3. Estado del arte
3.2. Aprovechamiento de biosólidos para uso agrícola
Optimization models describe a problem by using an objective function and at least one constraint. They are solved by algorithms implemented in mathematical solvers. Optimizing the objective function is subject to a number of constraints which represent technical and economic boundaries of the problem. Typical objectives of electricity models are the minimization of costs or the maximization of profits which result from electricity generation or transmission. Related to the problem structure and characteristics of variables, different mathematical methods exist to formulate and solve a problem. If the mathematical methods are transferred in models that can be used for a wide range of applications, these models can be classified as applied mathematics (Leuthold, 2010). Many electricity models have been developed by using methods and research from the field of Operations Research in which many methods for complex problems and powerful algorithms are developed to find suitable and comprehensive solutions based on a mathematical foundation.
The general mathematical formulation of linear programs (linear optimizations) and mixed integer linear programs are presented here to show the mathematical basis for the electricity market model developed in this thesis. Linear programs (LP) are a special class of optimization models which aims to minimize (maximize) a linear function F(x) with n variables subject to equalities and inequalities. Every linear program can be transformed to the standard form. The standard form of linear programs is given by the following equations (Castillo et al., 2002). Objective function:
minimize ( ) = ∑ ∗ (2-1)
Subject to:
= (2-2)
24 CHAPTER 2. Modeling fundamentals for electricity systems with renewable energy sources
With and as parameter matrices to the variables and the right hand side parameters and . Variables are multiplied by parameters . Maximization problems are obtained by multiplying the objective function by -1. Inequalities can be also converted from ≥ to ≤ by multiplying with -1.
If a mixed integer linear program (MIP) is used, at least one variable of is restricted to natural numbers (N = 0, 1, 2, 3,….). Often this approach is chosen if binary variables are used in dispatch problems to indicate operational status as online = 1 and offline = 0. Then the following constraint is necessary:
at least one ε N (2-4)
2.3.2 Objective functions of electricity models
Models which optimize operation management and operation planning based on economic decision parameters include economic values (costs, revenues, etc.) related to short-term operation. Technical restrictions are implemented by the constraints. Table 1 provides an overview on the wide range of cost and revenue items that can be considered in the objective function of short-term electricity models. The list is extended from findings in different papers (Hetzer et al., 2008; Boqiang and Chuanwen, 2009; Guo et al., 2012; Schroeder, 2012).
Table 1: Items considered in objective functions of short-term operation models
Class of cost/revenue Specific item of the electricity system
Operation costs • Variable operation costs (auxiliary materials, wear of components, replacement, cleaning) without fuel costs
• Costs for consumable goods such as fossil fuels (coal, oil, natural gas, uranium) or biomass
• Load change costs
• Start-up/shut-down costs
• Costs for external costs such as environmental emissions (CO2 emission allowances)
• Penalties for failing to meet the demand (curtailment of renewable energy sources)
Costs for provision of
energy • Costs for providing reserve capacity • Costs for providing spinning reserve (depending on kind of operating power plants)
• Costs for energy security which incorporates stochastic security Costs for electricity
transport • Costs for energy transport to consumers • Network charges Revenues for electricity
sales • Revenues from sold energy (electricity, heat) • Revenues from flexible demand (shut down of large consumers) • Revenues from provided spinning reserve or reserve capacity
• Revenues from compensations and subsidies related to sold energy
Expansion planning models take the whole life cycle of the infrastructure covered by the model into account. Expenses for new power plant and transmission lines reduced by obtained
CHAPTER 2. Modeling fundamentals for electricity systems with renewable energy sources 25
subsidies are implemented as well as costs for dismantling and recycling of power plants. In the objective function following items can be included, but final selection strongly depends on the problem and target of the model. Table 2 shows the findings which are extended from findings in different papers (Graeber, 2002; Remme, 2006; Weber, 2008; Lynch et al., 2012; Nagl et al., 2012).
Table 2: Items considered in objective functions of long-term planning models
Class of cost/revenue Specific item of the electricity system
Costs for infrastructure • Expenses of expansion (investment expenses for new power plants and transmission lines)
• Fix costs for operation (staff, rents, non-consumable equipment) • Dismantling and recycling costs of power plants
• Residual value after lifetime or use of power plants Operation costs • Variable operation costs (auxiliary materials, wear of
components, replacement, cleaning) without fuel costs
• Costs for consumable goods such as fossil fuels (coal, oil, natural gas, uranium) or biomass
• Costs for external costs such as environmental emissions (CO2
emission allowances)
• Taxes for operation or investment
• Penalties for failing to meet the demand (curtailment of renewable energy sources)
Costs for electricity
transport • Costs for energy transport to consumers Revenues for electricity
sales • Revenues from sold energy (electricity, heat) • Revenues from provided spinning reserve or reserve capacity • Revenues from compensations and subsidies related to sold
energy
Further revenues • Compensations and subsidies for investments
• Willingness to pay of consumers if elastic demand is considered
This list can be extended in relation to the market environment or problem which should be covered by the model. The optimization of electricity systems normally is carried out for economic aspects, such as the optimization of costs or profits. Therefore, cash flows which appear at different points in time have to be converted (discounted) to a base year to guarantee comparability of values from different years (e.g. by using the net present value approach). An optimization of other items and preferences in the energy system seems to be possible, but these items are often converted into economic values, see e.g. reduction of CO2 emissions which
are normally implemented by using a price for CO2 emissions. This price is set to include costs
of external effects by CO2 emissions or it represents costs for avoiding CO2 emissions in an
26 CHAPTER 2. Modeling fundamentals for electricity systems with renewable energy sources
2.3.3 Technical aspects of electricity systems as models constraints
The minimization or maximization of variables in the objective function is subject to further economic or technical constraints. Again, these constraints can be split into short-term operational constraints and constraints for expansion planning models which model long-term developments, new investment decisions, policy targets and security of supply. Depending on the model focus, the level of formulation of each issue is extended by using a very detailed description of the issue or is shortened or left out if the issue makes it more difficult to solve the problem. Table 3 summarizes potential constraints which are extended from findings of Slomski (1990) and Remme (2006).
Table 3: Technical and economic constraints in electricity models
Class of constraints Specific constraint in models
Potential constraints for
operation of power plants • Maximum turbine or generator capacity limits • Restricted capacity ranges (low capacities) • Turbine efficiencies
• Heat generation of power plants • Part-load behavior
• Temporal potential to change capacities • Start-up/shut-down times
• Minimum hours for offline and online of power plants • Revision cycles
• Required spinning reserve
• Constraints of availability of consumable goods such as fossil fuels Specific operational
constraints of RE power plants:
• Technical conversion from renewable energy to electricity • Generation profiles for RE technologies (exogenously generated) • Available resources (e.g. seasonal water availability)
• Included storage application Specific operational
constraints of storage power plants (pumped storage power plants, compressed air storage or storage batteries)
• Conversion efficiencies
• Storage losses (hourly, seasonal, etc.) • Inflow and Outflow of the storage • Storage level
• Storage maximum volume
• Lifetime depending on storage cycles Potential constraints for
grid operation and network flows
• Grid structure and flow mechanisms (DC-flows, AC-flows) • Maximum of net transfer capacity between nodes (regions) • Voltage levels
• Transmission efficiencies and losses • Distributions losses
Potential constraints of
demand • Timely demand and its geographical distribution • Demand-side management and its behavior in the power market (electricity which can shifted to other points in time)
• Import and export from the modeled area to the world around • New applications and players in the sector such electric vehicles
CHAPTER 2. Modeling fundamentals for electricity systems with renewable energy sources 27
Class of constraints Specific constraint in models
Potential constraints in expansion planning models
• Maximum new capacity and its geographical distribution • Lifetime of infrastructure (power plants, grid, etc.)
• Construction time of new infrastructure (including potential delays) • Maximum annual budget to be invested
• Security of supply in each time level
• Relations between short-term and long-term market requirements (e.g. nuclear power phase-out, availability of resources, market framework)
• Strategic targets of investment policies (e.g. investment in national key technologies)
• Policy targets of renewable energy sources
When developing a new optimization model for the electricity market all required constraints should be described qualitatively in the planning process. Then, this qualitative system characterization has to be implemented regarding logical, technical or economic relations into mathematical constraints in the model. Often, a problem or issue is formulated by a large range of constraints in the mathematical model to cover a technical or economic behavior adequately. It could be necessary to start with a model using a lower number of constraints to validate the model. Later, additional constraints are added to detail a specific element in the system and to restrict potential solutions of the optimization model.
2.3.4 Combining different objectives in energy scenarios
Energy scenarios support the decision making process by providing more information concerning potential development strategies and structures of future energy systems. In this regard, optimization models can find cost-efficient solutions to the problem by minimizing overall system costs, variable operation costs or investments in new infrastructure. As seen before, one noticeable drawback of optimization models and other modeling approaches is the limitation of including only options in the evaluation process which have been identified and investigated during the implementation and development process. If long-term time horizons are covered, the uncertainty of input data increases due to unknown prospective developments in terms of cost, technology parameters or economic and political framework conditions (Börjeson et al., 2006). Another problem emerges from the difficulty of monetizing some important decision variables and technical, economic and social implications of a future energy system in the objective function of the mathematical model. These unmodeled implications of the energy system can generate negative effects on societies, e.g. permanent storage of nuclear waste is a negative effect. This long-term environmental and economic damage for the society is often not included in the decision making process for nuclear power plants. External effects of conventional power plants such as CO2 emissions or other environmentally damaging
materials can satisfactorily be included in models by adding a penalty for each produced emission entity. Especially, CO2 emission allowances are commonly used in energy system
analyses. Other environmental policies (such as national targets for RES defined by many governments around the world) can be considered in models as exogenous constraints. The support of RE technologies is either implemented by modeling the specific support mechanism or by setting quotas for the contribution of renewable energy sources.
28 CHAPTER 2. Modeling fundamentals for electricity systems with renewable energy sources
Foley et al. (2010) propose in their paper to include further decision variables linked with socio-economic constraints in the model approaches of energy system analyses. According to the paper, socio-economic effects increasingly influence the decision making process. These additional effects should improve the economic or most cost-efficient solution, which is found by an energy model. The meaning and consequences of the solutions should be widened through the use of a multi-perspective. Due to the fact that some RE technologies still require political support in the form of an incentive programs or use a high amount of land, these requirements are justified either by environmental policy targets or socio-economic benefits. These benefits can be represented by new employment created by the installation of RE projects and component manufacturing. Other positive developments for the society are the increase of economic wealth or social security.
Today, socio-economic analyses of energy systems are carried out in separate studies. In these studies the scenario results of energy system modeling are used as inputs. This implies that socio-economic effects are discussed and evaluated after the optimization of the electricity systems. Consequently, these additional conclusions from the socio-economic analysis do not influence the outcomes of the energy system analysis as it is normally carried out by other experts. Adjustment of scenario assumptions or recalculation after linking with socio-economic parameters is often not possible. Technologies with a positive effect on a region or country on a socio-economic level could be underrated in pure cost-efficient scenarios.
Therefore, this thesis aims to combine a socio-economic objective for an energy scenario with the objective of cost minimization. In the optimization model for electricity market, it is necessary to analyze new scenario options which extensively vary the use of different technologies. Only a range of selected scenarios are then evaluated regarding their socio- economic effects caused by different technology paths.