Energy Security: A Robust Optimization Approach to Design a Robust European Energy Supply via TIAM-WORLD

Abstract

Energy supply routes to a given region are subject to random events, resulting in partial or total closure of a route (corridor). For instance, a pipeline may be subject to technical problems that reduce its capacity. Or, oil supply by tanker may be reduced for political reasons or because of equipment mishaps at the point of origin or again, by a conscious decision by the supplier in order to obtain economic benefits. The purpose of this article is to formulate a simplified version of the above issue that mainly addresses long-term uncertainties. The formulation is done via a version of the TIAM-WORLD Integrated Model, modified to implement the approach of robust optimization. In our case, the approach can be interpreted as a revival of chance-constrained programming under the name of distributionally robust, or ambiguous, chance-constrained programming. We apply the approach to improve the security of supply to the European Energy system. The resulting formulation provides several interesting features regarding the security of EU energy supply and has also the advantage to be numerically tractable.

Appendices

Appendix

A Proof of Proposition 1

Proposition 1 is a special case of a more general theorem in RO by taking into account that the variables CAP _k ’s are nonnegative. For the interested readers, we state now the general theorem and prove Proposition 1 as a corollary. In order to show the derivation, we shall use the concise notation

$$ z_0+ \sum\limits_k z_k \xi_k \le 0 $$

(6)

to represent the inequality (Eq. 4). The coefficients $\hat z$ are easily identified as

$$\begin{array}{rll} z_0 &= & \sum\limits_{k} ({\rm ACT}_{k } - {\rm CAP}_{k})\\ z_k & = & d_{k}\cdot {\rm CAP}_{k}. \end{array}$$

The main result can be formulated as

Theorem 1

Let η _k be K independent random variables with common support [ − 1,1] and known means E(η _k ) = ν _k . The probabilistic inequality $\hat z_0+ \sum_k \hat z_k \eta_k \le$ 0 is satisfied with probability at least (1 − ϵ) if there exists a vector $w\in \mathbb R^k$ such that the deterministic inequality

$$\begin{array}{rll} \hat z_0 &+& \sum\limits_k \hat z_k \nu_k + \sum\limits_k ( |w_k | - w_{k}\nu_{k}) \\ &+& \sqrt{2K \ln\frac 1\epsilon}\max\limits_k |\hat z_k- w_k| \le 0 \end{array}$$

(7)

is satisfied.

Note that the range of the random factors is now [ − 1, 1]. The above theorem is the formal statement of the theory for inequalities with random factors having known means ν _k and common range [ − 1, 1] as discussed in [3, example 2.4.9, p. 55]. For the sake of completeness, we propose a proof of Theorem 1. This technical derivation is given in the next section.

We show now how to prove Proposition 1 as a corollary of Theorem 1.

Proof

(Proposition 1) Let us start with (6) and define the variables η _k  = 2ξ _k  − 1. In view of Assumption 1 the range of η _k is [ − 1,1] and E(η _k ) = ν _k  = 2μ _k  − 1 ≤ 0. Inequality (6) becomes

$$ z_{0} +\frac 12 \sum\limits_{k}z_{k} + \frac 12 \sum\limits_{k}z_{k}\eta_{k} \le 0. $$

Let $\hat z_{0} = z_{0} +\frac 12\sum_{k}z_{k}$ and $\hat z_{k} = z_{k}/2$. The hypotheses of Theorem 1 for the inequality $\hat z_{0} + \sum_{k}\hat z_{k}\eta_{k} \le 0$ are verified. Hence,

$$\begin{array}{rll} \hat z_0 &+& \sum\limits_k \hat z_k \nu_k + \sum\limits_k ( |w_k | -w_{k}\nu_{k})\\ &+& \sqrt{2K \ln\frac 1\epsilon}\max\limits_k |\hat z_k- w_k| \le 0 \end{array}$$

is a sufficient condition to ensure the constraint satisfaction with probability at least (1 − ε). If we substitute ν _k , $\hat z_{0}$, and $\hat z_{k}$ by their values, we obtain the condition

$$\begin{array}{rll} z_{0} &+& \sum\limits_{k}\mu_{k} z_{k} + \sum\limits_k ( |w_k | +w_{k} -2\mu_{k} w_{k}) \\ &+& \sqrt{\frac K2 \ln\frac 1\epsilon}\max\limits_k | z_k - 2 w_k| \le 0. \end{array}$$

(8)

Recall that z _k  = d _k CAP _k  ≥ 0. We claim that only positive values w ≥ 0 need to be considered. Indeed, the theorem does not specify the value it should take. In particular, we can choose w so as to have to minimize the right-most component $\sum_k ( |w_k | +w_{k} -2\mu_{k} w_{k}) + \sqrt{\frac K2 \ln\frac 1\epsilon}\max_k | z_k - 2 w_k|$. If for some k′, w _k′ < 0, then |w _k′| + w _k′ = 0 and the contribution of term k′ in the summation is − 2μ _k′ w _k′ + max { z _k′ − 2 w _k′, max _{k ≠ k′}|z _k  − w _k |}. Clearly, this term can be made smaller by taking w _k′ = 0. Hence, we can assume w _k′ ≥ 0.

With w ≥ 0 inequality (8) becomes

$$\begin{array}{rll} z_{0} &+& \sum\limits_{k}\mu_{k} z_{k} + \sum\limits_k (1 - \mu_{k})2 w_{k}\\ &+& \sqrt{\frac K2 \ln\frac 1\epsilon}\max\limits_k | z_k - 2 w_k| \le 0. \end{array}$$

Writing u for 2 w in the above inequality, we obtain the condition

$$\begin{array}{rll} z_{0} &+& \sum\limits_{k}\mu_{k} z_{k} + \sum\limits_k (1 - \mu_{k}) u_{k}\\ &+& \sqrt{\frac K2 \ln\frac 1\epsilon}\max\limits_k | z_k - u_k| \le 0. \end{array}$$

Using the same argument as before, we easily prove that we can restrict our choice of u to u ≤ z. Hence, | z _k  − u _k | = z _k  − u _k  ≥ 0 and using the additional scalar variable v ≥ 0 we can transform our inequality into

$$\begin{array}{rll} z_{0} &+& \sum\limits_{k}\mu_{k} z_{k} + \sum\limits_k (1 - \mu_{k}) u_{k} + \sqrt{\frac K2 \ln\frac 1\epsilon}\cdot v \le 0 \\ z_k &-& u_k \le v,\; k=1,\dots,K \\ u &\ge& 0,v\ge 0. \end{array}$$

This concludes the proof of the proposition.□

B Proof of the Main Theorem

The scheme is based on an immediate consequence of Markov inequality.

Lemma 1

Let X be a real random variable. Then

$$ {\rm Prob}(X > 0) \le \inf\limits_{t > 0} E(\exp(tX))dF_X. $$

(9)

Proof

The proof is based on the following observations. First, for all t > 0, Prob(X > 0) = Prob(tX > 0). Second, $\int \mathbb I(tX > 0) dF \le \int \exp(tX) dF$ where $\mathbb I(a\ge 0)$ is the indicator function with $\mathbb I(a\ge 0)= 1$ is a ≥ 0 and 0 otherwise. □

The problem of interest is to bound the probability of $\hat z_0 + \sum_k \hat z_k\eta_k > 0$ when the η _k are independent random variables with support [ − 1,1] and mean E(η _k ) = ν _k . Setting $X = \hat z_0 + \sum_k \hat z_k\eta_k$ we get in the right-hand side of (9)

$$\begin{array}{rll} E(\exp(tX)) &=& E\left(\exp\left(t\hat z_0 + t\sum\limits_k \hat z_k\eta_k\right)\right)\\ &=& \exp(t \hat z_0 ) \prod\limits_k E(\exp( t \hat z_k\eta_k)). \end{array}$$

Here, we used, as announced, the property that the generating function of a sum of independent variables is the product of the individual generating functions.

Let us now turn our attention to the worst case distribution. To this end, let us perform changes in notation $\tau_k = t \hat z_k$ and s _k  = η _k . The next result holds the same for all indices k. So we drop for a while the subscript k in the proposition below.

Proposition 2

The worst distribution in $\mathcal F$, $\bar F =$ $\arg\max_{F \in \mathcal F} \int \exp(ts)dF(s)$ , is the two-point distribution, with s = 1 with probability $p = \frac{1 + \nu}{2}$ and s = − 1 with probability $q= 1-p = \frac{1 - \nu}{2}$.

The proposition is rather intuitive but not so easy to prove rigorously. We refer to Ref. [3] for a proof.

In view of the above proposition,

$$ E(\exp(\tau s)) \le g_{\nu}(\tau ) \!=\! p e^\tau + qe^{-\tau } \!=\! \frac{1 + \nu}{2} e^\tau + \frac{1 - \nu}{2}e^{-\tau }. $$

It is easier to work with the logarithm h _ν (τ) = logg _ν (τ).

$$\begin{array}{rll} h_{\nu}(\tau ) & = & \log(p e^\tau + qe^{-\tau })\\ h^{\prime}_{\nu}(\tau ) & = & \frac{p e^\tau - qe^{-\tau}}{p e^\tau + qe^{-\tau }}\\ h^{\prime\prime}_{\nu}(\tau ) & = & 1 - \left(\frac{p e^\tau - qe^{-\tau }}{p e^ \tau + qe^{-\tau }}\right)^2. \end{array}$$

Taylor expansion of order 2 yields

$$ h_{\nu}(\tau ) = h_{\nu}(0) + h_{\nu}^{\prime}(0)\tau + h_{\nu}^{\prime\prime}(\bar \tau)\frac {\tau^2}{2} $$

for some $\bar \tau$ between 0 et τ. One easily check that h _ν (0) = log(1) = 0, h′ _ν (0) = p − q = ν and for all t, 0 ≤ h′′ _ν (τ) ≤ 1. Hence,

$$ h_{\nu}(\tau ) \le \nu \tau +\frac {\tau^2}{2}. $$

Coming back to (9), we replace τ _k by its value and get

$$ \begin{array}{rll} && \inf\limits_{t > 0} E(\exp(tX))dF_X \\ && \;\; \le \inf\limits_t \exp\left(t\left(\hat z_0 + \sum\limits_k \nu_k \hat z_k\right) + \frac {t^2}{2}\sum\limits_k \hat z_k^2\right). \end{array}$$

If $\hat z_0 + \sum_k \nu_k \hat z_k< 0$, the infimum in t is a minimum and is achieved at

$$ t_{opt} = -\frac{\hat z_0 + \sum_k \nu_k \hat z_k}{ \sum_k \hat z_k^2} >0. $$

The hypothesis $\hat z_0 + \sum_k \nu_k \hat z_k < 0$ just means that the constraint is strictly satisfied in the average. We have thus

$$ {\rm Prob}\left(\hat z_0+ \sum\limits_k z_k \hat \xi_k > 0\right) \le \exp\left( -\frac{\left(\hat z_0 + \sum_k \nu_k \hat z_k\right)^2}{2\sum_k \hat z_k^2}\right). $$

A sufficient condition to have the right-hand side bounded by ϵ is

$$ \left(\hat z_0 + \sum\limits_k \nu_k \hat z_k\right)^2 \ge 2\ln\frac 1\epsilon ||\hat z||^2_2. $$

Since $\hat z_0 + \sum_k \nu_k \hat z_k \le 0$, the above condition becomes

$$ \hat z_0 + \hat z^T\nu + \sqrt{2\ln\frac 1\epsilon} \;||\hat z||_2 \le 0. $$

When this deterministic constraint is satisfied, we can guarantee

$$ {\rm Prob}\left(\hat z_0+ \sum\limits_k \hat z_k \eta_k > 0\right) \le \epsilon. $$

We just proved

Proposition 3

The deterministic constraint $\hat z_0 + \hat z^T\nu + \sqrt{2\ln\frac 1\epsilon} \;||\hat z||_2 \le 0$ implies ${\rm Prob}(\hat z_0+ \sum_k \hat z_k \eta_k > 0) \le \epsilon.$

The proposition can be strengthened if we take advantage of the known range of η. For any w and in view of − 1 ≤ η  ≤ 1

$$\begin{array}{rll} \hat z_0 &+& \hat z^T\eta = \hat z_0 + w^T\eta + (\hat z-w)^T\eta \le \hat z_0 + ||w||_1\\ &+& (\hat z-w)^T\eta . \end{array}$$

Set $\bar z_0 = \hat z_0 + ||w||_1$ and $\bar z = \hat z-w$, we have from Proposition 3

$$\begin{array}{rll} \bar z_0 &+& \bar z^T \nu + \sqrt{2\ln\frac 1\epsilon} \;||\bar z||_2\\ &\le& 0\; \Rightarrow\; {\rm Prob}\left(\bar z_0+ \sum\limits_k \bar z_k \eta_k > 0\right) \le \epsilon. \end{array}$$

Replacing $\bar z_0$ and $\bar z$ by their value, we have

$$\begin{array}{rll} \hat z_0 &+& \hat z^T \nu - w^T \nu + ||w||_1+ \sqrt{2\ln\frac 1\epsilon} \;||\hat z-w||_2\\ &\le& 0\; \Rightarrow\; {\rm Prob}\left(\hat z_0+ \sum\limits_k \hat z_k \eta_k > 0\right) \le \epsilon. \end{array}$$

Finally by bounding the ℓ₂-norm such that $||a||_2 \le \sqrt{K} \max_k|a_k|$ we prove Theorem 1

$$\begin{array}{rll} \hat z_0 &+& \hat z^T \nu - w^T \nu + ||w||_1+ \sqrt{2K\ln\frac 1\epsilon} \;\max\limits_k|\hat z_k-w_k|\\ &\le& 0\; \Rightarrow\; {\rm Prob}\left(\hat z_0+ \sum\limits_k \hat z_k \eta_k > 0\right) \le \epsilon. \end{array}$$

C Discussion on the Dynamic Issue

In the sketch of methods to handle uncertainty, we did not pay attention to the dynamic feature of the TIAM-WORLD model. In a multistage problem as this one, decisions are taken sequentially. Decisions at stage t need not be fixed prior to that stage, and at stage t the past history of realizations of the uncertain factors is known. This information should be used to design “optimal decisions.” Conceptually, this can be done by replacing actual decisions by contingent decisions, often named “recourses.” The framework of Stochastic Programming [8] is perfectly designed to handle this situation. Unfortunately, solving multistage uncertain linear programs even with moderate precision is already a challenge beyond all available optimization techniques. We refer to [3, pp. 411–413] for an enlightening discussion on the “dramatic theoretical gap between what multistage programming intends to achieve and what, if any, it achieves.”

A possible remedy to this dramatic state of affair is to restrict recourses to the class of affine decision rules. In this formulation, future decisions are affine functions of the past realizations of the uncertain coefficients. In so doing, the new variables in the mathematical programming formulation are not any more the actual decisions, but the coefficients of the affine functions. When embedded in a robust optimization framework, this approach leads to numerically tractable problems. Affine decision rules is not the panacea, because it is in essence sub-optimal (recourses are restricted to a very small subset of all possible functions), but it offers an attractive alternative that has proved very useful in solving practical problems.

Clearly, there is a need in our problem of interest to say what is to be done in the event of a corridor failure. In the light of the above discussion, this is definitely a major challenge. robust optimization with affine decision rules could be used. Some works on similar problems has been done in [1], but on instances of considerably smaller dimensions. In the present problem, introducing affine decision rules to decide on the supply routes and levels to match the actual realizations of capacities would spread uncertainty throughout the model. Straightforward application of the robust optimization machinery would convert the new uncertain program into a far too large model to be treated by existing computers and commercial solvers. Our choice has been to ignore the adjustment feature that is intrinsic to multistage problems under uncertainty. Therefore, we shall be content to use RO on various time periods, in order to reveal robust configurations of the corridors but falling short of indicating remedial actions for each outcome of the random events.