The Uncertainty in the TIAM Model

Energy Security: a robust optimization approach to design a robust European energy supply via TIAM

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.

In the SynsCOP15 project we have formulated the above issue that addresses mainly long term uncertainties. The formulation is done via a version of the TIAM Integrated Model, modified to implement the approach of Robust Optimization (Ben-Tal et al., 2009). In our case, the approach can be interpreted as a revival of Chance Constrained Programming (Charnes et al., 1958) under the name of Distributionally Robust, or Ambiguous, Chance Constrained Programming (Calafiore and El-Gahoui, 2006; Iyengar and Erdogan,2006). We apply the approach to improving the security of supply to the European Energy system.

1. Energy security issue for EU

Energy security is now considered as a priority in any energy policy and future energy strategy. In the United States, the Energy Independence and Security Act of 2007 and the proposed Clean Energy and Security Act of 2009 consider energy security at a pillar of the US energy policy. In Europe, the Green Paper on energy security (EC, 2000), the Green Paper on the European strategy for sustainable, competitive and secure energy (EC, 2006), several directives (for example: the 2006 directive on oil stocks forcing each Member State to maintain a minimum petroleum reserve, the 2004 and 2005 directives on measures to safeguard security of, respectively, natural gas supply and electricity supply), the current proposals for new regulations on investment projects in energy infrastructure (2009) and on gas supply security (2009) as well as the 20-20-20 Energy Policy (EC, 2007), aim at strengthening the European Community security of energy supply. Several concerns or fears are behind the importance given to energy security:

• The rapidly increasing World demands of energy, mainly driven by the emerging countries like China and India (IEA, 2007) may have important consequences on the availability and price of energy resources at the World level. Debates on the possibility of imminent peak oil add to these fears.
• The import dependence of Europe is expected to grow in a business-as-usual context, and energy imports might reach up to 65\% of the EU consumption by 2030 (EC, 2007). Moreover, this import dependence of Europe as well as many other importing countries or regions tends to concern a relatively small number of supplying countries (IEA, 2007).
• The Russian-Ukrainian gas crisis of January 2009, resulting in a disruption of gas supplies to EU via Ukraine, illustrates the increased transit risks.
• Uncertain geopolitical stability or strategy of supplying regions, like Middle East, Nigeria or Venezuela is of course of concern.
• Threats to energy security can result from system failures either of the supplier, or of the import country, for different reasons, such as natural events, terrorism, poor quality or conditions of the installations (operational failures from inadequate protection, generation capacity limitations due to under-investment) (Grubb et al., 2006). The 2003 blackout in Italy and the 2006 blackout that affected several European countries are examples.
• Environmental risk must be considered, related for example to the potential damage from accidents or any future policy implemented in the supplying or consuming countries and affecting the production and consumption of energy.

Based on this description, the risks related to energy supply can be geological (possible exhaustion of the resources), economic (fluctuations in the prices), technical (system failures - for different reasons), environmental (accidents or policies), or geopolitical (Behrens et al., 2009). Moreover, the management of risks will differ if the risks are external or internal to Europe. Internal EU risks generally mean coping with low-probability events, as well as appropriate investments in supply, storage, transmission and distribution of energy (Behrens et al., 2009). Finally, the time scale of different risks varies from short term (supply shortage due to technical failures for example) to long term (depletion of the resources, pricing, etc.) and appropriate actions will differ according to the time scale and nature of risks (IEA, 2007).

The proposed measures or strategies to increase the energy security can be classified in four categories (Behrens et al., 2009; EC, 2007; IEA, 2007; Frogatt and Levi, 2009):

• Diversification of the fuel mix, the geographic sources, the transportation routes and the diversification of suppliers; diversification can indeed be considered as a general ``insurance'' against heavy dependence, and against massive disruption.
• Definition of commercial agreement between suppliers and consumers, for example between the EU, Russia and Ukraine to secure gas supply from Russia via Ukraine to the EU.
• Appropriate investment in supply, storage, transport and distribution technologies to guarantee the quality of the energy system, to increase the available capacity of the production system and of the import and local network, and to promote an efficient management of, and recovery from, energy system disruptions.
• Decrease of the total energy demand (increase the efficiency of the energy system) and priority to energy sources considered less risky (reduction of foreign sources, reduction of sources with higher risk of accidents, etc.).

The impact of climate policies on energy security is considered as positive as regards the import dependence dimension of energy security, when considering the decrease of fossil fuel consumption and the growth of domestic renewable energy sources. However, the increase of nuclear generation raises issues of import dependence and availability of the resource, while the growth of renewable sources might affect negatively the reliability of the energy system energy security given their higher dependence on weather and intermittency (Grubb et al., 2006);. Indeed, trade-offs between energy security, climate change policies but also the climate resilience of the energy systems are to be found, where not only technologies but integrated policies promoting both greenhouse gas reduction and energy security (Brown and Huntington, 2008; IEA, 2007).

2. Robust optimization for handling uncertainty

To model global energy security we have introduced in TIAM a new constraint that concerns the total energy supply of EU and the obvious candidates for the random parameters of energy supply are the availability factor of the technology representing an import channel.  The requirement that the new constraint is to be satisfied for all possible realizations of its random component implies that the certain part should be chosen so as to match the worst possible case. This corresponds to the simultaneous failure of all corridors. This drastic requirement would exclude most solutions but the more conservative ones; it would not be deemed reasonable by the planners. Rather, one would like to have the uncertain constraint satisfied most of the time, at the risk of having it violated on rare occasions. The big issue is how to make this qualitative and vague requirement as a quantified and tractable entity.

The natural formulation for this requirement is to fix a lower bound on the probability that the constraint be satisfied. This idea was proposed as early as 1958 by Charnes and co-authors in (Charnes et al., 1958), and further discussed in (Miller and Wagner, 1965) and (Prékopa, 1970), under the name of Chance Constrained Programming (CCP). Unfortunately, this approach turned out to lead to untractable numerical issues in all situations, but a very few special cases (Ben-Tal et al., 2009). This formulation is not directly implementable in an optimization program such as TIAM.

Robust Opimization (RO) is an alternative proposal which essentially aims at overcoming the numerical issues raised by the computation of probabilities. The idea, similar to CCP, is to make sure that the constraint remains feasible for a set of ``relevant'' realizations of the random factors, at the risk of possible failure in some ``exceptional'' cases. But, contrary to CCP, RO defines the set of relevant realizations in an explicit way, e.g., as a polyhedron, rather than implicitly by means of a condition on a probability. The paradigm of Robust Linear Optimization goes back to (Soyster, 1973), but the field remained almost idle until the idea was revived circa 1997, independently and essentially simultaneously, in the frameworks of both Integer Programming (Kouvelis and G Yu, 1997) and Convex Programming (Ben-Tal and Nemirovski, 1998) and (El-Ghaoui and Lebret, 1997). The salient feature of RO is that it reformulates the uncertain constraint into plain inequalities, named the equivalent robust counterpart, which can be efficiently handled by convex optimization codes.

The more recent views on RO, as presented in the extensive monograph (Ben-Tal and Nemirovski, 2009) reconciles RO and CCP under the concept of Distributionally Robust (Calafiore and El-Gahoui, 2006), or Ambiguous (Iyengar and Erdogan, 2006) Chance Constrained Programming, in short ACCP. ACCP shares with RO the goal of leading to implementable and tractable formulations. To this end, it modifies in the CCP formulation as follows: the probability of satisfaction is not measured with respect to a specific probability distribution for each uncertain parameter, but with respect to a class that is described by few parameters (e.g., independent random variables with common support and known means). It turns out that this idea reconciles the concept of uncertainty set that underlies RO and the probabilistic statement in CCP. We shall briefly present RO from the viewpoint of ACCP and show how it can be implemented in our problem of interest.

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References

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