## 2 System Dynamics  
Figure 1 shows a power system with \(N\) control areas [1].  
The model of frequency response for any area i of N power system control areas can be described as [2]:  
\[\dot{\Delta f_{t}}=\left(\frac{1}{2H_{i}}\right)\Delta P_{mi}-\left(\frac{1}{2H_{ i}}\right)\Delta P_{Li}-\left(\frac{D_{i}}{2H_{i}}\right)\Delta f_{i}-\left( \frac{1}{2H_{i}}\right)\Delta P_{tie,i} \tag{1}\]  
While the dynamic of the governor can be expressed as:  
\[\Delta\dot{P}_{mi}=\left(\frac{1}{T_{ti}}\right)\Delta P_{gi}-\left(\frac{1}{T _{ti}}\right)\Delta P_{mi} \tag{2}\]  
And the dynamic of the turbine can be expressed as:  
\[\Delta\dot{P}_{gi}=\left(\frac{1}{T_{gi}}\right)\Delta P_{ci}-\left(\frac{1}{ R_{i}T_{gi}}\right)\Delta f_{i}-\left(\frac{1}{T_{gi}}\right)\Delta P_{gi} \tag{3}\]  
The total tie-line power change between area \(i\) and the other areas can be calculated as:  
\[\dot{\Delta P}_{tie,i}=2\pi.\left[\sum_{\begin{subarray}{c}j=1\\ j=i\end{subarray}}^{N}T_{ij}\Delta f_{i}-\sum_{\begin{subarray}{c}j=1\\ j\
eq i\end{subarray}}^{N}T_{ij}\Delta f_{j}\right] \tag{4}\]  
Where (\(\Delta\)) is a differential operator.  
Also, the frequency deviation is added to the tie-line flow deviation in the supplementary feedback control loop to maintain the net interchange power with neighboring areas according to scheduled values. The area control error (ACE) represents the linear combination of tie-line power changes and frequency for area i, and is known as,  
\[ACE_{i}=B_{i}\Delta f_{i}\ +P_{tie,i} \tag{5}\]  
Equations (1)-(4) represent the frequency response model for \(N\) power system control areas with one generator unit in each area, and can be combined in the following state space model:  
Figure 1: Dynamic model of a control area in an interconnected environment.  
Figure 2 shows the block diagram of a single-input single-output (SISO) linear time invariant system with CDM control.  
where:  
_D(s)_ is the denominator polynomial of the plant transfer function;  
_N(s)_ is the numerator polynomial;  
_A(s)_ is considered as the forward denominator polynomial;  
_F(s)_ and _B(s)_ are considered as the reference numerator and feedback numerator polynomials.  
In this method, d is the external disturbance signal, r is taken as the reference input to the system, u as the controller signal, and \(y\) is the output of the control system.  
\[y=\frac{N(s)F(s)}{P(s)}r+\frac{A(s)N(s)}{P(s)}d \tag{7}\]  
Where \(P(s)\) represents the characteristic polynomial of the closed-loop system and is defined by  
\[P(s)=A(s)D(s)+B(s)N(s) \tag{8}\]  
\(A\)(s) and \(B\)(s) are considered as the control polynomials and are defined as  
\[A(s)=\sum_{i=0}^{p}l_{i}s^{i}\;and\;B(s)=\sum_{i=0}^{q}k_{i}s^{i} \tag{9}\]  
For practical realization, the condition \(p\geq q\) must be satisfied. To get the characteristic polynomial _P(s)_, the controller polynomials from Equation (3) are substituted in Equation (2) and are given as  
\[P(s)=\sum_{i=0}^{p}l_{i}s^{i}\;D(s)+\sum_{i=0}^{q}k_{i}s^{i}\;N(s)=\sum_{i=0}^ {n}a_{i}s^{i}\;,a_{i}>0 \tag{10}\]  
Some parameters are needed for the design of the CDM, such as the stability indices (\(\gamma\)), the equivalent time constant (\(\tau\)), and the stability limits (\(\gamma_{i}^{*}\)). The relations between these parameters and the coefficients of the characteristic polynomial (a\({}_{i}\)) can be described as follows:  
\[\gamma_{i}=\frac{a_{i}^{2}}{a_{i+i}a_{i-1}}\;,i\in[1,n-1]\,,\gamma_{0}=\gamma _{n}=\infty \tag{11}\]  
Figure 2: A block diagram of a CDM control system.