Development of Methods for Sensitivity Analysis of Electrical Energy Networks and Systems within State Space

Abstract

This paper presents a new method that should make it possible to determine the sensitivity of uncontrolled electrical power grids with only a modest amount of effort. Unlike most of the methods described in the literature, this method is based on an implicit description of the electrical power grid in the state space, in which the system matrices are set up using mesh equations based on a complete tree. In this case, the sensitivity matrix is not derived from the Jacobian matrix, but from the weighted inverse of the impedance matrix. It will be shown that this makes it possible to determine the sensitivity of the network independently of the operating point, under the assumption that the impedances of the network remain constant.

1. Introduction

The ongoing transformation of the German energy grids requires a better determination of the grid status in order to be able to carry out necessary switching operations. This is becoming particularly important in view of the increasing integration of fluctuating generation plants and the rising number of electrical consumers, in addition to the ensuing requirements resulting from §14a EnWG. Completely metering the entire grid would be ideal; however, it is difficult to implement economically, technically and organizationally. According to [1], a complete monitoring of the grid state can be realized by recording 7% to 15% of the grid connections and the associated transformer outputs. For targeted control, it is also helpful to be able to estimate the effects of various switching operations as well as possible ones in advance. The sensitivity analysis of the network is particularly suitable for this. Numerous approaches for the necessary calculations can be found in the literature. Most are based on the Newton–Raphson method [2,3] and an analysis of the Jacobi matrix, like described here [4,5,6,7]. With this, it is possible to analyze the effects of changed input and output states, but also with changed transformer ratios [8] around an operating point. However, it is necessary to recalculate the Jacobian matrix at each operating point. Another possibility is offered by a state space based load flow simulation, in which the system equations are set up implicitly. This new procedure, as well as the possibility of analyzing the sensitivity with the help of the system equations, is examined in this paper for an uncontrolled case.

2. Load Flow Simulation Using State Space

Various approaches of analyzing electrical grids can be found in the literature. Some well-known methods include the current iteration method and the Newton–Raphson method. Both methods exhibit rapid convergence and map the network information completely. Another possibility is to analyze the grid in state space. These methods have already been described in detail in [9]. As the following analyses take place in state space, the basic terms required for this are briefly outlined again below.

In order to transfer the grids, to be calculated, into state space, a complete tree is first created, from which the state variables and system matrices are derived according to Figure 1. This process can be carried out using basic methods of network analysis as described in [10] or in [11]. In addition, the formulas of the state space equations are realized in a symbolic form, which allows the system parameters to be varied without having to re-establish the system matrices for each time step.

For very simple systems, such as those shown in Figure 2, the derivation of the corresponding differential equation system can be carried out manually with little effort. In the following, the basic principle of the methodology is demonstrated using the very simple example network from Figure 2.

Using Kirchhoff’s laws, the network analysis leads to the following differential equations, which are used to set up the corresponding systems of equations. All impedances are normalized to the angular frequency $\omega$. The time variable is therefore formally extended by $\omega$, as follows:

$$\stackrel{\mathbf{u}\left(\omega t\right)}{\overbrace{\left[\begin{array}{c}{i}_0\\ u_0\end{array}\right]}}=\stackrel{\mathbf{R}}{\overbrace{\left[\begin{array}{cc}1/R_p& -1\\ 1& R\end{array}\right]}}\stackrel{\mathbf{z}\left(\omega t\right)}{\overbrace{\left[\begin{array}{c}{u}_C\\ i_L\end{array}\right]}}+\stackrel{\mathbf{X}}{\overbrace{\left[\begin{array}{cc}1/X_C& 0\\ 0& X_L\end{array}\right]}}\stackrel{\dot{\mathbf{z}}\left(\omega t\right)}{\overbrace{\left[\begin{array}{c}{\dot{u}}_C\\ {\dot{i}}_L\end{array}\right]}}.$$

(1)

With

$i_0$	=	source current
$u_0$	=	source voltage
$i_L$	=	inductance current
$u_C$	=	capacitor voltage
R	=	series resistance
$R_p$	=	parallel resistance
$X_L$	=	$\omega ·L$ = impedance of the inductance
$X_C$	=	${\left(\omega ·C\right)}^{-1}$ = impedance of the capacitor

This system description can also be represented in matrix notation as follows:

$$\mathbf{u}\left(\omega t\right)=\mathbf{R}·\mathbf{z}\left(\omega t\right)+\mathbf{X}·\dot{\mathbf{z}}\left(\omega t\right).$$

(2)

where

$\mathbf{u}$	=	Vector containing all sources
$\mathbf{R}$	=	Resistances matrix
$\mathbf{X}$	=	Reactance matrix
$\mathbf{z}$	=	Vector of the state variables

To solve this system, the implicit system representation (2) can be transformed into the explicit system representation (3) by equating the coefficients.

$$\dot{\mathbf{z}}\left(\omega t\right)=\mathbf{A}·\mathbf{z}\left(\omega t\right)+\mathbf{B}·\mathbf{u}\left(\omega t\right),$$

(3)

with

$$\begin{array}{cc}\hfill \mathbf{A}& =-\mathbf{R}·{\mathbf{X}}^{-\mathbf{1}}\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathbf{B}& ={\mathbf{X}}^{-\mathbf{1}}.\hfill \end{array}$$

(5)

This is the standard form of the differential equation system. Therefore, according to [12], Equation (6) can be derived from Equation (2):

$$\stackrel{\dot{\mathbf{z}}\left(\omega t\right)}{\overbrace{\left[\begin{array}{c}{\dot{u}}_C\\ {\dot{i}}_L\end{array}\right]}}=\stackrel{\mathbf{A}}{\overbrace{\left[\begin{array}{cc}-X_C/R_p& X_C\\ -1/X_L& -R/X_L\end{array}\right]}}\stackrel{\mathbf{z}\left(\omega t\right)}{\overbrace{\left[\begin{array}{c}{u}_C\\ i_L\end{array}\right]}}+\stackrel{\mathbf{B}}{\overbrace{\left[\begin{array}{cc}{X}_C& 0\\ 0& 1/X_L\end{array}\right]}}\stackrel{\mathbf{u}\left(\omega t\right)}{\overbrace{\left[\begin{array}{c}{i}_0\\ u_0\end{array}\right]}}.$$

(6)

The differential equations from Equation (6) can be used to calculate the system states of the given grid in the time domain using suitable solution algorithms, e.g., to simulate transient or switch-on processes. Here, A represents the system matrix, B the input matrix, u the input vector and z the state vector If a steady-state system with constant periodic feeds is assumed, as is the case with energy technology considerations, these equations can also be Fourier-transformed. Thus, the following applies:

$$\dot{\mathbf{z}}\left(t\right)=\mathrm{j}\omega ·\mathbf{z}\left(\omega t\right)$$

(7)

The time-dependent solution is thus converted into a complex system of equations, as follows:

$$\underline{\mathbf{z}}=\mathbf{B}·{(\mathrm{j}\omega \mathbf{E}-\mathbf{A})}^{-1}·\underline{\mathbf{u}}.$$

(8)

With Equation (8), it is possible to carry out load flow simulations in state space. If there is no fault, it can be assumed that $f\approx 50$ applies. Under the further assumption that the impedances of the system remain constant, e.g., due to switching operations, the equation can be simplified as follows:

$$\underline{\mathbf{z}}=\underline{\mathbf{K}}·\underline{\mathbf{u}}.$$

(9)

where

$$\underline{\mathbf{K}}=\mathbf{B}·{(\mathrm{j}\omega \mathbf{E}-\mathbf{A})}^{-1}$$

(10)

This makes it possible to solve the load flow equation for each different load situation with just one mathematical operation. The $\underline{\mathbf{u}}$ vector includes all sources of the system and, in the case of constant PQ or U loads, the necessary calculations, which can be solved iteratively (see also [9]). A possible termination condition occur if the largest deviation of the vector $\underline{\mathbf{u}}$ would drop below a previously arbitrarily defined lower bound $ϵ$.

$$max\left(\left|{\underline{\mathbf{u}}}^{\upsilon }-{\underline{\mathbf{u}}}^{\upsilon -1}\right|\right)<ϵ.$$

(11)

The index $\upsilon$ in Equation (11) represents the iteration counter. Without systems that need be controlled, the entire calculation according to Equation (9) is reduced to single matrix/vector multiplication, which is less than with the Newton–Raphson method and thus represents a possible alternative for shortening long time-series calculations. With an average of four to seven repetitions, the number of iteration steps required is also low.

3. Sensitivity Analysis of Unregulated Electrical Energy Systems within State Space

For the evaluation of the sensitivity of uncontrolled electrical energy networks, the applicability of state space methods needs to be investigated and demonstrated on the basis of an example. For a better illustration, the methods should be developed and subsequently applied to a simple equivalent circuit diagram of an electrical energy network according to Figure 3. This consists of a slack, a power feeder, a load and two cables interconnecting the various components. In order to better understand the simulation in state space, the components are represented as two-terminal or four-terminal networks with their respective equivalent elements. In order to be able to set any desired operating point, the source $V_1$ of the slack regulates the voltage $U_{C1}$ to the desired nominal voltage, while the requested active and reactive power flows are set via the sources $V_2$ and $V_3$. Sources $V_2$ and $V_3$ could also be replaced by current sources using the same method. To make the network less complicated, the indices were simplified and the output capacitance $C_{31}$ of cable K1 and the input capacitance $C_{32}$ of cable K2 were combined in the resulting $\mathbf{X}$ matrix as cable capacitance $C_3$.

3.1. Establishing the System Equations

Based on the model network shown in Figure 3, the equations of state of the energy system are now derived in minimal form, as described in Section 2. According to the rank of the system, 8 state variables can be derived in vector notation, which together with the vector of the driving variables $\underline{\mathbf{u}}$ (sources/loads) describe the system in the implicit form of the state space:

$$\underline{\mathbf{u}}=\mathbf{R}·\underline{\mathbf{z}}+\mathbf{X}·\dot{\underline{\mathbf{z}}}$$

(12)

The matrices of the impedances ${\mathbf{R}}^{n\times n},{\mathbf{X}}^{n\times n}$ and the vectors of the input variables ${\underline{\mathbf{u}}}^{n\times m}$ and the associated state variables ${\underline{\mathbf{z}}}^{n\times m}$ are shown in the following. The index K identifies the elements of the cables.

$$\mathbf{R}=\left[\begin{array}{cccccccc}{R}_1& 1& 0& 0& 0& 0& 0& 0\\ -1& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 1& 1& 0\\ 0& -1& 1& R_{K1}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -1& 0& 1\\ 0& 0& -1& 0& 1& R_{K2}& 0& 0\\ 0& 0& -1& 0& 0& 0& R_3& 0\\ 0& 0& 0& 0& -1& 0& 0& R_2\end{array}\right]$$

(13)

$$\mathbf{X}=\left[\begin{array}{cccccccc}{L}_1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1/C_1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1/C_3& 0& 0& 0& 0& 0\\ 0& 0& 0& L_{K1}& 0& 0& 0& 0\\ 0& 0& 0& 0& 1/C_2& 0& 0& 0\\ 0& 0& 0& 0& 0& L_{K2}& 0& 0\\ 0& 0& 0& 0& 0& 0& L_3& 0\\ 0& 0& 0& 0& 0& 0& 0& L_2\end{array}\right]$$

(14)

$$\underline{\mathbf{u}}=\left[\begin{array}{c}{\underline{U}}_{V1}\\ 0\\ 0\\ 0\\ 0\\ 0\\ -{\underline{U}}_{V3}\\ -{\underline{U}}_{V2}\end{array}\right]$$

(15)

$$\underline{\mathbf{z}}=\left[\begin{array}{c}{\underline{i}}_1\\ {\underline{u}}_{C1}\\ {\underline{u}}_{C3}\\ {\underline{i}}_{K1}\\ {\underline{u}}_{C2}\\ {\underline{i}}_{K2}\\ {\underline{i}}_3\\ {\underline{i}}_2\end{array}\right]$$

(16)

As already described, load flow calculations in energy networks can usually be based on steady states with approximately constant frequency, which means that Equation (12) can be simplified according to (7), as follows:

$$\underline{\mathbf{u}}=\mathbf{R}·\underline{\mathbf{z}}+\mathrm{j}\mathbf{X}·\underline{\mathbf{z}}$$

(17)

If no switching operations take place, $\mathbf{R}$ and $\mathbf{X}$ can be assumed to be constant and therefore summarized. With ${\underline{\mathbf{K}}}^{-1}=\mathbf{R}+\mathrm{j}\mathbf{X}$, you then get

$${\underline{\mathbf{K}}}^{-1}=\left[\begin{array}{cccccccc}{R}_1+\mathrm{j}{L}_1& 1& 0& 0& 0& 0& 0& 0\\ -1& \mathrm{j}/C_1& 0& 1& 0& 0& 0& 0\\ 0& 0& \mathrm{j}/C_3& -1& 0& 1& 1& 0\\ 0& -1& 1& R_{K1}+\mathrm{j}{L}_{K1}& 0& 0& 0& 0\\ 0& 0& 0& 0& \mathrm{j}/C_2& -1& 0& 1\\ 0& 0& -1& 0& 1& R_{K2}+\mathrm{j}{L}_{K2}& 0& 0\\ 0& 0& -1& 0& 0& 0& R_3+\mathrm{j}{L}_3& 0\\ 0& 0& 0& 0& -1& 0& 0& R_2+\mathrm{j}{L}_2\end{array}\right]$$

(18)

with which the state variables can be calculated as follows:

$$\underline{\mathbf{z}}=\underline{\mathbf{K}}·\underline{\mathbf{u}}$$

(19)

3.2. Sensitivity Analysis of the Test Network

The sensitivity of a system is determined by varying the input variables $\underline{\mathbf{u}}$ from their operating point and analyzing the resulting changes in the state variables. For the first investigations, the parameters of the test network from Figure 3 were therefore initially specified in such a way that the operating point AP1 was set according to the following specifications. For this purpose, at a nominal voltage $U_N$ of 230 V, a power of 2 $\mathrm{k}$$\mathrm{W}$ should be generated via the source $V_2$ and a constant power of 4 $\mathrm{k}$$\mathrm{W}$ should be drawn via the source $V_3$. The differential power is applied by the source $V_1$ (slack). The impedances $R_{1\dots 3}$ and $L_{1\dots 3}$ are calculated from the corresponding short-circuit power ($S_k$) of the sources. Here, $|{\underline{S}}_{k1}|=40\mathrm{kVA}$, $|{\underline{S}}_{k2}|=2.42\mathrm{kVA}$ and $|{\underline{S}}_{k3}|=4.84\mathrm{kVA}$. The ratio P/Q should be 10/1. This results in the values shown in

Table 1. Values of the components for the equivalent circuit diagram according to Figure 3 for operating point AP1: $U_N=230$ V, $P_2=2 \mathrm{k}\mathrm{W}$, $P_3=4 \mathrm{k}\mathrm{W}$, $P/Q=10/1$, $|{\underline{S}}_{k1}|=40\mathrm{kVA}$, $|{\underline{S}}_{k2}|=2.42\mathrm{kVA}$, $|{\underline{S}}_{k3}|=4.48\mathrm{kVA}$.

$R_1=3.9477$	$L_1=0.3963$	$C_1=25,063.7706$	${\underline{U}}_{V1}=325.9371\underline{/0.17°}$
$R_2=65.7851$	$L_2=6.6033$	$C_2=25,063.77057$	${\underline{U}}_{V2}=422.5955\underline{/-179.99°}$
$R_3=32.8926$	$L_3=3.3017$	$C_3=12,531.8853$	${\underline{U}}_{V3}=36.2127\underline{/179.81°}$
$R_{K1}=0.211$	$L_{K1}=0.122$
$R_{K2}=0.211$	$L_{K2}=0.122$

for each component:

If the values from Table 1 are inserted into Equation (18), the absolute value of the matrix $\mathbf{K}$ for this operating point is calculated as follows:

$$|\underline{\mathbf{K}}|=\left[\begin{array}{cccccccc}0.0381& 0.8489& 0.8405& 0.0381& 0.8376& 0.0127& 0.0254& 0.0127\\ 0.8489& 3.3681& 3.3345& 0.1511& 3.3234& 0.0503& 0.1009& 0.0503\\ 0.8405& 3.3345& 3.5222& 0.8404& 3.5104& 0.0531& 0.1065& 0.0531\\ 0.0381& 0.1511& 0.8404& 0.0381& 0.8376& 0.0127& 0.0254& 0.0127\\ 0.8376& 3.3234& 3.5104& 0.8376& 3.7231& 0.9438& 0.1062& 0.0563\\ 0.0127& 0.0503& 0.0531& 0.0127& 0.9438& 0.0143& 0.0016& 0.0143\\ 0.0254& 0.1009& 0.1065& 0.0254& 0.1062& 0.0016& 0.0270& 0.0016\\ 0.0127& 0.0503& 0.0531& 0.0127& 0.0563& 0.0143& 0.0016& 0.0143\end{array}\right]$$

(20)

From this, the state variables at the operating point can also be calculated analytically using Equation (19).

$$|{\mathbf{z}}_{AP1}|=\left[\begin{array}{c}\hfill 2.9260\\ \hfill 230.0000\\ \hfill 229.3522\\ \hfill 2.9268\\ \hfill 230.0001\\ \hfill 2.9139\\ \hfill 5.8425\\ \hfill 2.9130\end{array}\right]$$

(21)

Once the variables shown above are set at the operating point, the control of the sources is deactivated so that they are assumed to be constant for further analysis, unless otherwise specified. This applies each time, after setting the operating point to a new one.

3.2.1. Sensitivity Analysis through Sequential Variation of Sources

In order to be able to analyze the effects of changes in the load and feed-in nodes, the influence of defined changes in the voltage sources on the state variables of the grid according to Figure 3 will be examined below. This is carried out by sequentially increasing each of the three voltage sources by 10 % from their operating point, as shown in Table 1. Only one source should be increased by 10 % at a time, while the others remain at their operating point. The operating point will be identified below by the index AP1. This results in the values shown in

Table 2. Results of the state variables of the previously set operating point, as well as the results when increasing a single voltage source by 10% in each case.

	$\left|{\underline{\mathit{z}}}_{\mathit{AP}1}\right|$	$\left|{\underline{\mathit{z}}}_1\right|$	$\left|{\underline{\mathit{z}}}_2\right|$	$\left|{\underline{\mathit{z}}}_3\right|$
State Variable	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{1},\mathbf{0}}$	$\mathbf{1}.\mathbf{1}·{\underline{\mathit{U}}}_{\mathit{V}\mathbf{1},\mathbf{0}}$	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{1},\mathbf{0}}$	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{1},\mathbf{0}}$
	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{2},\mathbf{0}}$	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{2},\mathbf{0}}$	$\mathbf{1}.\mathbf{1}·{\underline{\mathit{U}}}_{\mathit{V}\mathbf{2},\mathbf{0}}$	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{2},\mathbf{0}}$
	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{3},\mathbf{0}}$	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{3},\mathbf{0}}$	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{3},\mathbf{0}}$	$\mathbf{1}.\mathbf{1}·{\underline{\mathit{U}}}_{\mathit{V}\mathbf{3},\mathbf{0}}$
$\left|{\underline{i}}_1\right|$	3.5122	4.4411	2.9769	3.4699
$\left|{\underline{u}}_{C1}\right|$	230.0000	250.7080	232.1241	230.1679
$\left|{\underline{u}}_{C3}\right|$	229.2219	249.7235	231.4651	229.3992
$\left|{\underline{i}}_{K1}\right|$	3.5131	4.4420	2.9777	3.4707
$\left|{\underline{u}}_{C2}\right|$	229.8701	250.3030	232.2476	230.0469
$\left|{\underline{i}}_{K2}\right|$	2.9156	2.6066	3.5188	2.9129
$\left|{\underline{i}}_3\right|$	6.4304	7.0505	6.4982	6.3854
$\left|{\underline{i}}_2\right|$	2.9147	2.6056	3.5178	2.9120

.

In order to be able to evaluate the impact of a source on the corresponding state variables, the deviations of the state variables from the initial state are calculated according to Equation (22).

$$\Delta {\mathbf{z}}_x={\displaystyle \frac{{\mathbf{z}}_x-{\mathbf{z}}_{AP1}}{{\mathbf{z}}_{AP1}}}\mathrm{for}x=1,2,3$$

(22)

If we now apply Equation (22) to Table 2, we obtain the values listed in

Table 3. Deviations of the state variables from the operating point $\Delta \underline{\mathbf{z}}$ if one voltage source is increased by 10 % in each case.

	$\left|\mathit{\Delta }{\underline{\mathit{z}}}_1\right|$	$\left|\mathit{\Delta }{\underline{\mathit{z}}}_2\right|$	$\left|\mathit{\Delta }{\underline{\mathit{z}}}_3\right|$
State Variable	$\mathbf{1}.\mathbf{1}·{\underline{\mathit{U}}}_{\mathit{V}\mathbf{1},\mathbf{0}}$	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{1},\mathbf{0}}$	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{1},\mathbf{0}}$
	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{2},\mathbf{0}}$	$\mathbf{1}.\mathbf{1}·{\underline{\mathit{U}}}_{\mathit{V}\mathbf{2},\mathbf{0}}$	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{2},\mathbf{0}}$
	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{3},\mathbf{0}}$	${\underline{\mathit{U}}}_{\mathit{V}\mathbf{3},\mathbf{0}}$	$\mathbf{1}.\mathbf{1}·{\underline{\mathit{U}}}_{\mathit{V}\mathbf{3},\mathbf{0}}$
$\left|{\underline{i}}_1\right|$	0.3144	0.1830	0.0315
$\left|{\underline{u}}_{C1}\right|$	0.0892	0.0092	0.0017
$\left|{\underline{u}}_{C3}\right|$	0.0885	0.0098	0.0017
$\left|{\underline{i}}_{K1}\right|$	0.3144	0.1829	0.0315
$\left|{\underline{u}}_{C2}\right|$	0.0880	0.0103	0.0017
$\left|{\underline{i}}_{K2}\right|$	0.1050	0.2070	0.0020
$\left|{\underline{i}}_3\right|$	0.1051	0.0116	0.0168
$\left|{\underline{i}}_2\right|$	0.1051	0.2071	0.0020

. This shows that, for example, the variation of ${\underline{U}}_{V1}$ has a much stronger influence on the state variable ${\underline{i}}_1$ than ${\underline{U}}_{V2}$ or even ${\underline{U}}_{V3}$, whereas ${\underline{U}}_{V2,0}$ has a considerable impact on ${\underline{i}}_{K2}$ and ${\underline{i}}_2$.

According to Equation (19), it should therefore be possible to deduce a y-percent change in the state variables from an x-percent change in the sources (input variables), since there is a linear correlation between them. This should be demonstrated, for example, by varying the source $V_2$ in a range between $\pm 20\%$ with an equidistant step size of $\Delta U_{V2}=10\%$.

3.2.2. Relationship between State Variables and System Matrix

As shown above, it is possible to establish a relationship between the input and state variables. The results from Table 2 and Table 3 show how a variation of the sources affects the corresponding system variables. In the following, we will investigate whether it is possible to make a quantitative statement directly from the K-matrix as to how strongly which source influences a state variable. For this purpose, the state variables $|{\underline{\mathbf{z}}}_{1\dots 3}|$ resulting from the corresponding voltages ${\underline{U}}_{V1,0\dots V3,0}$ are to be compared with each other. This should be carried out according to the example of the state variable $u_{C1}$, which results from the calculation of its row vector (row 2), as follows:

$$\begin{array}{cc}\hfill |{\underline{u}}_{C1}|& =|\sum _{k=1}^n{\underline{\mathbf{K}}}_{(2,k)}·{\underline{\mathbf{u}}}_{\left(k\right)}|\hfill \\ \hfill & =\left[\begin{array}{cccccccc}0.8489& 3.3681& 3.3345& 0.1511& 3.3234& 0.0503& 0.1009& 0.0503\end{array}\right]·\left|\underline{\mathbf{u}}\right|\hfill \\ \hfill & =0.8489·|{\underline{U}}_{V1}|-0.0503·|{\underline{U}}_{V2}|-0.1009·|{\underline{U}}_{V3}|\hfill \end{array}$$

(23)

Here, the numbers in parentheses represent the respective row and column index of the matrix, or the row index of the vector.

Equation (23) can be used as a first approximation to determine the influence of the sources onto the state variable. If one compares the factor of ${\underline{U}}_{V1}$ in Equation (23) with the value of ${\underline{u}}_{CE1}$ in Table 3 for $|\Delta {\underline{z}}_1|$, ${\underline{U}}_{V1}$ has the strongest influence on the state variable in both cases; however, according to Table 3, the influence of ${\underline{U}}_{V3}$ with $0.00169$% is less than that of ${\underline{U}}_{V2}$ with $0.0092$, which does not correspond to Equation (23). A direct conclusion regarding the influence of the sources on the state variables via the impedance matrix is therefore not possible.

Another possibility would be to relate each element of the K-matrix to the sum of the elements of its respective associated row matrix and to thereby assign a weighting factor. The ratios within the system matrix then result in

$${\underline{\mathbf{K}}}_{verh(i,k)}={\displaystyle \frac{{\underline{\mathbf{K}}}_{(i,k)}}{{\sum }_{k=1}^n{\underline{\mathbf{K}}}_{(i,k)}}}$$

(24)

with row index $i=1,\dots ,n$ and column index $k=1,\dots ,n$. A possible dependency will also be discussed here using the example of the state variable ${\underline{u}}_{C1}$.

$$\left|{\underline{\mathbf{K}}}_{verh,(2,:)}\right|=\left[\begin{array}{cccccccc}0.0807& 0.3202& 0.3170& 0.0144& 0.3159& 0.0048& 0.0096& 0.0048\end{array}\right]$$

(25)

Equation (19) can be used to simplify (25) back to

$$\left|{\underline{u}}_{CE1}\right|=0.0807·|{\underline{U}}_{V1}|-0.0048·|{\underline{U}}_{V2}|-0.0096·|{\underline{U}}_{V3}|$$

(26)

Similar to the previous example, according to Equation (26), the influence of ${\underline{U}}_{V3}$ should be greater than that of ${\underline{U}}_{V2}$, but this is not shown in the results of Table 3, which means that it is not possible to draw a direct correlation between the influence of the input variables and the state variables.

3.2.3. Calculating the Sensitivity Matrix

As shown in Section 3.2.1, the input variables of the $\underline{\mathbf{u}}$ vector are linearly linked to the state variables according to Equation (19) via the control matrix $\underline{\mathbf{K}}$, so that corresponding factors can be formed according to Table 3. In order to be able to establish this relationship for all $\Delta \underline{\mathbf{u}}$, the so-called sensitivity matrix $\mathbf{S}$ is introduced below. With this, it is possible to determine the weighted influence of $\Delta \underline{\mathbf{u}}$ on $\Delta \underline{\mathbf{z}}$ directly. Only the information of one operating point is required for the calculation. It will be calculated using the example network from Figure 3. For this purpose, it is examined how a change in the sources affects the values of the system variables. As an example, the change in voltage ${\underline{u}}_{C1}$ is considered here if the source voltage ${\underline{U}}_{V1}$ changes by 10% from an initial state ${\underline{U}}_{V1,0}$ to a value ${\underline{U}}_{V1,1}$. Assuming a constant $\underline{\mathbf{K}}$ matrix, the change in the system variable $\Delta \underline{\mathbf{z}}$ will be as follows. Since only one system variable will be considered here for illustration purposes, only the corresponding line vector (line 2) of Equation (19) must be evaluated. This results in the following:

$${\underline{\mathbf{z}}}_{\left(2\right)}=\left[\begin{array}{cccccccc}{\underline{\mathbf{K}}}_{(2,1)}& {\underline{\mathbf{K}}}_{(2,2)}& {\underline{\mathbf{K}}}_{(2,3)}& {\underline{\mathbf{K}}}_{(2,4)}& {\underline{\mathbf{K}}}_{(2,5)}& {\underline{\mathbf{K}}}_{(2,6)}& {\underline{\mathbf{K}}}_{(2,7)}& {\underline{\mathbf{K}}}_{(2,8)}\end{array}\right]·\left[\begin{array}{c}{\underline{U}}_{V1}\\ 0\\ 0\\ 0\\ 0\\ 0\\ -{\underline{U}}_{V3}\\ -{\underline{U}}_{V2}\end{array}\right]$$

(27)

If you now want to evaluate the change in the state variables, the following applies:

$$\begin{array}{cc}\hfill \Delta \underline{{\mathbf{z}}_{\left(2\right)}}& ={\displaystyle \frac{{\underline{\mathbf{z}}}_{1\left(2\right)}-{\underline{\mathbf{z}}}_{0\left(2\right)}}{{\underline{\mathbf{z}}}_{0\left(2\right)}}}\hfill \\ \hfill & ={\displaystyle \frac{({\underline{\mathbf{K}}}_{(2,1)}·{\underline{U}}_{V1,1}-{\underline{\mathbf{K}}}_{(2,7)}·{\underline{U}}_{V3,0}-{\underline{\mathbf{K}}}_{(2,8)}·{\underline{U}}_{V2,0})-({\underline{\mathbf{K}}}_{(2,1)}·{\underline{U}}_{V1,0}-{\underline{\mathbf{K}}}_{(2,7)}·{\underline{U}}_{V3,0}-{\underline{\mathbf{K}}}_{(2,8)}·{\underline{U}}_{V2,0})}{{\underline{\mathbf{K}}}_{(2,1)}·{\underline{U}}_{V1,0}-{\underline{\mathbf{K}}}_{(2,7)}·{\underline{U}}_{V3,0}-{\underline{\mathbf{K}}}_{(2,8)}·{\underline{U}}_{V2,0}}}\hfill \\ \hfill & ={\displaystyle \frac{{\underline{\mathbf{K}}}_{(2,1)}·1,1·{\underline{U}}_{V1,0}-{\underline{\mathbf{K}}}_{(2,7)}·{\underline{U}}_{V3,0}-{\underline{\mathbf{K}}}_{(2,8)}·{\underline{U}}_{V2,0}-{\underline{\mathbf{K}}}_{(2,1)}·{\underline{U}}_{V1,0}+{\underline{\mathbf{K}}}_{(2,7)}·{\underline{U}}_{V3,0}+{\underline{\mathbf{K}}}_{(2,8)}·{\underline{U}}_{V2,0}}{{\underline{\mathbf{K}}}_{(2,1)}·U_{V1,0}-{\underline{\mathbf{K}}}_{(2,7)}·{\underline{U}}_{V3,0}-{\underline{\mathbf{K}}}_{(2,8)}·{\underline{U}}_{V2,0}}}\hfill \\ \hfill & =\underset{=\Delta {\underline{u}}_1}{\underbrace{(1,1-1)}}·\underset{={\underline{\mathbf{S}}}_{(2,1)}}{\underbrace{{\displaystyle \frac{{\underline{\mathbf{K}}}_{(2,1)}·{\underline{U}}_{V1,0}}{{\underline{\mathbf{K}}}_{(2,1)}·{\underline{U}}_{V1,0}-{\underline{\mathbf{K}}}_{(2,7)}·{\underline{U}}_{V3,0}-{\underline{\mathbf{K}}}_{(2,8)}·{\underline{U}}_{V2,0}}}}}\hfill \end{array}$$

(28)

For a ${\mathbf{K}}^{n\times n}$ matrix, all factors of the ${\mathbf{S}}^{n\times n}$ matrix can be calculated as follows. For all $i=1,\dots ,n$ and $k=1,\dots ,n$, the following applies:

$${\underline{\mathbf{S}}}_{(i,k)}={\displaystyle \frac{{\underline{\mathbf{K}}}_{(i,k)}·{\underline{\mathbf{u}}}_{0\left(k\right)}}{{\sum }_{k=1}^n{\underline{\mathbf{K}}}_{(i,k)}·{\underline{\mathbf{u}}}_{0\left(k\right)}}}$$

(29)

Using the scaling matrix $\underline{\mathbf{S}}$, it is thus possible to represent the percentage influence of any change in the input variables $\Delta \underline{\mathbf{u}}$ on the state variables $\Delta \underline{\mathbf{z}}$, which are to be examined. Here, $\Delta \underline{\mathbf{u}}$ refers to the vector ${\underline{\mathbf{u}}}_0$, which contains the values for the current operating point.

$$\Delta {\underline{\mathbf{z}}}_S=\underline{\mathbf{S}}·\Delta \underline{\mathbf{u}}$$

(30)

Using the example from Figure 3, the $\underline{\mathbf{S}}$ matrix results are as follows:

$$\left|\underline{\mathbf{S}}\right|=\left[\begin{array}{cccccccc}3.1444& 0& 0& 0& 0& 0& 0.3147& 1.8298\\ 0.8918& 0& 0& 0& 0& 0& 0.0159& 0.0924\\ 0.8854& 0& 0& 0& 0& 0& 0.0168& 0.0978\\ 3.1438& 0& 0& 0& 0& 0& 0.3146& 1.8293\\ 0.8799& 0& 0& 0& 0& 0& 0.0167& 0.1035\\ 1.0502& 0& 0& 0& 0& 0& 0.0200& 2.0702\\ 1.0514& 0& 0& 0& 0& 0& 0.1675& 0.1162\\ 1.0508& 0& 0& 0& 0& 0& 0.0200& 2.0708\end{array}\right]$$

(31)

If you calculate the change $\Delta \underline{\mathbf{z}}$ for a 10 % change in ${\underline{U}}_{V1}$ according to Equation (30), you get

$$\begin{array}{cc}\hfill \left|\Delta {\underline{\mathbf{z}}}_S\right|& =\left|\underline{\mathbf{S}}\right|·{\left[\begin{array}{cccccccc}0.1& 0& 0& 0& 0& 0& 0& 0\end{array}\right]}^T\hfill \\ \hfill & ={\left[\begin{array}{cccccccc}0.3144& 0.0892& 0.0885& 0.3144& 0.0880& 0.1050& 0.1051& 0.1051\end{array}\right]}^T\hfill \end{array}$$

(32)

For validation, the result from (32) is compared with the results from Table 3.

$$\begin{array}{cc}\hfill \Delta \left|{\underline{\mathbf{z}}}_{Diff}\right|& =\left|\Delta {\underline{\mathbf{z}}}_S\right|-\left|\Delta {\underline{\mathbf{z}}}_3\right|\hfill \\ \hfill & ={\left[\begin{array}{cccccccc}0.1887& 0.0028& 0.0014& 0.2279& 0.0111& 0.0076& 0.0056& 0.0056\end{array}\right]}^T·{10}^{-14}\hfill \end{array}$$

(33)

In the context of calculation accuracy, it can thus be shown that it is possible to use the $\underline{\mathbf{S}}$ matrix to calculate the impact caused by the change of an input variable to the change of the state variable and to directly quantitatively evaluate the influence of the sources by evaluating the elements of the $\underline{\mathbf{S}}$ matrix.

3.3. Analysis of the Validity of the Sensitivity Matrix

In the following, the range in which the sensitivity matrix is valid will be analyzed. This will be examined for various operating points at which both the power and the impedance ratio are changed.

3.3.1. Variation of the Nominal Power

To ensure that the validity of the $\underline{\mathbf{S}}$ matrix does not depend on the operating point, the calculations should be compared with each other for a varied power specification. It is also of interest how the reactive power affects the sensitivity matrix. For this purpose, the values are first calculated at a constant P/Q ratio and then, in order to be able to investigate the influence of the reactive power, at a changed P/Q ratio.

Constant P/Q Ratio

First, the active and reactive powers should be varied using the $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$Q$}\right.$ ratio of $\raisebox{1ex}{$10$}\!\left/ \!\raisebox{-1ex}{$1$}\right.$ set at operating point AP1. If the power of the sources is increased to $P_2=-3\mathrm{kW}$, $Q_2=-300\mathrm{var}$, $P_3=10\mathrm{kW}$ and $Q_3=1\mathrm{kvar}$, as shown in Figure 3, the new state variables according to Equation (34) are obtained as follows:

$$\left|{\underline{\mathbf{z}}}_{AP2}\right|={\left[\begin{array}{cccccccc}10.3133& 230.0000& 227.7088& 10.3143& 228.6855& 4.3955& 14.7116& 4.3946\end{array}\right]}^T$$

(34)

and the input variables $|{\underline{\mathbf{u}}}_{AP2}|$ result in

$$|{\underline{\mathbf{u}}}_{AP2}|={\left[\begin{array}{cccccccc}270.9181& 0& 0& 0& 0& 0& 258.6247& 519.2394\end{array}\right]}^T$$

(35)

The resulting state variables can also be calculated using the $\underline{\mathbf{S}}$ matrix. The calculation is then carried out according to Equation (30), where $\Delta \underline{\mathbf{u}}$ results from the percentage deviation of the resulting voltages in AP2 from the voltages in AP1.

$$\left|\Delta \underline{\mathbf{u}}\right|=\left|{\displaystyle \frac{{\underline{\mathbf{u}}}_{AP2}}{{\underline{\mathbf{u}}}_{AP1}}}-1\right|=\left[\begin{array}{c}0.1213\\ 0\\ 0\\ 0\\ 0\\ 0\\ 8.1418\\ 0.2287\end{array}\right]$$

(36)

From this, $\Delta {\underline{\mathbf{z}}}_{AP2,S}$ results in

$$|\Delta {\underline{\mathbf{z}}}_{AP2,S}|={\left[\begin{array}{cccccccc}2.5248& 0.0000& 0.0079& 2.5241& 0.0063& 0.5085& 1.5181& 0.5086\end{array}\right]}^T$$

(37)

and finally $|{\underline{\mathbf{z}}}_{AP2}|$ in

$$\begin{array}{cc}\hfill |{\underline{\mathbf{z}}}_{AP2,S}|& =|{\underline{\mathbf{z}}}_{AP1}·(\Delta {\underline{\mathbf{z}}}_{AP2,S}+1)|\hfill \\ \hfill & ={\left[\begin{array}{cccccccc}10.3133& 230.0000& 227.7088& 10.3143& 228.6855& 4.3955& 14.7116& 4.3946\end{array}\right]}^T\hfill \end{array}$$

(38)

The comparison of the results from (34) and (38) shows that the calculations are the same in terms of the calculation accuracy, and it is therefore possible to calculate the state variables at different operating points using the $\mathbf{S}$ matrix.

Changed P/Q Ratio

The next step is to investigate the possibility of correctly accounting for the influence of reactive power with the help of the $\mathbf{S}$ matrix. For this purpose, the active power P is once again set according to Table 1, but the ratio $\raisebox{1ex}{$\mathrm{P}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{Q}$}\right.$ is set to $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$1$}\right.$. This results in $Q_2=-1\mathrm{kvar}$ and $Q_3=2\mathrm{kvar}$. At this operating point, the state variables are calculated as follows:

$$|{\underline{\mathbf{z}}}_{AP3}|={\left[\begin{array}{cccccccc}3.2468& 230.0000& 229.2095& 3.2508& 229.9990& 3.2448& 6.5037& 3.2407\end{array}\right]}^T$$

(39)

and, accordingly, the voltages are set to

$$|{\underline{\mathbf{u}}}_{AP3}|={\left[\begin{array}{cccccccc}242.1243& 0& 0& 0& 0& 0& 81.5213& 436.9472\end{array}\right]}^T$$

(40)

To check whether the corresponding state variables can also be calculated from the changes in the voltages, we first calculate the changes in the input variables $\Delta \underline{\mathbf{u}}$ again, as follows:

$$\left|\Delta {\underline{\mathbf{u}}}_{AP3}\right|=\left|{\displaystyle \frac{{\underline{\mathbf{u}}}_{AP3}}{{\underline{\mathbf{u}}}_{AP1}}}-1\right|=\left[\begin{array}{c}0.0191\\ 0\\ 0\\ 0\\ 0\\ 0\\ 2.1259\\ 0.1814\end{array}\right]$$

(41)

and, finally, this is used to calculate the state variables.

$$\begin{array}{cc}\hfill |{\underline{\mathbf{z}}}_{AP3,S}|& =|{\underline{\mathbf{z}}}_{AP1}·(\Delta {\underline{\mathbf{z}}}_{AP3,S}+1)|\hfill \\ \hfill & ={\left[\begin{array}{cccccccc}3.2468& 230.0000& 229.2095& 3.2508& 229.9990& 3.2448& 6.5037& 3.2407\end{array}\right]}^T\hfill \end{array}$$

(42)

As shown before, the comparison of the results of (39) and (42) shows that the calculations are the same in terms of their calculation accuracy, i.e., the ratio of $\raisebox{1ex}{$\mathrm{P}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{Q}$}\right.$ does not matter. This also shows that the complex $\underline{\mathbf{S}}$ matrix can also be used to correctly represent complex variables such as reactive power.

3.3.2. Variation of the $R_k/X_k$ Ratio

The next step is to test the extent to which the $\underline{\mathbf{S}}$ matrix remains valid even after changes to $\underline{\mathbf{K}}$ (e.g., due to switching operations). For this purpose, $\raisebox{1ex}{$\mathrm{P}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{Q}$}\right.$ is changed back to $\raisebox{1ex}{$10$}\!\left/ \!\raisebox{-1ex}{$1$}\right.$ and the $\raisebox{1ex}{$\mathrm{P}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{X}$}\right.$ ratio of the short-circuit impedances is changed to $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$1$}\right.$. This leads to the following change in the elements:

$$\begin{array}{cc}\hfill R_1& =3.5470\hfill \\ \hfill L_1& =1.7777\hfill \\ \hfill R_2& =59.1074\hfill \\ \hfill L_2& =29.6243\hfill \\ \hfill R_3& =29.5537\hfill \\ \hfill L_3& =14.8121\hfill \end{array}$$

(43)

For these values, the following state vector results for this operating point, according to Equations (18) and (19), are

$$|{\underline{\mathbf{z}}}_{AP4}|={\left[\begin{array}{cccccccc}10.3133& 230.0000& 227.7088& 10.3143& 228.6855& 4.3955& 14.7116& 4.3946\end{array}\right]}^T$$

(44)

and the resulting input vector

$$|{\underline{\mathbf{u}}}_{AP4}|={\left[\begin{array}{cccccccc}268.6435& 0& 0& 0& 0& 0& 285.4331& 510.7429\end{array}\right]}^T$$

(45)

With $|{\underline{\mathbf{u}}}_{AP4}|$, the resulting state variables can be determined using the S-matrix, analogous to Equations (36) and (38).

$$|{\underline{\mathbf{z}}}_{AP4,S}|={\left[\begin{array}{cccccccccc}10.3315& 229.8905& 228.1133& & 10.3292& 228.8655& 4.3915& & 14.7159& 4.3939\end{array}\right]}^T$$

(46)

As expected, with deviations of up to 36% between Equations (44) and (46), it is therefore not possible to calculate the state variables using the $\mathbf{S}$ matrix of the first operating point $AP1$. Instead, the matrix must be set up again according to Equation (29).

$$|{\underline{\mathbf{S}}}_{AP3}|=\left[\begin{array}{cccccccc}0.9900& 0& 0& 0& 0& 0& 0.7032& 0.6269\\ 0.9919& 0& 0& 0& 0& 0& 0.1251& 0.1115\\ 0.9910& 0& 0& 0& 0& 0& 0.1341& 0.1195\\ 0.9903& 0& 0& 0& 0& 0& 0.7031& 0.6268\\ 0.9831& 0& 0& 0& 0& 0& 0.1331& 0.1268\\ 0.7727& 0& 0& 0& 0& 0& 0.1046& 1.6578\\ 0.4640& 0& 0& 0& 0& 0& 0.5241& 0.0560\\ 0.7738& 0& 0& 0& 0& 0& 0.1047& 1.6580\end{array}\right]$$

(47)

Starting from this $\underline{\mathbf{S}}$ matrix, it is then possible to carry out the same calculations as shown above. It can therefore be seen that a $\underline{\mathbf{S}}$ matrix is only valid if $\underline{\mathbf{K}}$ remains constant.

3.3.3. Examining the System Using Current Sources

In order to check the general validity of the $\underline{\mathbf{S}}$ matrix, the influence of current sources should also be investigated. For this purpose, the voltage sources $V_2$ and $V_3$ from Figure 3 are replaced by the current sources, $I_2$ and $I_3$. The specified powers should initially remain the same. The resulting matrices are as follows.

$$\mathbf{R}=\left[\begin{array}{cccccc}{R}_1& 1& 0& 0& 0& 0\\ -1& 0& 0& 1& 0& 0\\ 0& 0& 0& -1& 0& 1\\ 0& -1& 1& R_{K1}& 0& 0\\ 0& 0& 0& 0& 0& -1\\ 0& 0& -1& 0& 1& R_{K2}\end{array}\right]$$

(48)

$$\mathbf{X}=\left[\begin{array}{cccccc}{L}_1& 0& 0& 0& 0& 0\\ 0& 1/C_1& 0& 0& 0& 0\\ 0& 0& 1/C_3& 0& 0& 0\\ 0& 0& 0& L_{K1}& 0& 0\\ 0& 0& 0& 0& 1/C_2& 0\\ 0& 0& 0& 0& 0& L_{K2}\end{array}\right]$$

(49)

$$\underline{\mathbf{u}}=\left[\begin{array}{c}{\underline{U}}_{V1}\\ 0\\ {\underline{I}}_3\\ 0\\ {\underline{I}}_2\\ 0\end{array}\right]\underline{\mathbf{z}}=\left[\begin{array}{c}{\underline{i}}_1\\ {\underline{u}}_{C1}\\ {\underline{u}}_{C3}\\ {\underline{i}}_{K1}\\ {\underline{u}}_{C2}\\ {\underline{i}}_{K2}\end{array}\right]$$

(50)

As the impedances of the grid have changed, and the $\underline{\mathbf{S}}$ matrix must be recalculated according to Equation (29). With this new matrix, the results should be calculated analogously to

Table 4. Results of the calculations of $\Delta \underline{\mathbf{z}}$ for different variations of the source $V_2$. The results between two adjacent columns differ by $\Delta {\underline{U}}_{V2}$ in each case.

	$0.8\mathit{·}{\underline{\mathit{U}}}_{\mathit{V}2}$	$0.9\mathit{·}{\underline{\mathit{U}}}_{\mathit{V}2}$	$1.0\mathit{·}{\underline{\mathit{U}}}_{\mathit{V}2}$	$1.1\mathit{·}{\underline{\mathit{U}}}_{\mathit{V}2}$	$1.2\mathit{·}{\underline{\mathit{U}}}_{\mathit{V}2}$	$\mathit{\Delta }{\underline{\mathit{U}}}_{\mathit{V}2}$
$|\Delta {\underline{i}}_1|$	0.366	0.183	0	0.183	0.366	0.183
$|\Delta {\underline{u}}_{c1}|$	0.0185	0.0092	0	0.0092	0.0185	0.0092
$|\Delta {\underline{u}}_{c3}|$	0.0196	0.0098	0	0.0098	0.0196	0.0098
$|\Delta {\underline{i}}_{k1}|$	0.3659	0.1829	0	0.1829	0.3659	0.1829
$|\Delta {\underline{u}}_{c2}|$	0.0207	0.0103	0	0.0103	0.0207	0.0103
$|\Delta {\underline{i}}_{k1}|$	0.414	0.207	0	0.207	0.414	0.207
$|\Delta {\underline{i}}_3|$	0.0232	0.0116	0	0.0116	0.0232	0.0116
$|\Delta {\underline{i}}_2|$	0.4142	0.2071	0	0.2071	0.4142	0.2071

and Equation (30). As can be seen from

Table 5. Results of the calculations of $\Delta \underline{\mathbf{z}}$ and $\Delta {\underline{\mathbf{z}}}_S$ for different variations of the source ${\underline{I}}_2$.

	$0.8\mathit{·}{\underline{\mathit{I}}}_2$		$0.9\mathit{·}{\underline{\mathit{I}}}_2$		$1.0\mathit{·}{\underline{\mathit{I}}}_2$		$1.1\mathit{·}{\underline{\mathit{I}}}_2$		$1.2\mathit{·}{\underline{\mathit{I}}}_2$
	$\Delta \underline{\mathbf{z}}$	$\Delta {\underline{\mathbf{z}}}_{\mathbf{S}}$	$\Delta \underline{\mathbf{z}}$	$\Delta {\underline{\mathbf{z}}}_{\mathbf{S}}$	$\Delta \underline{\mathbf{z}}$	$\Delta {\underline{\mathbf{z}}}_{\mathbf{S}}$	$\Delta \underline{\mathbf{z}}$	$\Delta {\underline{\mathbf{z}}}_{\mathbf{S}}$	$\Delta \underline{\mathbf{z}}$	$\Delta {\underline{\mathbf{z}}}_{\mathbf{S}}$
$|{\underline{i}}_1|$	0.2014	0.2014	0.1007	0.1007	0	0	0.1007	0.1007	0.2014	0.2014
$|{\underline{u}}_{c1}|$	0.0112	0.0112	0.0056	0.0056	0	0	0.0056	0.0056	0.0112	0.0112
$|{\underline{u}}_{c3}|$	0.0118	0.0118	0.0059	0.0059	0	0	0.0059	0.0059	0.0118	0.0118
$|{\underline{i}}_{k1}|$	0.2013	0.2013	0.1006	0.1006	0	0	0.1006	0.1006	0.2013	0.2013
$|{\underline{u}}_{c2}|$	0.0124	0.0124	0.0062	0.0062	0	0	0.0062	0.0062	0.0124	0.0124
$|{\underline{i}}_{k2}|$	0.1999	0.1999	0.1	0.1	0	0	0.1	0.1	0.1999	0.1999

, the results for the calculations of the changes in the state values via the $\mathbf{S}$ matrix are also the same here as those via the calculation of $\Delta \underline{\mathbf{z}}$. This means that these calculations are possible regardless of the type of source.

4. Conclusions

An approach was presented that makes it possible to analyze uncontrolled electrical energy networks in the state space representation using a sensitivity matrix for arbitrary operating points. This can be applied regardless of the active and reactive powers source types, as long as the admittance matrix (${\underline{\mathbf{K}}}^{-1}$) remains constant. If the value changes, e.g., due to switching operations, the sensitivity matrix must be recalculated. The required matrix can be calculated very easily directly from the resulting states of an operating point, without the need for additional supporting points. Any changes made to the input variables $\Delta \underline{\mathbf{u}}$ can also be taken into account. Based on the $\underline{\mathbf{S}}$ matrix, it is therefore quite easy to determine the dependencies of the input parameters (sources) on the grid under investigation, both via a calculation and via a direct comparison of the resulting factors, as these reflect the influence in a weighted manner. With this information, it is also possible to evaluate the controllability of the system, as well as the observability, using appropriately positioned measuring devices that contain output equations. It also appears possible to use the matrix for controlled input variables (e.g., PQ sources). We will investigate this and possibly other aspects in a future study.