Multi-energy Grid Expansion Planning Under Uncertainty: A Robust Optimization Approach

This paper has been developed as part of Integrated Network Planning (iNeP) of the research project Northern German Living Lab (Norddeutsches Reallabor), which is supported by the German Federal Ministry of Economic Affairs and Climate Action (BMWK) under agreement no. 03EWR007O2.

Marwan Mostafa,
Johannes Heise,
Davood Babazadeh and
Christian Becker

Institute of  Electric Power and Energy Technology,

Hamburg University of Technology, Hamburg, Germany

Integrating the gas and district heating with the electrical grid in a multi-energy grid has been shown to provide flexibility and prevent bottlenecks in the operation of electrical distribution grids. This integration however presents new challenges, including uncertainties in demand prediction, energy prices, and renewable energy availability. In response to these challenges, this paper proposes a novel approach to apply robust optimization methods in the integrated planning of multi-energy grids, to reduce the risk of investment in grid expansion and to optimize the use of different carbon-neutral energy carriers. The uncertainty in energy prices is modeled using interval uncertainty with a proportional deviation. This allows planners, operators and regulators to prioritize the expansion of specific grids in certain areas of a city. By minimizing a cost function subject to various constraints, the strategy ensures robustness against uncertainties in energy prices. This robust optimization approach is applied to Hamburg as a case study. The study concludes that district heating expansion in high-density areas is a low-risk investment for carbon neutrality. In less dense areas, electrification supports decentralized heat pumps. Meanwhile, hydrogen gas grids are viable where electric expansion is impractical. Increased uncertainty leads to more conservative solutions. This novel approach can be implemented promptly and practically by grid planners and is an important component of a new holistic integrated planning process for multi-energy grids.

Robust optimization, Integrated planning, Multi-energy grids, Uncertainty management, Sector coupling

[A,1]\(i\)Cell index
[A,2]\(j\)Heating technology index
[A]\(de\)Decentralized heat pumps in households
[A]\(dg\)Decentralized hydrogen boilers in households
[A]\(ce\)District heating with centralized electric heat pumps
[A]\(cg\)District heating with centralized hydrogen boilers

[B,1]\(N\)Total number of cells
[B,2]\({H}\)Total heating energy demand per year in kWh/a
[B,3]\({P}\)Total heating load power in kW
[B,4]\({L}\)Total length of streets in m
[B,5]\({W}\)Binary flag for existing district heating grid
[B,6]\({C^{l}}\)Cost of grid expansion in €/m or €/kW
[B,7]\({C^{p}}\)Cost of heating power generator in €/kW
[B,8]\({\hat{C}^{f}}\)Predicted price of energy carrier
[B,90]\({\tilde{C}^{f}}\)Uncertain price of energy carrier
[B,91]\({\Delta^{f}}\)Maximum deviation in price of energy carrier
[B,92]\({\eta}\)Efficiency of heating power generator
[B,93]\({COP}\)Coefficient of performance of electric heat pumps
[B]\({\sigma}\)Scale factor for grid expansion
[B]\({\bar{\Gamma}}\)Upper limit on grid expansion

[C]\({J}\)Set of heating technologies, \({J = \{ce,cg,de,dg\}}\)
[C]\({U^{f}}\)Uncertainty set of energy carrier price

[D]\({X}\)Binary decision variable

§ INTRODUCTION

Achieving carbon neutrality in energy systems requires innovative approaches in the integrated planning of distribution grids, including electricity, gas, and district heating. A main challenge in achieving this goal is the optimization of the use of different energy carriers and the mitigation of investment risks associated with grid expansion. This paper presents a robust optimization (RO) methodology addressing the uncertainties of future energy prices. While initial work focuses on energy price volatility, subsequent work aims to extend to a broader spectrum of uncertainties.

§.§ Motivation and Problem Description

The city of Hamburg has set the target of carbon neutrality by 2045. To meet this target, an integrated planning process was proposed in <cit.>. In this process, the optimal grid expansion roadmap guaranteeing the security of a carbon-neutral energy supply is to be found. The first step in this process is to forecast future heating energy demand as heating is the dominant end-use of the transported energy through the grid. The heating energy demand and load distribution are aggregated in cells which offer a comprehensive view of energy systems, incorporating energy demand and various socio-economic factors. This method provides detailed regional demand forecasts and structures <cit.>. In this study, a 1 km2 grid represents these cells.

It was found in <cit.> that the provision of heat pumps at the low-voltage level cannot be implemented without significant grid expansion. In addition to heat pumps, the integration of additional loads, notably electric vehicles, into the grid due to increased electrification in all sectors intensifies the existing challenges. It was demonstrated that a centralized solution through local or district heating can alleviate the time-critical expansion in the electrical grid. Additionally, centralized heat supply offers the opportunity to integrate short-term and seasonal storage. These, in turn, allow for a notably enhanced flexibility in grid operation, presenting a significant advantage. At the coupling elements between the heating and electrical grids, the provided flexibility can be used to support the operation of the electrical grid during periods of high utilization.

The next important step is to identify potential areas for centralized heat supply and at the same time reduce the immense expansion needs of the electrical grid. In this paper, four heating technologies associated with four different grid expansion options are studied and the decision to invest in regionally expanding one of the four grid options is taken. The grid associated with the lowest total economic cost must be chosen. This decision is made under uncertain energy prices and during a dynamic and crucial transition phase in the grid as investing in expanding the wrong grid might hinder achieving the carbon neutrality goal, increasing the investment risk considerably.

§.§ Robust Optimization

Scenario-based stochastic optimization methods are widely used to handle uncertainties in power generation and energy prices in the planning of electrical distribution grids <cit.>. According to <cit.>, RO can be more efficient than these approaches in solving planning problems. This is because unlike in stochastic optimization where the solution space depends on the number of uncertain parameters and the number of scenarios, in RO the size of the solution space depends on the number of uncertain parameters only. It is shown in <cit.> that a two-stage RO approach for the short-term planning of an electrical distribution grid is an efficient tool for solving complex planning problems. Most applications of RO have focused on the optimization of energy systems <cit.>, sizing of batteries and renewable generation <cit.>.
Moreover, the implementation of RO in the planning is limited to uncertainties in the dispatching and renewable generation and considers only the electrical distribution grid.

This work's aim is to expand these methods to a multi-energy distribution grid and increase the number of uncertain parameters. The goal of the RO problem is to find the optimal grid expansion decision that minimizes a cost function subject to various constraints while ensuring robustness against uncertainty. This RO technique provides a solution that is optimal for the worst-case scenario and feasible for all realizations of uncertainties. This approach provides multi-energy grid planning steps that are robust against the uncertainty in the parameters. It guarantees a satisfactory performance under all possible scenarios ensuring that the subsequent planning steps in the multi-energy grid are not significantly degraded even under unfavorable conditions.

In Section <ref>, the problem is formulated as a deterministic optimization problem. The proposed RO problem is introduced in Section <ref>. Section <ref> presents the numerical result. In Section <ref>, conclusions are drawn and an outlook for the expansion of this model is discussed.

§ DETERMINISTIC MODEL

Aggregated data for heating in Hamburg in the year 2045. (a) Peak load power based on standard load profiles in kW, (b) Heating energy demand per year kWh/a, (c) Coverage of the current district heating grid.

A deterministic cost optimization model to choose the cost-optimal grid expansion in each cell is presented. Fig. <ref> shows the input data to this model represented in (a) aggregated data for peak heating load power based on standard load profiles for Germany <cit.>, (b) total heating energy consumption per year and (c) current district heating grid coverage.

The objective function (OF) of the proposed model formalized in (<ref>) minimizes the total economic cost associated with implementing one of four technologies:  decentralized household heat pumps, decentralized household hydrogen boilers, district heating with centralized electric heat pumps and district heating with hydrogen boilers.
The cost represented in (<ref>)  consists of three components: the cost of energy, the cost of the heating power generator and the cost of grid expansion. The energy cost depends on the total yearly energy demand in kWh/a and the efficiency of the heating technology (COP for heat pumps). The cost of power generators depends on the load power and the cost of heat generators per kW. The cost of grid expansion for gas and district heating depends on the length of the streets in a cell modified by a factor $\sigma$ derived from empirical analysis of the length of the existing grids compared to the length of the streets. This reflects the fact that gas and district heating grids are laid along the streets. Electrical distribution grid expansion however depends on the grid capacity in kW instead of the length of the streets.
In (<ref>) and (<ref>) the binary integer decisions are constrained such that only one grid is selected for each cell.
To consider the existing district heating grid in our solution, a constraint is added to guarantee that one of the two centralized heating technologies will be used in a cell if it already has a district heating grid. This if-statement can be expressed as a linear program using the Big M method in (<ref>) and (<ref>), where $M$ is a large number.
Lastly, the investment in electric grid expansion is limited in (<ref>) to a maximum value $\Bar{\Gamma}_{de}$ to reflect practical expansion limits within the given timeframe and other electrification needs in the city.

\begin{equation}
OF \;\equiv\; min\ \sum_{i=0}^{N}\sum_{j \in J}  X_{i,j}\cdot \hat{C}_{i,j}  \label{eq1:a}
\end{equation}

\begin{multline}
\hat{C}_{i,j} = \underbrace{\frac{\hat{C}^{f}_{j}\cdot H_{i}}{\eta_{j}}}_{\text{Energy cost}} + \underbrace{{C}^{p}_{j}\cdot P_{i}}_{\text{Generator cost}}\\ + \underbrace{{C}^{l}_{j}\cdot \sigma_{j}\cdot \{
\begin{array}{ll}
L{i} \quad &  j \neq de \\
P_{i} \quad & j = de \
\end{array}
\}}_{\text{Grid expansion cost}}  \quad;\forall i, \forall j \in J \label{eq1:b}
\end{multline}

subject to:

\begin{equation}
X_{i,j} \in \{0,1\} \qquad;\forall i, \forall j \in J\label{eq1:c}
\end{equation}

\begin{equation}
\sum_{j \in J} X_{i,j} = 1 \qquad;\forall i \label{eq1:d}
\end{equation}

\begin{equation}
P_{i} \geq Q  - M \left(1 - X_{i,j} + W_{i} \right)\quad;\forall i , \forall j \in \{ce,cg\} \label{eq1:e}
\end{equation}

\begin{equation}
X_{i,j}  \leq  1 - W_{i} \quad;\forall i, \forall j \in \{de,dg\}  \label{eq1:f}
\end{equation}

\begin{equation}
\sum_{i=0}^{N}X_{i,de} \cdot P_{i} \leq \Bar{\Gamma}_{de}  \label{eq1:g}
\end{equation}

§ ROBUST MODEL

As mentioned in section <ref>, multiple factors can introduce uncertainty in our problem. However, this work focuses on modeling the uncertainty of energy prices only. This is because energy costs constitute the major share of the total economic cost and historically, energy prices have fluctuated greatly due to technological advancements and geopolitical events.

§.§ Uncertainty Set Realization

The proportional interval uncertainty set presented in (<ref>) represent the uncertainty associated with the fluctuation in energy carrier prices. The uncertain values can deviate from the predicted or nominal values by a deviation factor $\Delta$ which is given as a percentage of the predicted value.

\begin{equation}
U^{f}_{j}= \{ \Tilde{C}^{f}_{j} \in \mathbb{R}_{+}:\Big|\frac{\Tilde{C}^{f}_{j}- \hat{C}^{f}_{j}}{\hat{C}^{f}_{j}}\Big| \leq \Delta^{f}_{j}\} \quad; \forall j \in J \label{eq2}
\end{equation}

For the purpose of this research, only two energy carriers, electricity and hydrogen, are taken into account, resulting in
$\Delta^{f}_{de}=\Delta^{f}_{ce}=\Delta^{f}_{electricity}$ and $\Delta^{f}_{dg}=\Delta^{f}_{cg}=\Delta^{f}_{hydrogen}$.
This results in a box-constrained robust uncertainty set <cit.>.

§.§ Robust Formulation

Expanding the deterministic model in (<ref>) with the uncertainty set in (<ref>) results in the robust formulation in (<ref>). The resulting formulation is a max-min problem. The outer maximization problem represents the realization of the uncertainty and achieves the worst-case scenario for energy carrier prices. The inner minimization problem has the objective of choosing the decision variables that achieve the minimum cost under this worst-case scenario.

\begin{equation}
OF \;\equiv\; \max_{\tilde{C}^{f}} \min_{X}\ \sum_{i=0}^{N}\sum_{j \in J}  X_{i,j}\cdot \tilde{C}_{i,j}  \label{eq3:a}
\end{equation}

\begin{multline}
\tilde{C}_{i,j} = \frac{\tilde{C}^{f}_{j}\cdot H_{i}}{\eta_{j}} + {C}^{p}_{j}\cdot P_{i}\\ + {C}^{l}_{j}\cdot \sigma_{j}\cdot \{
\begin{array}{ll}
L{i} \quad &  j \neq de \\
P_{i} \quad & j = de \
\end{array}
\}  \quad;\forall i, \forall j \in J \label{eq3:b}
\end{multline}

subject to:

\begin{equation}
\tilde{C}^{f}_{j} \in U^{f}_{j} \qquad; \forall j \in J   \label{eq3:c}
\end{equation}

\begin{equation}
\eqref{eq1:c}\mbox{--}\eqref{eq1:g}, \eqref{eq2}  \label{eq3:e}
\end{equation}

§ RESULTS AND DISCUSSION

The study in this paper is conducted for a future scenario of the year 2045. In each cell, buildings are classified into different categories such as single-family houses, apartment buildings, commercial buildings, mixed-use buildings and public buildings. Based on the size of the buildings, state of renovation and population density in a cell, the heat energy and power demand are forecasted and aggregated in each cell <cit.>. The existing power generation capacities are assumed not to be carbon-neutral and are therefore not considered.

§.§ Data Set

Alongside the previously mentioned aggregated data in fig. <ref>, the problem parameters are complemented with the data in table <ref>. The cost of heating power generators is devided over their lifetime of 20 years. These costs are estimated based on internal studies done by industry experts from our project partners. The maximum deviation in the price of electricity is set to 50 %. In contrast, the maximum deviation in the price of green hydrogen is set to 200 % to reflect the greater uncertainty in the predicted prices of green hydrogen, due to the lack of a green hydrogen market today and the unavailability of its mass production.

Summary of data set parameters

Parameter   Value   Parameter   Value

$COP_{ce}$    3.8   $COP_{de}$    3.0

$\eta_{cg,dg}$   88%   $\bar{\Gamma}_{de}$   1.2 GW

$C_{dg}^{l}$   610 €/m   $C_{ce,cg}^{l}$   2160 €/m

$C_{de}^{l}$   320 €/kW   $\sigma_{ce,cg}$   1.1

$\sigma_{dg}$   1.2   $\sigma_{de}$   6.0

$C_{ce}^{p}$   798 €/kW   $C_{de}^{p}$   1270 €/kW

$C_{cg}^{p}$   79 €/kW   $C_{dg}^{p}$   235 €/kW

$\hat{C}_{ce}^{f}$   15.2 ct/kWh   $\hat{C}_{de}^{f}$   23.7 ct/kWh

$\hat{C}_{cg}^{f}$   10.0 ct/kWh   $\hat{C}_{dg}^{f}$   10.7 ct/kWh

$\Delta_{electricit}^{f}$   50%   $\Delta_{hydrogen}^{f}$   200%

§.§ Numerical Results

The robust mixed-integer linear programming (MILP) formulation was solved using the MOSEK solver on Apple M1 16 GB RAM in Python.
The results shown in fig. <ref> show that the bulk of heat demand should be covered by district heating using centralized heat pumps. In lower-density areas, the electrical distribution grid should be expanded to cover the heating demand using decentralized household heat pumps. In areas with very low load density, on the other hand, the hydrogen gas grid should be developed as heating using decentralized household gas boilers is the most cost-effective solution. District heating using centralized hydrogen boilers is not chosen in any cell.

Solution of the robust optimization problems showing the distribution of heating technologies.

§.§ Post-event Analysis

To analyze the effect of the size of the uncertainty set on the results, a post-event analysis was conducted. In the analysis, the optimization problem was solved for increasing values of maximum deviation in the hydrogen price from 0% to 200% and the maximum deviation in the price of electricity was kept constant at 50%. The result of the analysis shown in fig. <ref> shows that at 0% deviation the dominant chosen technologies are by far those depending on hydrogen as an energy carrier but as soon as uncertainty is introduced the decentralized heat pumps dominate the low-density areas and the maximum electric grid expansion of 1.2 GW is utilized. From a deviation value of 70% and higher, the district heating solution with electric heat pumps becomes dominant in covering most of the demand. This clearly shows that investing in the district heating grid is a low-risk decision since it is cost-effective in most of the higher-load density areas in the city regardless of the deviation value and the energy carrier used.

The impact of the size of the uncertainty set on the investment cost in the energy infrastructure, which includes the grid expansion and generator costs is shown in fig. <ref>. We can see an inherent characteristic of RO. As the size of the uncertainty set increases, the solution becomes more conservative and therefore the investment costs needed increase. The analysis indicates that when the deviation grows to 10% the costs increase by 23% compared to the initial value at 0% deviation. Beyond 70% deviation, the costs increase by 38% over the initial value.

Effect of the size of the uncertainty set due to increasing deviation from the predicted hydrogen prices on the installed grid capacity.

Effect of the size of the uncertainty set on the investment cost in infrastructure.

§ CONCLUSION AND OUTLOOK

This paper proposed a novel robust optimization approach to model the uncertainty of energy prices in grid expansion planning to aid decision-making in a new planning process for integrated multi-energy grids. The uncertainty in energy prices is modeled using interval uncertainty sets with a proportional deviation resulting in a box-constrained uncertainty. The new method provides significant benefits for strategic, integrated grid planning. Primarily it helps grid planners and regulating authorities make strategic decisions to prioritize the expansion of the individual grids in specific areas of the city with reduced investment risk despite the uncertainty. This approach can additionally be implemented directly and pragmatically by grid planners in their current planning processes. Furthermore, it is a main component of an innovative holistic integrated planning process for multi-energy grids.

According to the results expanding the district heating grid is a low-risk investment in areas with high load density. Conversely, in areas where load density is below 10 MW/km2, the key to carbon neutrality in heating lies in the expansion of the electric grid, which will facilitate the adoption of decentralized household heat pumps. In areas where load density is low and where prioritizing electric grid expansion is not feasible or limited, the development of a hydrogen gas grid emerges as a strategic move. The results also show that the solution became more conservative as the uncertainty set became larger.

The proposed approach shows potential for expansion by modeling more uncertain parameters and heating technologies as it has a short runtime despite the large amount of data. The next steps are to expand this method to model the more uncertain parameters as well as implement more detailed modeling of the uncertainties using distributionally robust and chance-constrained optimization methods.

§ ACKNOWLEDGMENT

The authors of this paper thank their industrial project partners Hamburger Energiewerke GmbH, Gasnetz Hamburg GmbH and Stromnetz Hamburg GmbH for their technical support and insight into the processes of grid operation and planning.

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abstractTo meet ever-stricter climate targets and achieve the eventual decarbonization of the energy supply of German industrial metropolises, the focus is on gradually phasing out nuclear power, then coal and gas combined with the increased use of renewable energy sources and employing hydrogen as a clean energy carrier. While complete electrification of the energy supply of households and the transportation sector may be the goal, a transitional phase is necessary as such massive as well as rapid expansion of the electrical distribution grid is infeasible. Additionally, German industries have expressed their plans to use hydrogen as their primary strategy in meeting carbon targets. This poses challenges to the existing electrical, gas, and heating distribution grids. It becomes necessary to integrate the planning and developing procedures for these grids to maximize efficiencies and guarantee security of supply during the transition. The aim of this paper is thus to highlight those challenges and present novel concepts for the integrated planning of the three grids as one multi-energy grid.
eventtitle2022 NEIS Conference
titleIntegrated Planning of Multi-energy Grids: Concepts and Challenges

eventtitleETG Congress
titleA multifaceted approach with high regional resolution to a coordinated scenario framework as a basis for integrated grid planning in Hamburg

Belgrade, Serbia

eventtitle2023 IEEE Belgrade PowerTech
titleCoupled Multi-Energy Grid Planning - Paving the Way from Isolated to Integrated Planning

abstractIn pursuit of identifying the most accurate and efﬁcient uncertainty modelling (UM) techniques, this paper provides an extensive review and classiﬁcation of the available UM techniques for probabilistic power system stability analysis. The increased penetration of system uncertainties related to renewable energy sources, new types of loads and their ﬂuctuations, and deregulation of the electricity markets necessitates probabilistic power system analysis. The abovementioned factors signiﬁcantly affect the power system stability, which requires computationally intensive simulation, including frequency, voltage, transient, and small disturbance stability. Altogether 40 UM techniques are collated with their characteristics, advantages, disadvantages, and application areas, particularly highlighting their accuracy and efﬁciency (as both are crucial for power system stability applications). This review recommends the most accurate and efﬁcient UM techniques that could be used for probabilistic stability analysis of renewable-rich power systems.
titleA Review of Uncertainty Modelling Techniques for Probabilistic Stability Analysis of Renewable-Rich Power Systems
akami et al. - 2022 - A Review of Uncertainty Modelling Techniques for P.pdf:application/pdf

abstractDistribution expansion planning (DEP) is used to determine the best expansion plan of a distribution system. Considering the uncertainties of loads and of power productions of wind distributed energy resources (DERs) and incorporating AC power ﬂow equations make the DEP problem increasingly challenging. In this context, this paper presents a new adaptive robust distribution expansion planning model to identify the timing of feeder reinforcements in addition to the location, capacity, and installation time of dispatchable and wind DERs. The proposed approach characterizes the uncertain nature of loads and power productions of wind DERs through polyhedral uncertainty sets and the robustness of the solution can be controlled by means of a budget of uncertainty. Since AC power ﬂow equations rather than DC ones are considered, the proposed model is a nonlinear and non-convex min–max–min optimization problem, which cannot be solved directly by commercial optimization packages. Hence, a trilevel decomposition algorithm using both primal and dual cutting planes is introduced to solve the problem. Additionally, to attain a smoother optimization problem, the non-convex AC power ﬂow equations are convexiﬁed. The working of the proposed model is illustrated using 33-bus and 123-bus distribution networks.
issn0885-8950, 1558-0679
journaltitleIEEE Transactions on Power Systems
shortjournalIEEE Trans. Power Syst.
titleAdaptive Robust Expansion Planning for a Distribution Network With DERs
daptive Robust Expansion Planning for a Distribut.pdf:C
Amjady et al. - 2018 - Adaptive Robust Expansion Planning for a Distribut.pdf:application/pdf

Springer International Publishing

booktitleHandbook of Optimization in Electric Power Distribution Systems
noteSeries Title: Energy Systems
titleElectric Distribution Network Planning Under Uncertainty
ópez et al. - 2020 - Electric Distribution Network Planning Under Uncer.pdf:application/pdf

Springer International Publishing

booktitleRobust Optimal Planning and Operation of Electrical Energy Systems
titleRobust Short-Term Electrical Distribution Network Planning Considering Simultaneous Allocation of Renewable Energy Sources and Energy Storage Systems
ohammadi-ivatloo and Nazari-Heris - 2019 - Robust Optimal Planning and Operation of Electrica.pdf:application/pdf

abstractIn this paper, a robust distribution system expansion planning (DSP) approach is presented to supply the load growth locally and move toward nearly zero energy local distribution areas (LDAs). In the proposed approach, a distribution system operator (DSO) is responsible for secure and optimum operation of LDAs. Therefore, investors on distribution system upgrades use this approach to maximize the proﬁt on investments by determining the installation year of new distribution feeders and energy resources, distributed energy resource (DER) placements and sizes considered by corresponding DSOs. The accurate AC power ﬂow solution is used and mathematical methods are developed to model the DSP as a quadratically constrained programming (QCP) problem. The Benders decomposition is applied to investigate the reliability and the optimality of the proposed plan and correspondingly modify the investment plan as required. The uncertainty of renewable DER forecast errors, locational marginal price (LMP) of an LDA at a transmission bus, and transactive power associated with a load bus is modeled using robust optimization. The proposed transactive DSP (TDSP) approach is implemented on the IEEE 33-bus distribution test system and the results are analyzed and validated. The proposed numerical results show the optimality and the robustness of the proposed approach.
issn0885-8950, 1558-0679
journaltitleIEEE Transactions on Power Systems
shortjournalIEEE Trans. Power Syst.
titleRobust Coordinated Distribution System Planning Considering Transactive DSO's Market
abiri-Renani et al. - 2023 - Robust Coordinated Distribution System Planning Co.pdf:application/pdf

abstractElectric power systems and the companies and customers that interact with them are experiencing increasing levels of uncertainty due to factors such as renewable energy generation, market liberalization, and climate change. This raises the important question of how to make optimal decisions under uncertainty. This paper aims to provide an overview of existing methods for modeling and optimization of problems affected by uncertainty, targeted at researchers with a familiarity with power systems and optimization. We also review some important applications of optimization under uncertainty in power systems and provide an outlook to future directions of research.
journaltitleElectric Power Systems Research
shortjournalElectric Power Systems Research
shorttitlePower systems optimization under uncertainty
titlePower systems optimization under uncertainty: A review of methods and applications
oald et al. - 2023 - Power systems optimization under uncertainty A re.pdf:application/pdf

abstractThis article presents a two-stage adaptive robust optimization (ARO) for optimal sizing and operation of residential solar photovoltaic (PV) systems coupled with battery units. Uncertainties of PV generation and load are modeled by user-deﬁned bounded intervals through polyhedral uncertainty sets. The proposed model determines the optimal size of PV–battery system while minimizing operating costs under the worst-case realization of uncertainties. The ARO model is proposed as a trilevel min–max–min optimization problem. The outer min problem characterizes sizing variables as “here-and-now” decisions to be obtained prior to uncertainty realization. The inner max–min problem, however, determines the operation variables in place of “wait-and-see” decisions to be obtained after uncertainty realization. An iterative decomposition methodology is developed by means of the column-and-constraint technique to recast the trilevel problem into a single-level master problem (the outer min problem) and a bilevel subproblem (the inner max–min problem). The duality theory and the Big-M linearization technique are used to transform the bilevel subproblem into a solvable single-level max problem. The immunization of the model against uncertainties is justiﬁed by testing the obtained solutions against 36 500 trial uncertainty scenarios in a postevent analysis. The proposed postevent analysis also determines the optimum robustness level of the ARO model to avoid over/under conservative solutions.
issn1551-3203, 1941-0050
journaltitleIEEE Transactions on Industrial Informatics
shortjournalIEEE Trans. Ind. Inf.
titleTwo-Stage Robust Sizing and Operation Co-Optimization for Residential PV–Battery Systems Considering the Uncertainty of PV Generation and Load
Aghamohamadi et al. - 2021 - Two-Stage Robust Sizing and Operation Co-Optimizat.pdf:application/pdf

abstractTo explore the benefit of energy storage for countering high-level wind power fluctuations, a two-stage distributionally robust optimization model is proposed for wind farms and storage units (SUs) jointly operated power systems. First, the 1-norm and ∞ -norm confidence sets are presented to model the fluctuations of wind power output, then a two-stage distributionally optimization model is formed to minimize system total cost with secure operation constraints, where the ON-OFF status of generators and SUs are determined in the first stage by day-ahead dispatching, while the power output of generators, wind power curtailment, load shedding and SUs charging/discharging power are optimized in the second stage. Afterward, the columnand-constraint generation (CCG) algorithm is presented to solve the proposed two-stage model. Finally, the influence of confidence level of confidence sets and SU capacities on system total cost is analyzed, and the effectiveness of the proposed model is also validated by the case studies.
journaltitleIEEE Access
shortjournalIEEE Access
titleA Two-Stage Distributionally Robust Optimization Model for Wind Farms and Storage Units Jointly Operated Power Systems
i et al. - 2021 - A Two-Stage Distributionally Robust Optimization M.pdf:application/pdf

Frankfurt (Main)

titleRepresentative VDEW Load Profiles (Repraesentative VDEW-Lastprofile)

abstractA robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library.
issn0030-364X, 1526-5463
journaltitleOperations Research
shortjournalOperations Research
titleThe Price of Robustness
ertsimas and Sim - 2004 - The Price of Robustness.pdf:application/pdf

abstractIn view of the relatively large energy consumption of national building stocks, many cities and municipalities start to prepare energetic building stock models to monitor energy efficiency and plan policies at city or regional scales. In many cases, data on individual buildings is not available. A usual approach to this is the “archetype” approach – classifying the building stock into energetic types (archetypes). This classification is usually based on non-energetic properties available in digital cadastres (construction type, year of construction etc.) and can be a large source of error. We present our research into the difficulties and pitfalls associated with such an approach using the city of Hamburg as an example. In the end, we compare the modelled estimates with consumption data at three different levels to evaluate model performance.
journaltitleEnvironmental and Climate Technologies
titleAssigning Energetic Archetypes to a Digital Cadastre and Estimating Building Heat Demand. An Example from Hamburg, Germany
igital Cadast.pdf:C
Dochev et al. - 2020 - Assigning Energetic Archetypes to a Digital Cadast.pdf:application/pdf