Constraint: emissions depending on combustion temperature, thus associated

with technologies (like N O_x in transportation, heating, industry, etc.)

X

k∈DM T ch∈DM D1(k)

EnvEmis2T ch,P oll(t)∗CapT ch(t)∗CapF actT ch(t)≤EnvEmisM ax2P oll(t)

Choosing the appropriate modelling region can be straightforward when it is ex-plicitly defined by geography as a country or region, or it can be more general according to a different criterion, such as electric grid, or an energy system. An energy system is a set of items destined to produce, transmit and consume energy. The boundaries of an energy system can be physical or logical. A farm, a cottage or a polar station with no integration into an electric grid may represent an autonomous energy system.

In MARKAL Nyon, we have chosen as modelled region the Nyon city and all the en-ergy forms were imported. MARKAL Nyon is described on page 99 and Social MARKAL Nyon on page 123.

In TIMES Romania, we chose as modelled region the whole country but as we de-sired to concentrate the modelling effort only on the demand side, all the energy forms were imported in form of virtual energy imports of final energy carriers. Social TIMES Romania is described on page 132.

A "classical" country-level MARKAL model is a perfect-foresight energy-system model that provides a detailed representation of both currently available and future energy sup-ply technology alternatives, and a simplified representation of demand technologies. This representation, called Reference Energy System (RES), can be in tabular or in visual form.

Figure 3.5: The form of the MARKAL problem matrix. Grey areas symbolise the non-zero coefficients in the matrix and the right-hand side. White areas are filled with zeros. Adapted from User’s Guide for MARKAL (Fishbone et al., 1983).

When moving to a local-scale model (region, canton, municipality, community), then the emphasis on modelling effort and degree of detail is moved to the demand side which is modelled in more detail. In this situation, several original MARKAL assumptions may no more be satisfied:

• Linearity of technologies

• Inelasticity of demand to prices

• Perfect information and perfect economic rationality

Linearity of technologies: if instead of an important number of technologies, there only can be zero, one, or at most two installations like in case of a municipal waste incinerator, then the problem is no more linear as we are in presence of one or more integer variables because there can be no 1.5 installations, only zero, one or maybe two.

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Figure 3.6: Integer programming (IP) polytope with linear programming (LP) relaxation, taken from http://en.wikipedia.org/wiki/Integer_programming A problem with a few integer

variables among linear variables is called mixed integer program-ming (MIP). This situation typ-ically arises when moving from a country scale down to region, city or community. If a country has dozens of installations, adding or subtracting one of them still leaves the remainder to be a number which is big enough to consider the problem linear. However when considering lump investments that have to be selected "all or noth-ing" like in case of an urban heating network, we have one or more vari-ables that can have values only zero or one and we speak aboutbinary programming.

Figure 3.7: Equilibrium of the supply and demand ac-cording to neoclassic theory. Q0is the equilibrium pro-duction quantity andP0 is the equilibrium price.

In a simple example, allow-ing only integer variables means that the system is subject to ad-ditional constraints and the feasi-bility region is reduced to a poly-hedron with next closest smaller integer values. However in dy-namic programming, there may be an optimal solution where there is an earlier investment in some technology in order to have suf-ficient additional capacity to sat-isfy the demand in every pe-riod. The objective function is the sum of long-term dis-counted cost contributions from each period and not only a lo-cal minimum in one point in time.

supply

Figure 3.8: Equilibrium of the supply and demand in the original MARKAL model. The supply function generated by the model is stepwise because of finite number of discrete technologies. Demand is inelastic, remains constant regardless of the supply and of the cost to satisfy the demand.

Inelasticity of demand to prices: in contrast with the case of integer or binary programming, in-elastic demand stems from the hy-pothesis of infinite size market where even a country is too small a player to influence through its consumption the price of the good, in our case the prices of imported energy agents.

This assumption is also realistic in case of an autarcy when there is no need to import energy agents and no need to consider the market forces.

In case of a regional, urban or com-munity model, this assumption is well satisfied. However in case of a model for an important country such as USA

or multi-regional model for EU or the entire world, it is reasonable to consider elasticity to energy prices. The same when studying the effects of important changes of prices like in case of petrol peak.

Figure 3.9: MARKAL equilibrium in the modified model with elastic demands. More steps bring a need to find more demand levels and more computational effort necessary. and the modelling effort mostly goes to the supply side. When reduc-ing the scale of the model, demand side becomes more important and it is worth to consider the correction for user’s behaviour concerning some demand technologies or final energy forms.

When the model is created with emphasis on increasing number of de-cision makers deciding frequently on

small amounts, like consumers who do not always behave rationally and do not always look for information, then there may be a systematic error on the demand side for some technologies whose acquisition and use is plagued by schemes of behaviour which are not in conformity with perfect economic rationality supposed by the model.

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