Saverio Bolognani
Senior Scientist at ETH Zurich
Fast scenario-based decision making
in unbalanced distribution networks
Saverio Bolognani, Florian Dörfler
Automatic Control Laboratory
Future power distribution grids
From fit-and-forget to
active power distribution networks
- Worst case analysis at planning stage
(e.g. California "15% rule")
- A little bit of sensing and actuation
- Real time control
Distribution Network operation under uncertainty
Exogeneous inputs
- Power generation
- Power demands
- Substation voltage
Grid
state
x_\text{exo}
x
Control inputs
- Tap changer
- Volt/VAR control
- Voltage regulators
- Active power control
- Load shedding
- Generation curtailment
x_\text{control}
x \in \mathcal X
Operational constraints
- Over/under voltage limits
- Line current limits
- Transformer loading
- Voltage stability
Distribution network model
Grid state (nodal variables)
x = (v,\theta, p, q)
v_i = \begin{bmatrix}
v_i^a \\ v_i^b \\ v_i^c
\end{bmatrix}, \quad i = 1, \ldots, N
Power flow equations
v_h e^{j\theta_h} \sum_k \bar{Y}_{hk} v_k e^{-j\theta_k}- p_h - j q_h = 0 \quad \forall h
G(x) = 0
Implicit nonlinear model
Uncertain and control inputs
x_\text{exo} = d
Exogenous stochastic inputs
x_\text{control} = c
Control inputs available to the DNO
x_\text{exo} = \begin{bmatrix}
\cdots & p_i & q_i & \cdots
\end{bmatrix}
e.g. load
i
x_\text{control} = \begin{bmatrix}
\cdots & p_k & q_k & \cdots
\end{bmatrix}
e.g. microgenerator
k
V x \le w
Operational constraints
- Nodal constraints (under- / over- voltage, power limits)
-
Proxy for line capacity, transformer current limits, etc.
Chance constrained decision problem
\min \ \ \ J(c)
\text{subject to}\quad\text{Prob}_d\left[
Vx \le w\right] > 1 - \epsilon
Complexity of the problem: feasible region.
An analytical derivation of the feasible region is hopeless
- general probability distribution for
- nonlinear dependence of on and
\mathcal F = \left\{ c \,|\, \text{Prob}_d\left[
Vx \le w\right]
> 1-\epsilon \right\}
d
c
x
d
Cost of control
Probability of satisfying constraints
\left\{
\begin{matrix}
G(x) = 0\\x_\text{exo} = d\\x_\text{control}=c
\end{matrix}
\right.
Nonlinear model
Chance constraints in Power Systems
L. Roald, M. Vrakopoulou, F. Oldewurtel, G. Andersson (2014)
"Risk-Constrained Optimal Power Flow with Probabilistic Guarantees"
D. Bienstock, M. Chertkov, S. Harnett (2014)
"Chance-Constrained OPF: Risk-Aware Network Control under Uncertainty"
T. Summers, J. Warrington, M. Morari, J. Lygeros (2014)
"Stochastic OPF based on convex approximations of chance constraints"
M. Vrakopoulou, M. Katsampani, K. Margellos, J. Lygeros, G. Andersson (2013)
"Probabilistic security-constrained AC optimal power flow"
DC model /
power balance eqs.
Planning /
day ahead dispatch
N-1 criterion
replacement
Scenario approach
Approximate the constraint
via the set of constraints
where are samples (realizations) of
- Based on stochastic models (fitted to the data)
- Directly from the history of the system
\text{Prob}_d\left[
Vx(c,d) \le w\right] > 1-\epsilon
Vx(c,d^{(i)}) \le w \qquad \forall d^{(i)}, i = 1,\ldots, M(\epsilon)
d^{(i)}
d.
G.C. Calafiore, M.C. Campi (2006)
"The scenario approach to robust control design"
Scenario approach
How many?
\sum_{i=0}^{n_c-1}
{{M(\epsilon)}\choose{i}}
\epsilon^i
(1-\epsilon)^{M(\epsilon)-i}
\le \beta
\begin{matrix}
\epsilon & M(\epsilon) \\
0.1 & 143 \\
0.05 & 291 \\
0.01 & 1475
\end{matrix}
\beta = 10^{-3}, \quad n_c = 5
- Does not depend on the grid size
- Depends on the number of controls
Computational complexity
For each sample, compute
and then derive the approximate feasible region
\mathcal F^{(i)} = \left\{ c \,|\, Vx(c,d^{(i)}) \le w \right\}
\hat {\mathcal F} = \bigcap_{i} \mathcal F^{(i)}
Computing each feasible region based on NL grid eqs.
is a computationally very intensive task
K. Dvijotham, K. Turitsyn (2015)
"Construction of power flow feasibility sets"
Linear manifold approximant
G(x) = 0
Power flow manifold
x^* \ |\ G(x^*) = 0
Nominal state
S. Bolognani and F. Dörfler (2015)
“Fast power system analysis via implicit linearization of the power flow manifold”
\nabla G(x^*) ^T (x- x^*) = 0
Tangent space
A_{x^*} x = b_{x^*}
Under-determined
system of linear equations
Linear manifold approximant
- Sparse
- Structure preserving
- No assumption on
- X/R ratio
- constant voltage
- radial network
- Implicit linear relation
Compare with DC, LinDistFlow, ...
\left\{
\begin{matrix}
G(x) = 0\\x_\text{exo} = d\\x_\text{control}=c
\end{matrix}
\right.
\left\{
\begin{matrix}
Ax = b\\x_\text{exo} = d\\x_\text{control}=c
\end{matrix}
\right.
Sparse implicit linear model
Implicit nonlinear model
Linear manifold approximant
General case
Extended state
\tilde x = (v,\theta,\text{Re}(i), \text{Im}(i),p,q)
\tilde G(x) = \left[
\begin{matrix}
\tilde G^\text{edges}(\tilde x)\\
\tilde G^\text{nodes}(\tilde x)
\end{matrix}
\right]
\tilde G^\text{edges}(x) = Yu - i
\tilde G_h^\text{nodes}(x) = u_h \bar i_h - s_h
u_h := v_h e^{j\theta_h}
From complex-value to real-valued equations
f(x) = Ax
\frac{\partial f}{\partial x} = \langle A \rangle, \quad
\langle A \rangle := \begin{bmatrix} \text{Re}A & -\text{Im}A \\ \text{Im}A & \text{Re}A \end{bmatrix}
complex-valued
real-valued
\langle \bar A \rangle = N \langle A \rangle N, \quad
N := \begin{bmatrix} I & 0 \\ 0 & -I \end{bmatrix}
Linear manifold approximant
\frac{\partial
\begin{bmatrix}
\text{Re}(u) \\ \text{Im}(u)
\end{bmatrix}
}{
\partial
\begin{bmatrix}
v \\ \theta
\end{bmatrix}
}
=
P(v,\theta)
:=
\begin{bmatrix}
\text{diag}(\cos\theta) & -\text{diag}(v) \text{diag}(\sin\theta) \\
\text{diag}(\sin\theta) & \text{diag}(v) \text{diag}(\cos\theta)
\end{bmatrix}
Evaluate the partial derivatives of
\tilde G(\tilde x)
\left.
\frac{\partial \tilde G}{\partial \tilde x}
\right|_{\tilde x^*} =
\begin{bmatrix}
\langle Y \rangle P(v^*,\theta^*) & -I & 0 \\
N \langle \text{diag } i^* \rangle N P(v^*,\theta^*) & \langle \text{diag } v^* \rangle N & -I
\end{bmatrix}
\nabla G^T(x^*) =
\begin{bmatrix}
\Big( N \langle \text{diag } i^* \rangle N + \langle \text{diag } v^* \rangle N \langle Y \rangle \Big) P(v^*,\theta^*) &
-I
\end{bmatrix}
Eliminate currents and obtain
IEEE 13 Test feeder
- Relatively highly loaded
- Substation voltage regulator
- Overhead and underground lines
- Variety of phasing
- Shunt capacitor banks
- Unbalanced loads
Quality of the approximation
Approximate feasible region
\mathcal F^{(i)} = \left\{ c \,|\, Vx(c,d^{(i)}) \le w \right\}
\hat{\mathcal F}^{(i)} = \text{Proj}_c \hat{\mathcal F}_x^{(i)}
\hat{\mathcal{F}}_x^{(i)} = \left\{ x \,|\, \begin{matrix}
Ax = b \\
x_\text{exo} = d^{(i)} \\
Vx \le w
\end{matrix}
\right\}
Feasible
control inputs
i-th feasible region for the scenario approach
Feasible
states
Approximate feasible region
Linearization point matters!
- Zero-load profile
- Numerical solution in typical conditions
- Measured state
Polytope operations
\hat F^{(i)} = \text{Proj}_c \hat F_x^{(i)}
\hat F_x^{(i)} = \left\{ x \,|\, \begin{matrix}
Ax = b \\
x_\text{exo} = d^{(i)} \\
Vx \le w
\end{matrix}
\right\}
\hat F = \bigcap_i \hat F^{(i)}
-
Convenient representation for optimization
- Linear inequalities in reduced dimension
-
Many redundant constraints
- Fourier-Motzkin elimination
- Fast operations on polytopes
i = 1,\ldots, N
i = 1,\ldots, N
Chance-constrained feasible region
Disturbances
- Power demand of spot loads
Operational constraints
- Under- and over-voltage limits
Decision variables
- Generation curtailment @671
- Tap changer position @630
5% confidence
M(\epsilon) = 181
\epsilon = 0.05
Microgenerator
Tap changer
Simulations
MultiParametric Toolbox for Matlab by IfA
- Towards real time chance-constrained operation
- Next step:
- Slow data - Forecast of uncertain inputs (offline)
- Fast data - Real time measurements (online)
Thank you for your attention.
Fast scenario-based decision making in unbalanced distribution networks
By Saverio Bolognani
Fast scenario-based decision making in unbalanced distribution networks
PSCC 2016, Genova
