end{matrix}$ (28)

Where: ${P_{inj - n_{x}} = P_{{wg}_{x}} - P_{{Cn}_{x}} Q_{inj - n_{x}} = Q_{{wg}_{x}} - Q_{{Cn}_{x}}$ (29)

Then, the sensitivity of a machine M_i to the real P_{inj-n_x} and reactive Q_{inj-n_x} powers injected in a bus n_x is expressed as follow: ${\frac{\partial P_{M_{i}}}{\partial P_{inj - n_{x}}} = \frac{Y (n_{i}, n_{x}) . V_{n_{i}} . cos (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}})}{\sum_{j = 1}^{j = N_{\to x}} Y (n_{x}, n_{j}) . V_{n_{j}} . cos (α_{n_{x}} - α_{n_{j}} - θ_{n_{x} n_{j}})} \frac{\partial Q_{M_{i}}}{\partial Q_{inj - n_{x}}} = \frac{Y (n_{i}, n_{x}) . V_{n_{i}} . sin (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}})}{\sum_{j = 1}^{j = N_{\to x}} Y (n_{x}, n_{j}) . V_{n_{j}} . sin (α_{n_{x}} - α_{n_{j}} - θ_{n_{x} n_{j}})}$ (30)

In the case where the node n_i is not directly connected to a wind farm, then we write: ${\begin{matrix} P_{M_{i}} = \frac{Y_{ix} . V_{i} . {cos}_{ix}}{\sum_{j = 1}^{j = N_{\to x}} Y_{xj} . V_{j} . {cos}_{xj}} . P_{inj - n_{x}} + Γ_{n_{i} \to n_{j} (j \neq x)}^{P_{inj}} \\ Q_{M_{i}} = \frac{Y_{ix} . V_{i} . {sin}_{ix}}{\sum_{j = 1}^{j = N_{\to x}} Y_{xj} . V_{j} . {sin}_{xj}} . Q_{inj - n_{x}} + Γ_{n_{i} \to n_{j} (j \neq x)}^{Q_{inj}} \end{matrix}$ (31)

Where: ${Y_{ix} . V_{i} . {cos}_{ix} = Y (n_{i}, n_{x}) . V_{n_{i}} . cos (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}}) Y_{ix} . V_{i} . {sin}_{ix} = Y (n_{i}, n_{x}) . V_{n_{i}} . sin (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}})$ (32)

On the path to the wind farm: ${\begin{matrix} P_{M_{i}} \\ = \frac{Y_{ix} . V_{i} . {cos}_{ix}}{Y_{xy} . \frac{P_{inj - n_{y}}}{Γ_{n_{y} \to n_{j}}^{Y_{yj} . V_{j} . {cos}_{yj}}} . {cos}_{xy} + Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {cos}_{xj}}} . \\ P_{inj - n_{x}} + Γ_{n_{i} \to n_{j} (j \neq x)}^{P_{inj}} \\ Q_{M_{i}} \\ = \frac{Y_{ix} . V_{i} . {sin}_{ix}}{Y_{xy} . \frac{Q_{inj - n_{y}}}{Γ_{n_{y} \to n_{j}}^{Y_{yj} . V_{j} . {sin}_{yj}}} . {sin}_{xy} + Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {sin}_{xj}}} . \\ Q_{inj - n_{x}} + Γ_{n_{i} \to n_{j (j \neq x)}}^{Q_{inj}} \end{matrix}$ (33)

Where: ${\begin{matrix} Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {cos}_{xj}} = \sum_{j = 1}^{j = N_{\to x}} Y_{xj} . V_{j} . {cos}_{xj (j \neq y)} \\ Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {sin}_{xj}} = \sum_{j = 1}^{j = N_{\to x}} Y_{xj} . V_{j} . {sin}_{xj (j \neq y)} \end{matrix}$ (34)

And: ${\begin{matrix} Γ_{n_{y} \to n_{j}}^{Y_{yj} . V_{j} . {cos}_{yj}} = Y_{yt} . \frac{P_{inj - n_{t}}}{Γ_{n_{t} \to n_{j}}^{Y_{tj} . V_{j} . {cos}_{tj}}} . {cos}_{yt} + Γ_{n_{y} \to n_{j} (j \neq t)}^{Y_{yj} . V_{j} . {cos}_{yj}} \\ Γ_{n_{y} \to n_{j}}^{Y_{yj} . V_{j} . {sin}_{yj}} = Y_{yt} . \frac{Q_{inj - n_{t}}}{Γ_{n_{t} \to n_{j}}^{Y_{tj} . V_{j} . {sin}_{tj}}} . {sin}_{yt} + Γ_{n_{y} \to n_{j} (j \neq t)}^{Y_{yj} . V_{j} . {sin}_{yj}} \end{matrix}$ (35)

So: ${\begin{matrix} P_{M_{i}} = \frac{Y_{ix} . V_{i} . {cos}_{ix}}{Y_{xy} . \frac{P_{inj - n_{y}}}{Y_{yt} . \frac{P_{inj - n_{t}}}{Γ_{n_{t} \to n_{j}}^{Y_{tj} . V_{j} . {cos}_{tj}}} . {cos}_{yt} + Γ_{n_{y} \to n_{j} (j \neq t)}^{Y_{yj} . V_{j} . {cos}_{yj}}} . {cos}_{xy} + Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {cos}_{xj}}} . P_{inj - n_{x}} \\ + Γ_{n_{i} \to n_{j} (j \neq x)}^{P_{inj}} \\ Q_{M_{i}} = \frac{Y_{ix} . V_{i} . {sin}_{ix}}{Y_{xy} . \frac{Q_{inj - n_{y}}}{Y_{yt} . \frac{Q_{inj - n_{t}}}{Γ_{n_{t} \to n_{j}}^{Y_{tj} . V_{j} . {sin}_{tj}}} . {sin}_{yt} + Γ_{n_{y} \to n_{j} (j \neq t)}^{Y_{yj} . V_{j} . {sin}_{yj}}} . {sin}_{xy} + Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {sin}_{xj}}} . Q_{inj - n_{x}} \\ + Γ_{n_{i} \to n_{j (j \neq x)}}^{Q_{inj}} \end{matrix}$ (36) ${\begin{matrix} \frac{\partial P_{M_{i}}}{\partial P_{inj - n_{x}}} \\ = \frac{Y (n_{i}, n_{x}) . V_{n_{i}} . cos (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}})}{\sum_{j = 1}^{j = N_{\to x}} Y (n_{x}, n_{j}) . V_{n_{j}} . cos (α_{n_{x}} - α_{n_{j}} - θ_{n_{x} n_{j}})} \\ \frac{\partial Q_{M_{i}}}{\partial Q_{inj - n_{x}}} \\ = \frac{Y (n_{i}, n_{x}) . V_{n_{i}} . sin (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}})}{\sum_{j = 1}^{j = N_{\to x}} Y (n_{x}, n_{j}) . V_{n_{j}} . sin (α_{n_{x}} - α_{n_{j}} - θ_{n_{x} n_{j}})} \end{matrix}$ (37) In the case where the node n_i is not directly connected to a wind farm, then we write: ${\begin{matrix} P_{M_{i}} \\ = \frac{Y_{ix} . V_{i} . {cos}_{ix}}{\sum_{j = 1}^{j = N_{\to x}} Y_{xj} . V_{j} . {cos}_{xj}} . P_{inj - n_{x}} + Γ_{n_{i} \to n_{j} (j \neq x)}^{P_{inj}} \\ Q_{M_{i}} \\ = \frac{Y_{ix} . V_{i} . {sin}_{ix}}{\sum_{j = 1}^{j = N_{\to x}} Y_{xj} . V_{j} . {sin}_{xj}} . Q_{inj - n_{x}} + Γ_{n_{i} \to n_{j} (j \neq x)}^{Q_{inj}} \end{matrix}$ (38)

Where: ${\begin{matrix} Y_{ix} . V_{i} . {cos}_{ix} \\ = Y (n_{i}, n_{x}) . V_{n_{i}} . cos (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}}) \\ Y_{ix} . V_{i} . {sin}_{ix} \\ = Y (n_{i}, n_{x}) . V_{n_{i}} . sin (α_{n_{i}} - α_{n_{x}} - θ_{n_{i} n_{x}}) \end{matrix}$ (39)

On the path to the wind farm:

${\begin{matrix} P_{M_{i}} \\ = \frac{Y_{ix} . V_{i} . {cos}_{ix}}{Y_{xy} . \frac{P_{inj - n_{y}}}{Γ_{n_{y} \to n_{j}}^{Y_{yj} . V_{j} . {cos}_{yj}}} . {cos}_{xy} + Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {cos}_{xj}}} . \\ P_{inj - n_{x}} + Γ_{n_{i} \to n_{j} (j \neq x)}^{P_{inj}} \\ Q_{M_{i}} \\ = \frac{Y_{ix} . V_{i} . {sin}_{ix}}{Y_{xy} . \frac{Q_{inj - n_{y}}}{Γ_{n_{y} \to n_{j}}^{Y_{yj} . V_{j} . {sin}_{yj}}} . {sin}_{xy} + Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {sin}_{xj}}} . \\ Q_{inj - n_{x}} + Γ_{n_{i} \to n_{j (j \neq x)}}^{Q_{inj}} \end{matrix}$ (40) Where: ${\begin{matrix} Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {cos}_{xj}} = \sum_{j = 1}^{j = N_{\to x}} Y_{xj} . V_{j} . {cos}_{xj (j \neq y)} \\ Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {sin}_{xj}} = \sum_{j = 1}^{j = N_{\to x}} Y_{xj} . V_{j} . {sin}_{xj (j \neq y)} \end{matrix}$ (41)

So: ${\begin{matrix} P_{M_{i}} = \\ \frac{Y_{ix} . V_{i} . {cos}_{ix}}{Y_{xy} . \frac{P_{inj - n_{y}}}{Y_{yt} . \frac{P_{inj - n_{t}}}{Γ_{n_{t} \to n_{j}}^{Y_{tj} . V_{j} . {cos}_{tj}}} . {cos}_{yt} + Γ_{n_{y} \to n_{j} (j \neq t)}^{Y_{yj} . V_{j} . {cos}_{yj}}} . {cos}_{xy} + Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {cos}_{xj}}} \\ . P_{inj - n_{x}} + Γ_{n_{i} \to n_{j} (j \neq x)}^{P_{inj}} \\ Q_{M_{i}} = \\ \frac{Y_{ix} . V_{i} . {sin}_{ix}}{Y_{xy} . \frac{Q_{inj - n_{y}}}{Y_{yt} . \frac{Q_{inj - n_{t}}}{Γ_{n_{t} \to n_{j}}^{Y_{tj} . V_{j} . {sin}_{tj}}} . {sin}_{yt} + Γ_{n_{y} \to n_{j} (j \neq t)}^{Y_{yj} . V_{j} . {sin}_{yj}}} . {sin}_{xy} + Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {sin}_{xj}}} \\ . Q_{inj - n_{x}} + Γ_{n_{i} \to n_{j (j \neq x)}}^{Q_{inj}} \end{matrix}$ (43)

Thus, the sensitivity $\frac{\partial P_{M_{i}}}{\partial P_{{wg}_{x}}}, \frac{\partial Q_{M_{i}}}{\partial Q_{{wg}_{x}}}$ of the of active and reactive powers P_{M
_i} and Q_{M
_i} generated by each power plant, at a bus n_i, versus the active and reactive powers P_{wg
_x} and Q_{wg
_x} generated at a bus n_x is calculated according the following relations [16]: ${\begin{matrix} \frac{\partial P_{M_{i}}}{\partial P_{inj - n_{t}}} = \frac{\frac{Y_{ix} . Y_{xy} . Y_{yt} . V_{i} . {cos}_{ix} . {cos}_{xy} . {cos}_{yt} . P_{inj - n_{x}} . P_{inj - n_{y}}}{Γ_{n_{t} \to n_{j}}^{Y_{tj} . V_{j} . {cos}_{tj}}}}{{[(Y_{yt} . \frac{P_{inj - n_{t}}}{Γ_{n_{t} \to n_{j}}^{Y_{tj} . V_{j} . {cos}_{tj}}} . {cos}_{yt} + Γ_{n_{y} \to n_{j} (j \neq t)}^{Y_{yj} . V_{j} . {cos}_{yj}}) . (Y_{xy} . \frac{P_{inj - n_{y}}}{Y_{yt} . \frac{P_{inj - n_{t}}}{Γ_{n_{t} \to n_{j}}^{Y_{tj} . V_{j} . {cos}_{tj}}} . {cos}_{yt} + Γ_{n_{y} \to n_{j} (j \neq t)}^{Y_{yj} . V_{j} . {cos}_{yj}}} . {cos}_{xy} + Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {cos}_{xj}})]}^{2}} \\ \frac{\partial Q_{M_{i}}}{\partial Q_{inj - n_{t}}} = \frac{\frac{Y_{ix} . Y_{xy} . Y_{yt} . V_{i} . {sin}_{ix} . {sin}_{xy} . {sin}_{yt} . Q_{inj - n_{x}} . Q_{inj - n_{y}}}{Γ_{n_{t} \to n_{j}}^{Y_{tj} . V_{j} . {sin}_{tj}}}}{{[(Y_{yt} . \frac{Q_{inj - n_{t}}}{Γ_{n_{t} \to n_{j}}^{Y_{tj} . V_{j} . {sin}_{tj}}} . {sin}_{yt} + Γ_{n_{y} \to n_{j} (j \neq t)}^{Y_{yj} . V_{j} . {sin}_{yj}}) . (Y_{xy} . \frac{Q_{inj - n_{y}}}{Y_{yt} . \frac{P_{inj - n_{t}}}{Γ_{n_{t} \to n_{j}}^{Y_{tj} . V_{j} . {sin}_{tj}}} . {sin}_{yt} + Γ_{n_{y} \to n_{j} (j \neq t)}^{Y_{yj} . V_{j} . {sin}_{yj}}} . {sin}_{xy} + Γ_{n_{x} \to n_{j} (j \neq y)}^{Y_{xj} . V_{j} . {sin}_{xj}})]}^{2}} \end{matrix}$ (44)

And whether we focus on the study of the sensitivity of the powers P_{M
_i} and Q_{M
_i} generated by a conventional power plant M_i to the powers P_{wg
_x} and Q_{wg
_x} exchanged by all wind farms wf_x connected to the super grid, Fig. 7,we must define the paths Γn connecting the power plant to each wind farm.

Where: $Γ n : x_{n} \to y_{n} \to z_{n} \to t_{n} \to \dots \to n_{{wg}_{n}}$ (45)n_{wg
_n} the node to which is connected the wind farm wg_n.

And thereby, we establish the following relationships: ${\begin{matrix} P_{M_{i}} = \sum_{k = 1}^{k = N_{wf}} \frac{Y_{{ix}_{k}} . V_{i} . {cos}_{{ix}_{k}}}{Y_{x_{k} y_{k}} . \frac{P_{inj - n_{y_{k}}}}{Y_{y_{k} t_{k}} . \frac{P_{inj - n_{t_{k}}}}{Γ_{n_{t_{k}} \to n_{j}}^{Y_{t_{k} j} . V_{j} . {cos}_{t_{k} j}}} . {cos}_{y_{k} t_{k}} + Γ_{n_{y_{k}} \to n_{j} (j \neq t_{k})}^{Y_{y_{k} j} . V_{j} . {cos}_{y_{k} j}}} . {cos}_{x_{k} y_{k}} + Γ_{n_{x_{k}} \to n_{j} (j \neq y_{k})}^{Y_{x_{k} j} . V_{j} . {cos}_{x_{k} j}}} . P_{inj - n_{x_{k}}} + Γ_{n_{i} \to n_{j} (j \neq x)}^{P_{inj}} \\ Q_{M_{i}} = \sum_{k = 1}^{k = N_{wf}} \frac{Y_{{ix}_{k}} . V_{i} . {sin}_{{ix}_{k}}}{Y_{x_{k} y_{k}} . \frac{Q_{inj - n_{y_{k}}}}{Y_{y_{k} t_{k}} . \frac{Q_{inj - n_{t_{k}}}}{Γ_{n_{t_{k}} \to n_{j}}^{Y_{t_{k} j} . V_{j} . {sin}_{t_{k} j}}} . {sin}_{y_{k} t_{k}} + Γ_{n_{y_{k}} \to n_{j} (j \neq t_{k})}^{Y_{y_{k} j} . V_{j} . {sin}_{y_{k} j}}} . {sin}_{x_{k} y_{k}} + Γ_{n_{x_{k}} \to n_{j} (j \neq y_{k})}^{Y_{x_{k} j} . V_{j} . {sin}_{x_{k} j}}} . Q_{inj - n_{x_{k}}} + Γ_{n_{i} \to n_{j} (j \neq x)}^{Q_{inj}} \end{matrix}$ (46)

N_wf is the total number of wind farms connected to the super grid.

In this case of study, we write mathematically: ${\begin{matrix} Δ P_{M_{i}} = \sum_{k = 1}^{k = N_{wf}} \frac{\partial P_{M_{i}}}{\partial P_{inj - n_{{wf}_{k}}}} . Δ P_{inj - n_{{wf}_{k}}} \\ Δ P_{M_{i}} = \sum_{k = 1}^{k = N_{wf}} \frac{\partial P_{M_{i}}}{\partial P_{inj - n_{{wf}_{k}}}} . Δ P_{inj - n_{{wf}_{k}}} \end{matrix}$ (47)

We have adopted the following notations: ${\begin{matrix} ξ_{P_{M_{i}}} = \frac{\partial P_{M_{i}}}{\partial P_{inj - n_{{wf}_{k}}}} \\ ξ_{Q_{M_{i}}} = \frac{\partial Q_{M_{i}}}{\partial Q_{inj - n_{{wf}_{k}}}} \end{matrix}$ (48)

And: ${\begin{matrix} Γ (ξ_{P_{M_{i}}}) = Δ P_{M_{i}} \\ Γ (ξ_{Q_{M_{i}}}) = Δ Q_{M_{i}} \end{matrix}$ (49)

Controlling the behaviour of a conventional power plant could be affected by a malfunction of one or more wind farms requires an instantaneous acquisition of active and reactive powers of the power plant and the magnitude of this impact in order to take an appropriate decision ensuring a strengthening of stability of these plants on the one hand and better integration of renewable energy on the other.

For this purpose, we have implemented a smart strategy based on fuzzy controllers [15–18 , 20].

To consider the uncertainty of the active and reactive wind generation, in the turbine and exciter fuzzy control. Indeed, during wind speed changes, the power to be injected by wind farms will be affected in terms of oscillations containing double the frequency of the studied network [19].

Indeed, any electrical magnitude considered in a positive synchronous reference frame (PSRF) running to the pulsation 1, or with respect to a negative synchronous reference frame (NSRF), turning pulsation, is none other than the sum of a constant term and a term fluctuates at twice the mains frequency, Fig. 10. Under the effect of wind speed changes for example, the active and reactive powers injected in a node i are expressed as follows [19]:

${\begin{matrix} P_{M_{i}} = \sum_{j = 1}^{j = N_{\to i}} Y (n_{i}, n_{j}) . [V_{n_{i} 0} + V_{n_{i} 2 f}] . \\ [V_{n_{j} 0} + V_{n_{j} 2 f}] . cos (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \\ Q_{M_{i}} = \sum_{j = 1}^{j = N_{\to i}} Y (n_{i}, n_{j}) . [V_{n_{i} 0} + V_{n_{i} 2 f}] . \\ [V_{n_{j} 0} + V_{n_{j} 2 f}] . sin (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \end{matrix}$ (50)

If we consider that:

$V_{{dpn}_{x}}^{+}, V_{{qpn}_{x}}^{+}$ the positive sequence of the (d, q) voltage at the node n_x in the PSRF frame,

$V_{{dnn}_{x}}^{-}, V_{{qnn}_{x}}^{-}$ the negative sequence of the (d, q) voltage at the node n_x in the NSRF frame,

The reference powers $P_{M_{i}}^{*}, Q_{M_{i}}^{*}$ , Figs. 8 and 9, will be evaluated as follows [19]: ${\begin{matrix} P_{M_{i}} = \sum_{j = 1}^{j = N_{\to i}} Y (n_{i}, n_{j}) . {\tilde{V}}_{n_{i}} . {\tilde{V}}_{nj} . cos (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \\ Q_{M_{i}} = \sum_{j = 1}^{j = N_{\to i}} Y (n_{i}, n_{j}) . {\tilde{V}}_{n_{i}} . {\tilde{V}}_{nj} . sin (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \end{matrix}$ (51) Where: $\begin{matrix} {\tilde{V}}_{n_{i}} = \sqrt{{[V_{{dpn}_{i}}^{+} - V_{{dnn}_{i}}^{-}]}^{2} + {[V_{{qpn}_{i}}^{+} - V_{{qnn}_{i}}^{-}]}^{2}} \\ {\tilde{V}}_{n_{j}} = \sqrt{{[V_{{dpn}_{j}}^{+} - V_{dnnj}^{-}]}^{2} + {[V_{qpnj}^{+} - V_{qnnj}^{-}]}^{2}} \end{matrix}$

5 Fuzzy studied model

The topology of each fuzzy controller to integrate, was based on an interaction between two input variables, characterized successively by an error ɛ_{x
_M
_i} and an instantaneous variation of the error $\frac{\partial ɛ_{{x_{M}}_{i}}}{\partial t}$ , to synthesize a fuzzy vector K_{x
_M
_i} which represents an estimate of level change of active and reactive powers (P_{M
_i}, Q_{M
_i}) exchanged between the machine M_i and the super grid [5], at the moment following the connection/disconnection of one or more wind farms [8 –14].

We defined the degree of membership of the error ɛ_{x
_M
_i} to a membership function mfunc_r by $μ_{{mfunc}_{r}}^{ɛ_{{x_{M}}_{i}}}$ and $μ_{{mfunc}_{r}}^{\frac{\partial ɛ_{{x_{M}}_{i}}}{\partial t}}$ the degree of membership of the error variation to another membership function [18]. Let then: $π_{R_{k}} = μ_{{mfunc}_{r}}^{ɛ_{{x_{M}}_{i}}} . μ_{{mfunc}_{r}}^{\frac{\partial ɛ_{{x_{M}}_{i}}}{\partial t}}$ (52)

A surface S_{R
_k} swept by the control vector K_{x
_M
_i} for a rule R_k is given by: $S_{R_{k}} = π_{R_{k}} . {mfunc}_{r}$ (53)

The overall area swept by the fuzzy vector K_{x
_M
_i} after use of all the rules is formulated as follows: $S_{K_{{x_{M}}_{i}}} = \frac{\sum_{k}^{k = N_{R_{k}}} S_{R_{k}}}{N_{R_{k}}}$ (54)

With: N_{R
_k} number of rules that are used. Thus, the fuzzy vector K_{x
_M
_i} is none other than the abscissa of the center of gravity of the overall surface S_{K
_{x
_M
_i}} swept by the control vector K_{x
_M
_i} and deduced in accordance with the relationship (46). $K_{{x_{M}}_{i}}^{nf} = \frac{\int_{- 1}^{1} π_{R_{k}} . {mfunc}_{r} (K_{{x_{M}}_{i}}) . K_{{x_{M}}_{i}} . \partial K_{{x_{M}}_{i}}}{\int_{- 1}^{1} π_{R_{k}} . {mfunc}_{r} (K_{{x_{M}}_{i}}) . \partial K_{{x_{M}}_{i}}}$ (55) $K_{{x_{M}}_{i}}^{nf}$ is an un-fuzzy vector.

6 New rescaling technique

A new technique of scaling will be of great importance for the magnitudes of the state variables that may possibly exceed the extreme limits quoted. In other words, all sizes to be treated x_{M
_i} giving rise to an error ɛ_{x
_M
_i} and a variation of error $\frac{\partial ɛ_{{x_{M}}_{i}}}{\partial t}$ , must undergo a transfer at the base of fuzzy variables evidenced by the interval [-1, 1], to generate the fuzzy input required for processing by the designated controller in accordance with the following system of equations [18]: ${\begin{matrix} ɛ_{{x_{M}}_{i}}^{f} = B_{ɛ} . ɛ_{{x_{M}}_{i}} + [1 - B_{ɛ}] \\ \frac{\partial ɛ_{{x_{M}}_{i}}^{f}}{\partial t} = B_{\frac{\partial ɛ}{\partial t}} . \frac{\partial ɛ_{{x_{M}}_{i}}}{\partial t} + [1 - B_{\frac{\partial ɛ}{\partial t}}] \\ B_{ɛ} = \frac{2}{G_{max}^{ss} [{mfunc}_{r} (ɛ_{{x_{M}}_{i}})] - G_{min}^{ss} [{mfunc}_{r} (ɛ_{{x_{M}}_{i}})]} \\ B_{\frac{\partial ɛ}{\partial t}} = \frac{2}{G_{max}^{ss} [{mfunc}_{r} (\frac{\partial ɛ_{{x_{M}}_{i}}}{\partial t})] - G_{min}^{ss} [{mfunc}_{r} (\frac{\partial ɛ_{{x_{M}}_{i}}}{\partial t})]} \end{matrix}$ (56)

Where: ${\begin{matrix} G_{max}^{ss} ({mfunc}_{r} (ɛ_{{x_{M}}_{i}})) \\ = max {\frac{1}{σ_{ɛ} . \sqrt{2 . π}} . exp [- \frac{(ɛ_{{x_{M}}_{i}} - μ_{ɛ})^{2}}{2 . σ_{ɛ}^{2}}]} \\ G_{min}^{ss} ({mfunc}_{r} (ɛ_{{x_{M}}_{i}})) \\ = min {\frac{1}{σ_{ɛ} . \sqrt{2 . π}} . exp [- \frac{(ɛ_{{x_{M}}_{i}} - μ_{ɛ})^{2}}{2 . σ_{ɛ}^{2}}]} \\ G_{max}^{ss} ({mfunc}_{r} (\frac{\partial ɛ_{{x_{M}}_{i}}}{\partial t})) \\ = max {\frac{1}{σ_{\partial ɛ} . \sqrt{2 . π}} . exp [- \frac{(\frac{\partial ɛ_{{x_{M}}_{i}}}{\partial t} - μ_{\partial ɛ})^{2}}{2 . σ_{\partial ɛ}^{2}}]} \\ G_{min}^{ss} ({mfunc}_{r} (\frac{\partial ɛ_{{x_{M}}_{i}}}{\partial t})) \\ = min {\frac{1}{σ_{\partial ɛ} . \sqrt{2 . π}} . exp [- \frac{(\frac{\partial ɛ_{{x_{M}}_{i}}}{\partial t} - μ_{\partial ɛ})^{2}}{2 . σ_{\partial ɛ}^{2}}]} \end{matrix}$ (57)

Where μ_iσ_i denote respectively the mean and standard deviation of the error and variation of error ɛ_{x
_M
_i} and $\frac{\partial ɛ_{{x_{M}}_{i}}}{\partial t}$ . Similarly, the control vector K_{x
_M
_i} must undergo a transfer to the basic quantities studied, rescaled, to assign the value $K_{{x_{M}}_{i}}^{nf}$ and this means the system of equations: ${\begin{matrix} K_{{x_{M}}_{i}}^{nf} = B_{K_{{x_{M}}_{i}}^{nf}} . K_{{x_{M}}_{i}} + [1 - B_{K_{{x_{M}}_{i}}^{nf}}] \\ B_{K_{{x_{M}}_{i}}^{nf}} = \frac{G_{max}^{ss} [{mfunc}_{r} (K_{{x_{M}}_{i}})] - G_{min}^{ss} [{mfunc}_{r} (K_{{x_{M}}_{i}})]}{2} \end{matrix}$ (58)

Where: ${\begin{matrix} G_{max}^{ss} [{mfunc}_{r} (K_{{x_{M}}_{i}})] \\ = max {\frac{1}{σ_{K_{{x_{M}}_{i}}} . \sqrt{2 . π}} . exp [- \frac{(K_{{x_{M}}_{i}} - μ_{K_{{x_{M}}_{i}}})^{2}}{2 . σ_{K_{{x_{M}}_{i}}}^{2}}]} \\ G_{min}^{ss} [{mfunc}_{r} (K_{{x_{M}}_{i}})] \\ = min {\frac{1}{σ_{K_{{x_{M}}_{i}}} . \sqrt{2 . π}} . exp [- \frac{(K_{{x_{M}}_{i}} - μ_{K_{{x_{M}}_{i}}})^{2}}{2 . σ_{K_{{x_{M}}_{i}}}^{2}}]} \end{matrix}$ (59)

Through this new technique of rescaling, we assigned a dynamic behavior to the fuzzy controller in order to ensure better tracking of the variable to control [18, 20].

7 Simulations and Results

The studied grid is the IEEE 118 bus test power system, Fig. 15.

With a view to assessing the impact of the commissioning of one or more wind farms on a super grid, we simulated the case of connecting a first wind farm wf₁ with a capacity of 0.78pu, in the base S_base = 100MVA, and this at the moment 50 seconds, Fig. 16.

The simulations were conducted with the following considerations:

The wind farm wf¹ is connected to bus n¹² and wf² is connected to bus n⁷.

We note that the active powers developed by different machines become fluctuating during the moments following the commissioning of the wind farm wf₁. These fluctuations arise because the different power plants are trying to regain a new stable equilibrium taking into account the contribution of new wind farm connected to the network. Indeed, referring to the relationship (43), changes in active and reactive powers are the consequences of superimposed terms P_{M_i2f} and Q_{M_i2f} the initial values P_{M
_i} and Q_{M
_i}.

Where: ${\begin{matrix} P_{M_{i} 2 f} = \sum_{j = 1}^{j = N_{\to i}} Y (n_{i}, n_{j}) . \\ [V_{n_{i} 0} . V_{n_{j} 2 f} + V_{n_{i} 2 f} . V_{n_{j} 0} + V_{n_{i} 2 f} . V_{n_{j} 2 f}] . \\ cos (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \\ Q_{M_{i} 2 f} = \sum_{j = 1}^{j = N_{\to i}} Y (n_{i}, n_{j}) . \\ [V_{n_{i} 0} . V_{n_{j} 2 f} + V_{n_{i} 2 f} . V_{n_{j} 0} + V_{n_{i} 2 f} . V_{n_{j} 2 f}] . \\ sin (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \end{matrix}$

Consequently we recorded oscillating modes of voltages at different nodes, Fig. 17. We tested the case of the commissioning of the wind farm from 50 to 70 seconds, (tc_wf = 50s and td_wf = 70s). Where: tc_wf and td_wf are respectively the times of commissioning and decommissioning of the wind farms.

These oscillations are described by: $V_{n_{i}} = V_{n_{i} 0} + V_{n_{i} 2 f}$ fluctuations in voltage levels V_{n
_k} can be explained through the following relationship which reflect the sensitivity of these voltages to active and reactive powers (P_{wg
_x}, Q_{wg
_x}) injected by the wind farms. $V_{n_{k}} = V_{n_{k 0}} + \frac{\partial V_{n_{k}}}{\partial P_{{wg}_{x}}} . Δ P_{{wg}_{x}}$

Where: $V_{n_{k 0}} = \frac{P_{inj - n_{k 0}}}{\sum_{j = 1}^{j = N_{\to k}} Y (n_{k}, n_{j}) . V_{n_{j 0}} . cos (α_{n_{k 0}} - α_{n_{j 0}} - θ_{n_{k 0} n_{j 0}})}$

We also note that the oscillations of the voltage levels are increasing at the commissioning of several wind farms at the same time, Fig. 18.

Figure 19 represents the temporal evolution of active powers of the wind farms wf₁ and wf₂ when they are brought into operation successively during:

$\begin{matrix} {wf}_{1} : {tc}_{wf 1} \leftrightarrow {td}_{wf 1} : 50 \to 65 s \\ {wf}_{2} : {tc}_{wf 2} \leftrightarrow {td}_{wf 2} : 70 \to 80 s . \end{matrix}$

Figures 20 and 21 represents the temporal evolution of active powers of the machines M₁₁, M₃₀, M₅₀ and the voltage levels when the wind farms wf₁ and wf₂ are brought into operation during the time intervals indicated above.

We clearly notice that at the moment of connection or disconnection of one or more wind farms, the voltages at different nodes fluctuate worsening gradually as the number of farms becomes important. Taking into account the sensitivity of powers generated by the different power plants to real and reactive powers injected by wind farms, we corrected voltage levels as well as real and reactive powers developed by these power plants through control strategies indicated by the Figs. 22 and 23. Active and reactive powers were stabilized of the fact that strategy adopted has eliminated origins of fluctuations:

${\begin{matrix} P_{M_{i}} = P_{M_{i} 0} - \\ \sum_{j = 1}^{j = N_{\to i}} \begin{matrix} Y (n_{i}, n_{j}) . [V_{n_{i} 0} . V_{n_{j} 2 f} + V_{n_{i} 2 f} . V_{n_{j} 0} + V_{n_{i} 2 f} . V_{n_{j} 2 f}] \\ . cos (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \end{matrix} \\ Q_{M_{i}} = Q_{M_{i} 0} - \\ \sum_{j = 1}^{j = N_{\to i}} \begin{matrix} Y (n_{i}, n_{j}) . [V_{n_{i} 0} . V_{n_{j} 2 f} + V_{n_{i} 2 f} . V_{n_{j} 0} + V_{n_{i} 2 f} . V_{n_{j} 2 f}] \\ . sin (α_{n_{i}} - α_{n_{j}} - θ_{n_{i} n_{j}}) \end{matrix} \end{matrix}$

By acting on the various turbines and the exciters of different power plants, as shown in Figs. 8 and 9, we were able to mitigate the peaks of the voltage levels at the instants following the connection of wind farms. Thus, we were able to bring the electrical network studied in stable condition during the connection or disconnection phases of wind farms.

8 Conclusion

In this work, we are interested in implementing a smart strategy based fuzzy controllers with the aim to ensure better integration of renewable energy in the networks of large sizes called super grids. Given the randomness of wind energy, the engagement of the operating wind farms must take into account the impact of the latter on the network status variables considered and those of conventional power plants. The machines are virtually insensitive to the impact of these defects and therefore, we were able to avoid the network studied the phenomenon of voltage collapse. We have paid particular attention to the margins of variation experienced by active and reactive powers generated by the different power plants of a super grid in the event of connection or disconnection of one or several wind sites.

Based on an analytical study of the results obtained, we can pronounce on the greater potential of the smart strategy of the fact that following the connection or disconnection of wind farms to a large network, algebraic variables and powers active and reactive evolve transiently in acceptable margins and can not in any way affect the quality of service provided to consumers. The objectives of this work have been achieved and the prospects remains promising.

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