Safety Guarantees for the Electricity Grid with Significant Renewables Generation

Abstract

This work presents a study of the frequency dynamics of the electricity grid under significant presence of generation from renewable sources. A safety requirement, namely ensuring that frequency does not deviate excessively from a reference level, is formally studied by means of probabilistic model checking of a finite-state abstraction of the grid dynamics. The dynamics of the electric network comprise a model of the frequency evolution, which is in a feedback connection with a model of renewable power generation by a heterogeneous population of solar panels. Each panel switches independently between two states (ON and OFF) in response to frequency deviations, and the power generated by the population of solar panels affects the network frequency response. A power generation loss scenario is analysed and its consequences on the overall network are formally quantified in terms of probabilistic safety. We thus provide guarantees on the grid frequency dynamics under several scenarios of solar penetration and population heterogeneity.

Appendices

A Definition of Kernel Continuous Regions

We want to underline the discontinuity of the kernel density $t_{\omega }(\cdot |q)$ caused by the presence of the $\delta (\cdot )$ functions. Let us define $g(\varDelta f) = - \alpha _1 \varDelta f - \alpha _2 \varDelta \phi - \beta _1 \varDelta x - \beta _2 \varDelta \xi$, $h_1(x) = - (1-a)x - b \varepsilon y$, $h_2(y) = - b(1-x-\varepsilon y)$ and $l(P_{PV}) = - \bar{P}Nx$. The transition kernel density can be written as

$$\begin{aligned} t_{\omega }(q'|q) = \left\{ \begin{aligned}&t_{f}(\varDelta f' - g(\varDelta f) ) \cdot&\quad \text {if } \varDelta \phi ' = \varDelta f \wedge x' = h_1(x) \\&\quad \cdot t_{P}(P_{PV}' - l(P_{PV}))&\wedge \ y' = h_2(y) \wedge \xi ' = P_{PV} \\&0&\text {otherwise} \end{aligned} \right. \end{aligned}$$

defining the continuous regions $\mathcal {C} = \{ \varDelta \phi ' = \varDelta f \wedge x = h_1(x) \wedge y' = h_2(y) \wedge \xi ' = P_{PV} \}$. Note that in the abstraction framework, regions $\mathcal {C}$ assume the discretised form $\mathcal {C}_d = \{ \varDelta \bar{\phi } ' = \varDelta \bar{f}_i \wedge \bar{x} = h_1(\bar{x}) \wedge \bar{y}' = h_2(\bar{y}) \wedge \bar{\xi }' = \bar{P}_j \}.$

B Probabilistic Safety for Partially Degenerate Models

Let us show that for a partially degenerate stochastic model the safety probability computation depends only on the stochastic state. Consider the model

$$\begin{aligned} \left\{ \begin{aligned}&x(k+1) = f(z(k)) + \omega (k) \\&y(k+1) = x(k), \end{aligned} \right. \end{aligned}$$

where $\omega (k) \sim \mathcal {N}(0, \sigma )$ and where $z = (x, \ y)^T$ denotes the complete state vector. Let us denote with $t_{\omega }(\cdot )$ the density of the Gaussian kernel. The one-step transition probability kernel can be split as follows:

$$\begin{aligned} \begin{aligned} P(x(k+1)|z(k))&= t_{\omega }( f(z(k)) ), \\ P(y(k+1)|z(k))&= P(y(k+1)|x(k)) = \delta (y(k+1) - x(k)), \end{aligned} \end{aligned}$$

where $\delta (z-p)$ represent the Dirac delta function of variable z, centred at point p. Let us consider a safe set $A = A_x \times A_y$, where $A_x$ and $A_y$ denote its projections on variables x and y, respectively. Define the value function at time step H as $V_H(z) = \mathbf {1}_{A}(z)$ and compute the one-step backward recursion:

$$\begin{aligned} \begin{aligned}&V_{H-1}(z) = \int _{A} V_H(z') P(z'|z)dz' = \int _{A} P(z'|z) dz' = \\&= \int _{A_y} \int _{A_x} t_{\omega }(dx' | f(z)) \delta (dy'-x) = \mathbf {1}_{A_y}(z) \int _{A_x} t_{\omega }(dx'| f(z)) = \\&= \int _{A_x} t_{\omega }(dx'|f(z)), \end{aligned} \end{aligned}$$

showing that the computation of the safety probability depends solely on the stochastic kernel affecting the dynamics of variable x.

C Value Function Continuity for Probabilistic Safety

In the following, we consider a generation-loss incident scenario; the load-loss case can be derived analogously. Recall the value function definition from Sect. 3 and compute the backward Bellman equation as

$$\begin{aligned} V_k(q) = \mathbf {1}_{\mathcal {L}}(q) \int _{\mathcal {Q}} V_{k+1}(\tilde{q}) t_s(\tilde{q}|q) d\tilde{q}, \quad \text {with } V_H(q) = \mathbf {1}_{\mathcal {L}}(q) . \end{aligned}$$

We show that the value functions are continuous within the continuity regions of the state space, thus there must exist a constant $\gamma$ so that

$$\begin{aligned} |V_k(q) - V_k(\tilde{q})| \le \gamma \, \Vert q- \tilde{q} \Vert . \end{aligned}$$

(13)

To enhance the readability let us define $g(\varDelta f) = - \alpha _1 \varDelta f - \alpha _2 \varDelta \phi - \beta _1 \varDelta x - \beta _2 \varDelta \xi$, $h(P_{PV}) = - \bar{P}Nx$ and $\varDelta f = \rho$, $P_{PV} = \psi$. We now show the validity of Equation (13) by finding a value for $\gamma$. From the definition of $V_k(q)$, we obtain:

$$\begin{aligned} \begin{aligned}&\left| \int _{\mathcal {Q}} V_{k+1}(q) t_{f}(\underline{\rho } - g(\rho )) t_{P}( \underline{\psi } - h(\psi )) d (\underline{\rho }) d (\underline{\psi }) \right. \\&\left. - \int _{\mathcal {Q}} V_{k+1}(\tilde{q}) t_{f}(\underline{\rho } - g(\tilde{\rho })) t_{P}( \underline{\psi } - h(\tilde{\psi })) d (\underline{\rho }) d (\underline{\psi }) \ \right| \le \\&\left| \int _{\mathcal {F}} V_{k+1}(q) t_{f}(\underline{\rho } - g(\rho )) d (\underline{\rho }) \cdot \int _{\mathcal {P}} V_{k+1}(q) t_{P}( \underline{\psi } - h(\psi )) d (\underline{\psi }) \right. \\&\left. - \int _{\mathcal {F}} V_{k+1}(\tilde{q}) t_{f}(\underline{\rho } - g(\tilde{\rho })) d (\underline{\rho }) \cdot \int _{\mathcal {P}} V_{k+1}(\tilde{q}) t_{P}( \underline{\psi } - h(\tilde{\psi })) d (\underline{\psi }) \right| , \end{aligned} \end{aligned}$$

where $\mathcal {F}$ and $\mathcal {P}$ denote the domain of frequency and power, respectively. In order to continue, we introduce a useful lemma.

Lemma 1

Assume $A, B, C, D \in [0,1]$, then $|AB - CD| \le |A-C| + |B-D|$.

Proof

Assume $A>C$, then

if $AB-CD > 0$,

$|AB - CD| \le |CB - CD| = C |B-D| \le |B-D| \le |B-D| + |A-C|.$

if $AB-CD<0$,

$|AB - CD| \le |AB - AD| = A |B-D| \le |B-D| \le |B-D| + |A-C|.$

Analogously for $A \le C$.    $\square$

Thanks to this Lemma, we can write

$$\begin{aligned} \begin{array}{c} \left| \int _{\mathcal {F}} V_{k+1}(q) t_{f}(\underline{\rho } - g(\rho )) d (\underline{\rho }) \cdot \int _{\mathcal {P}} V_{k+1}(q) t_{P}( \underline{\psi } - h(\psi )) d (\underline{\psi }) \right. \\ \left. - \int _{\mathcal {F}} V_{k+1}(\tilde{q}) t_{f}(\underline{\rho } - g(\tilde{\rho })) d (\underline{\rho }) \cdot \int _{\mathcal {P}} V_{k+1}(\tilde{q}) t_{P}( \underline{\psi } - h(\tilde{\psi })) d (\underline{\psi }) \ \right| \le \\ \left| \int _{\mathcal {F}} t_{f}(\underline{\rho } - g(\rho )) d (\underline{\rho }) \cdot \int _{\mathcal {P}} t_{P}( \underline{\psi } - h(\psi )) d (\underline{\psi }) \right. \\ \left. - \int _{\mathcal {F}} t_{f}(\underline{\rho } - g(\tilde{\rho })) d (\underline{\rho }) \cdot \int _{\mathcal {P}} t_{P}( \underline{\psi } - h(\tilde{\psi })) d (\underline{\psi }) \ \right| \le \\ \int _{\mathcal {F}} \left| t_{f}(\underline{\rho } - g(\rho )) - t_{f}(\underline{\rho } - g(\tilde{\rho })) \right| \ d (\underline{\rho }) + \\ + \int _{\mathcal {P}} \left| t_{P}( \underline{\psi } - h(\psi )) - t_{P}( \underline{\psi } - h(\tilde{\psi })) \right| d(\underline{\psi }) . \end{array} \end{aligned}$$

Let us focus on the first integral:

$$\begin{aligned} \begin{array}{c} \int _{\mathcal {F}} | t_{f}(\underline{\rho } - g(\rho )) - t_{f}(\underline{\rho } - g(\tilde{\rho }))| d (\underline{\rho }) = \frac{1}{\sigma _{f}} \int _{\mathcal {F}} \left| \varPhi \left( \frac{\underline{\rho } - g(\rho )}{\sigma _f}\right) - \varPhi \left( \frac{\underline{\rho } - g(\tilde{\rho })}{\sigma _f}\right) \right| d\underline{\rho } \\ = \int _{\mathcal {F}} \left| \varPhi \left( u - \frac{\alpha _1(\rho - \tilde{\rho })}{2 \sigma _f} \right) - \varPhi \left( u + \frac{\alpha _1(\rho - \tilde{\rho })}{2 \sigma _f} \right) \right| d \underline{\rho } \le \frac{2 \alpha _1}{\sqrt{2\pi } \sigma _f} \ |\rho - \tilde{\rho }| , \end{array} \end{aligned}$$

and similarly for the second integral. Therefore,

$$\begin{aligned} |V_k(q) - V_k(\tilde{q})| \le \frac{2 \alpha _1}{\sqrt{2\pi } \sigma _f} \ |\rho - \tilde{\rho }| + \frac{2 a_{max}}{\sqrt{2\pi } \sigma _P} \ |\psi - \tilde{\psi }|. \end{aligned}$$