Control problems in

low-inertia power systems

Scenario

  • Fewer synchronous generators to provide
    • "electro-mechanical" inertia
    • real-time power balancing
  • More generation from renewable sources
    • fluctuating power generation
    • reduced reliability in case of large disturbances
  • More power converters
    • higher flexibility
    • hard operational constraints

Typical frequency response analysis

Based on electro-mechanical synchronous generators

  • Definition of angles / frequency
    • Rotor angle / speed
  • Specifications
    • Generators' frequency / ROCOF limits
  • Dynamical models
    • ​2nd order Swing equations and more

Still valid?

Definition of angle / frequency

Three-phase voltages

u_a(t), u_b(t), u_c(t)

Balanced operation

u_a(t) + u_b(t) + u_c(t) = 0

Vector notation

\mathbf{u}(t) = \frac{2}{3} \left[ u_a(t) + u_b(t) e^{j \frac{2\pi}{3}} + u_c(t) e^{j \frac{4\pi}{3}} \right]
\mathbf{u}(t) \in \mathbb{C}

Vector notation (rotating frame)

Synchronous power system

\theta_{dq} = \int_0^t \omega d\tau = \omega t
\mathbf{u^{dq}}(t) = \mathbf{u}(t) e^{-j \theta_{dq}}
\begin{bmatrix} u_d \\ u_q \end{bmatrix} = \begin{bmatrix} \cos \theta_{dq} & \sin \theta_{dq} \\ - \sin \theta_{dq} & \cos \theta_{dq} \end{bmatrix} \begin{bmatrix} u_\alpha \\ u_\beta \end{bmatrix}
\mathbf{u}(t) = U e^{j (\theta_0 + \omega t)}
\mathbf{u^{dq}}(t) = U e^{j \theta_0}

Phasor notation

Time varying frequency

\mathbf{u}(t) = U(t) e^{j \theta(t)}
\theta(t) = \theta_0 + \int_0^t \omega(\tau) d\tau
  • No sinusoidal assumption
  • At each time, magnitude and angle are well defined
  • We can use the vector notation in ODEs
  • Frequency: tracking a ramp signal

Time varying frequency

u_a, u_b, u_c
T_{abc}^{dq}
\hat \theta(t)
u_d
u_q
\hat U(t)
\frac{k_1}{s}
k_2
\frac{1}{s}
\hat \omega(t)

Specifications

Frequency response to a fault

  • "Features" inspired by electro-mechanical dynamics
    • ROCOF (inertia)
    • bounded frequency deviation (primary control)
    • frequency restoration (secondary control)

Specifications

Frequency response to a fault

  • "Features" inspired by electro-mechanical dynamics
    • ROCOF (inertia)
    • bounded frequency deviation (primary control)
    • frequency restoration (secondary control)

Not very different from step response analysis

of a PID controller (and a nonlinear plant)

P

I

D

\displaystyle p_G = p_G^\text{ref} + k_D \dot \delta \omega + k_P \delta \omega + k_I \int \delta \omega

Specifications

Small-signal amplification

  • Disturbance model
    • Power demand of loads
    • Fluctuation of renewable generation
  • Amplification from disturbance to output
    • Norm of frequency deviation (which norm?)

Frequency in low inertia power systems

  • Definition of angles / frequency
     
  • Specifications
    • ​Response to large disturbances
    • Small signal response (system norms)
       
  • Dynamical models
    • ​Phasor-free models