Política económica

Portfolio management and production planning models optimize an objective func-tion, which can describe the total costs or the profit of a whole energy system, par-ticularly of an energy company. However, the profit maximizing approach fits the objectives of an operator of power plants better. As operators of power plants are private companies, they have to cover their costs as well gain profits on the short-term and long-short-term. Otherwise investors will be not willing to invest in these companies. In short-term power production models, the profit function G is de-fined as the difference between total expected revenues R from electricity or other energy sales and total expected costs C of generation. Some models also consider a correction term for stock changesΔS if the company is also dealing with other energy carriers, such as heat or fuel (see Weber (2005)). G is maximized to find out the optimal solution for unit commitment and power production.

Table3.2.:Overviewofstochasticmodelingapproachesforenergymarkets ModelandAu- thor ModelTypeFundamental Model/Application

UncertainParameterStochasticProcess Finan.Fund.SMPPLSOIDM TsengandBarz (2002)ElectricityandfuelpricesMeanreversion(MR) Muche(2007)Electricityprice,coal priceandCO2price AR(1)processderivedfrom MR Weber(2005)Electricitypriceandfuel prices

Mean-reversionprocess Göbelt(2001)Electricitypriceandfuel prices,energydemand

— Fletenetal. (2002)Electricityspotandcon- tractprices,inflowinto waterreservoir

— Schmoeller (2005)Electricityandfuel prices,inflowintowater reservoirsandreserve power ARMA(p,q)process, ARIMA(0,1,0),VARIMA (0,2,0) Swider(2006)Windintermittency—

3.3. Optimization models applied to power production and investment planning

Table3.2.:Overviewofstochasticmodelingapproachesforenergymarkets ModelandAu- thor ModelTypeFundamental Model/Application

UncertainParameterStochasticProcess Finan.Fund.SMPPLSOIDM Olsinaetal. (2007)Loaddemand,available capacity,windspeed

Gauss-MarkovProcess, two-stateMarkovmo.,uni- variatestochasticprocess Hundtetal. (2008)Electricityprices,coal andgasprices

Mean-reversionprocess Dupacovaetal. (2003)(Load)demand,inflow tohydro-reservoirsand prices

Cluster-analyticapproaches FelixandWeber (2007)GaspricesMeanreversion Bowdenand Payne(2008)ElectricitypricesARIMAmodel,EGARCH model Karakatsaniand Bunn(2008)ElectricitypricesARmodel,LinearRegres- sion Ladurantayeetal. (2009)ElectricitypricesPeriodicAR(1)model

Table3.2.:Overviewofstochasticmodelingapproachesforenergymarkets ModelandAu- thor ModelTypeFundamental Model/Application

UncertainParameterStochasticProcess Finan.Fund.SMPPLSOIDM Kreyetal.(2007)FuelpricesMultivariateAR(1)model Fletenand Kristoffersen (2008)

Waterinflow,electricity prices

ARMAmodel Yangetal.(2008)Electricityprice,fuel pricesandcarbonprice GeometricBrownianMo- tion Kumbaroglu etal.(2008)Electricityprice,fuel prices

GeometricBrownianMo- tion Blythetal. (2007)CarbonpriceGeometricBrownianMo- tion Kanudiaand Loulou(1998)CO2-emissions,electric- ityandfuelsupply,ca- pacities

—

3.3. Optimization models applied to power production and investment planning

The total revenues of the power plant system consist of expected revenues of electricity sales R^EL_t,s on the spot and OTC market, heat sales R^HT_t,s on the OTC market and (re)sales of fuels R^FU_t,s on the OTC market. The revenues are calcu-lated for each scenario s and time step t. The sales can be calcucalcu-lated from the sold quantities multiplied with the time- and state-dependent prices of each commod-ity.

The total costs of the system or company are made up of power plant operating costs C_u,t,sof each unit u and costs C_OTC,t,s(for fuel, heat or electricity purchase) resulting from OTC contracts. Both cost components can be divided into variable costs and fixed costs. Thus, the total cost function is formulated as follows:

C=

∑

The variable operation costs C_u^var_,t,sconsider continuous operation as well as start-up and shut-down costs of power generation, whereas binary variables determine the operation mode, the start-up or shut down action. The variable contract costs C^var_OTC,t,s are determined by the mathematical product of the time-varying prices

pOTCand amounts YOTCfor each purchase contract of electricity, heat and fuel.

C_OTC^var _,t,s= p^EL_OTC,t,sY_OTC,t,s^EL + p^HT_OTC,t,sY_OTC,t,s^HT + p^FUOTC,t,sY_OTC^FU_,t,s [3.19]

The last component of the objective function, the stock changeΔS, corresponds to the fuel storage changes, which can be determined by the difference between the storage level for each fuel type f at the beginning Sf(1) and at the end of the planning period Sf(T) multiplied with the appropriate fuel prices:

ΔS=

∑

f∈F

p_f,1Sf,1− pf,TSf,T [3.20]

However, beside these more general models, which optimize trade portfolios combined with power generation planning, there are models which maximize the profit of only electricity generation. The generation costs are described as the plant operation costs above. Some models are based on linear cost functions for the variable operation costs, but some consider a more detailed cost structure.

Troncoso et al. (2008) use a genetic algorithm to solve a non-linear model for the optimal short-term electricity production. Thereby the total cost of electricity production cost is minimized assuming a non-linear cost function. A quadratic cost function is also used instead of a linear function for the operation costs C^var_u,t,s of each plant u at time t and state s by Tseng et al. (see Tseng and Barz (2002)), whereas the start-up costs C^SU_u,t,sare no longer fixed ones, but they depend on the time SD_u,t,spassed since the beginning of the last shut down:

C_u,t,s^var(X_u,t,s^EL) = pf,t,s

a0+ a1X_u,t,s^EL + a2X_u,t,s^EL2

[3.21]

C_u,t,s^SU (Uu,t,s) =

⎧⎨

⎩

pf,t,sb^u₁

1− e^SDu,t,s

+ b^u₂ U_u,t,s= 1

0 Uu,t,s= 0 [3.22]

The first summand of the first term for the start-up costs represents the fuel costs in the start-up time; the second one b^u₂covers other costs for start-up (e.g. labour).

The binary variable U_u,t,sindicates the shut down status of a plant at time t and state s (1= plant is offline, 0 = plant is online).

Based on these cost functions, Tseng and Barz maximize the total profit func-tion (Eq. 3.23).

3.3. Optimization models applied to power production and investment planning

Gt₀→T(Uu,t,s,Xu^EL,t,s) =

∑

t=0

∑

s∈St

Prs_u∈U

∑

−Cu^SU,t,s(Uu,t,s) [3.23]

This profit function can also be formulated as a recursive term, in which the profit G_t→Tbetween time t and T is calculated with the help of the expected profit Gt+1→T between t+ 1 to T adding the expected profit attained at time step t.

Therefore it is enough to maximize the profit in time step t and state s adding the expected profit G∗t+1→T:

G∗s,t→T(Uu,t,s,X_u,t,s^EL) = max 

p^EL_t,sX_u,t,s^EL −C_u,t,s^var(X_u,t,s^EL) −C_u,t,s^SU (Uu,t,s) +

s∈S

∑

Prs,t→s^,t+1G∗s,t+1→T(Uu,t,s,X_u,t+1,s^EL )

 [3.24]

So the calculation has to be done backwards against the time axis. The recursive computation ends at the single root state at time t0 resulting in the total profit maximum during the planning horizon t0to T .

At last, it is worth mentioning that these approaches for short- and mid-term power production planning can be extended to optimize an energy system in the long-term planning horizon.