[PSA] Chapter 11 - Automatic Generation Control and the New Market Environment

11.0 Introduction

• AGC(Automatic generation control) - including
  1. Load frequency control
  2. economic dispatch

11.1 Power Control System Modeling

• Speed governor controls generator speed by adjusting steam valve position change
• Linearlizing governor linkage (steam valve position change is proportional to mechanical power change)
• Assuming small perturbation of every term

• Setting scaling factor $K_GK_T=1$, We can approximate turbine-govenor model

\[\Delta\hat P_M = \frac{1} {(1+T_Gs)(1+T_Ts)} \left( \Delta\hat P_C-\frac{1}{R}\Delta\hat\omega \right)\]

• For large-signal application, can add saturation that steam valve can move within close and open status

Related Droop Control equation

• For steady state, we can assume $s=0$, $G(0)=1$
• Fixing out power command $\Delta P_c=0$

\[\Delta\omega = -R\Delta P_M\]

• $R$ : Regulation, $1/R$ : Gain in governor block ($\Delta \omega \rightarrow \Delta P_M$)

11.2 Application to single machine-infinte bus system

• With speed governor, $\Delta P_M$ depends on $\Delta \omega$, $\Delta P_C$

\[\omega = \dot \theta = \omega_0 + \dot \delta \Rightarrow \Delta \omega = \Delta \dot \delta \tag{11.7}\]

11.3 Simplified Analysis of Power control system

• Two inputs : $\Delta V_{ref}$, $\Delta P_C$,
• Can take two outputs : $\Delta \vert V_a \vert$ and/or $\Delta \delta$
• If we are interested in transient analysis, we can completely ignore power control loop (since the feedback is too slow to react to transient change) (i.e. $\Delta P_M=0$)
• If we are interested in stpe response to $\Delta P_C$, with $\Delta V_{ref}=0$, we can easily leave out voltage control loop systems

11.4 Power control, Multigenerator case

• At each bus,

\[P_{Gi} = P_{Di}+P_i \tag{11.8}\]

• Assume an operating point,

\[P_{Gi}^0 = P_{Di}^0+P_i^0\tag{11.9}\]

Then we can get relationship between increment happening near operating point

\[\Delta P_{Gi} = \Delta P_{Di}+\Delta P_i\tag{11.10}\]

If we consider linearized model of swing equation,

\[M_i\Delta\ddot\delta_i + D_i \Delta \dot \delta_i + \Delta P_{Gi} = \Delta P_{Mi} \tag{11.11}\]

We can also linearize $P_{Di}$. Affected by both frequency and voltage magnitude, we can assume there’s no voltage magnitude change → no need to consider

• For small changes in frequency, we assume that

\[P_{Di} = P_{Di}^0 + {\partial P_{Di}(\omega^0) \over \partial \omega_i}\Delta \omega_i + \Delta P_{Li} \tag{11.13}\]

• $\Delta P_{Li}$ : External load added / removed (externally imposed load disturbance) (e.g. switching electronics)
• For Phase $a$, instantaneous voltage is

\[v_i(t) = \sqrt 2 \vert V_i \vert \cos{[\omega^0 t + \theta_i^0+\Delta\theta_i(t)]} \tag{11.13}\]

• $\omega_0 $ : Nominal frequency, $2\pi60$
• $\theta_i^0$ : angle of bus $i$ at operating point compared to slack bus
• Taking time derivative of the argument of the cosine function,

\[\omega_i = \omega^0 + \Delta \dot \theta_i \tag{11.14}\]

Thus

\[\Delta \omega_i = \omega_i-\omega^0 = \Delta \dot \theta _i \tag{11.15}\]

Define $D_{Li} \triangleq \partial P_{Di}(\omega^0) / \partial \omega_i$,

\[\Delta P_{Di} = D_{Li}\Delta\dot \theta_i + \Delta P_{Li} \tag{11.16}\] \[M_i\Delta \ddot \delta_i + D_i\Delta \dot \delta_i + D_{Li} \Delta \dot \delta_i + \Delta P_{Li} + \Delta P_i = \Delta P_{Mi} \tag{11.17}\]

• To calculate $\Delta P_i$, Assume that line admittance parameters are purely imaginary (lossless), get

\[P_i = \sum_{k=1}^N \vert V_i \vert \vert V_k \vert B_{ik} \sin(\theta_i-\theta_k) \tag{11.18}\]

With the same method above, define operation quantities and express them with delta

By simple trigonometry,

\[\begin{aligned} P_i^0+\Delta P_i = \sum_{k=1}^{n}|V_i||V_k||B_{ik}| \Big[ &\sin(\theta_i^0-\theta_k^0) \cos(\Delta\theta_i-\Delta\theta_k) \\ +&\cos(\theta_i^0-\theta_k^0) \sin(\Delta\theta_i-\Delta\theta_k) \Big] \end{aligned} \tag{11.19}\]

• Assuming small angle difference, $\cos{(\Delta\theta_i - \Delta \theta_k)}\approx 1, \sin(\Delta \theta_i-\Delta \theta_k) \approx \Delta\theta_i-\Delta \theta_k$

\[\Delta P_i = \sum_{k=1}^{n} \left[ |V_i||V_k||B_{ik}| \cos(\theta_i^0-\theta_k^0) \right] (\Delta\theta_i-\Delta\theta_k) \tag{11.20}\]

• Define Stiffness / Synchronizing Power Coefficient

\[T_{ik} \triangleq \vert V_i \vert \vert V_k \vert \vert B_{ik} \vert \cos{(\theta_i^0-\theta_k^0)}\]

Then

\[\Delta P_i = \sum_{k=1}^n T_{ik} (\Delta \theta_i-\Delta\theta_k) \tag{11.21}\] \[T_{ik} = \frac{\partial P_i} {\partial(\theta_i-\theta_k)}\]

• $T_{ik}$ : power flow stiffness or sensitivity against angle difference
• We can assume $\Delta \delta_i = \Delta \theta_i$ for small angle change, we can express

\[\Delta P_i = \sum_{k=1}^n T_{ik} (\Delta \delta_i-\Delta\delta_k)\]

• Merge two coefficients, $\tilde D_i \triangleq D_i + D_{Li}$

\[M_i\Delta\ddot{\delta}_i + \widetilde D_i\Delta\dot{\delta}_i + \Delta P_i = \Delta P_{Mi}-\Delta P_{Li} \tag{11.22}\]

• $M_i \Delta \ddot \delta_i$ : Power required ofr rotor acceleration
• $\tilde D_i \Delta \dot \delta_i$ : Damping effect of generator and load
• $\Delta P_i$ : Emitting power to the transmission network
• $\Delta P_{Mi}$ : Delta of turbine mechanical power