# Optimal Power Dispatch in Energy Systems Considering Grid Constraints

^{*}

## Abstract

**:**

_{p}per household can lead to curtailment of ∼60 MWh per year due to line loading. Furthermore, the proposed method eliminates grid violations due to the addition of new sectors and reduces the energy curtailment up to $45\%$. With the optimization of the heat pump operation, an increase of $7\%$ of the self-consumption was achieved with similar results for the combination of battery systems and electrical vehicles. In conclusion, a safe and optimal operation of a complex energy system is fulfilled. Efficient control strategies and more accurate plant sizing could be derived from this work.

## 1. Introduction

#### 1.1. Background

#### 1.2. Optimization of Energy Systems

## 2. Methodology

#### 2.1. Overview: Iterative Process

- Energy system structure: Here, the energy sectors to be considered are defined as well as the relevant technologies and their models. Natural renewable resources and demands time series must also be included. Time steps have to be big enough to make valid the steady-state assumption of the different energy sectors [18]. In this work, only the electric and heat sector are considered and a time step of an hour during a year is evaluated. Although several technologies, markets, and demands could be present in a distributed energy system [25], only the ones presented in Table 1 were considered in this study. Furthermore, the energy system is seen as a system aggregator from the grid perspective. This means that the system can buy and sell electricity in the market.
- Linear unit commit: Using a holistic approach, an abstraction of the energy system structure is created in oemof-Solph. This abstraction contains all possible energy flows between sources and sinks, and between energy sectors through the coupling technologies as depicted in Figure 2.Costs and additional constraints can be associated with each of the energy flows [23]. Here, the system will only consider the economical constraints of the energy flows. Therefore, the objective function is the cost minimization. From Equation (1), the following objective equation is derived for the considered energy system:$$\begin{array}{cc}\hfill min:& \sum _{t\in T}({p}_{elec\_ext}\left(t\right)\xb7{c}_{day\_ahead}\left(t\right)\hfill \end{array}$$$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{p}_{in\_chp}\left(t\right)\xb7{c}_{gas}+{p}_{in\_boiler}\left(t\right)\xb7{c}_{gas})\xb7\tau $$$${p}_{elec\_gen}\left(t\right)={p}_{pv}\left(t\right)+{p}_{elec\_chp}\left(t\right)$$$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{p}_{elec\_loads}\left(t\right)={p}_{elec\_demand}\left(t\right)+{p}_{battery}\left(t\right)+{p}_{ev}\left(t\right)+{p}_{elec\_heat\_pump}\left(t\right)$$In Equations (7) and (8), the parameters ${p}_{pv}$ and ${p}_{elec\_chp}$ refer to the PV generation and the CHP electric power generation within the system, whereas the parameters ${p}_{elec\_demand}$, ${p}_{battery}$, ${p}_{ev}$, and ${p}_{elec\_heat\_pump}$ refer to the electric demand, the battery power consumption, electrical vehicle charging, and the heat pump demand, respectively. Every storage unit (including EVs) has been considered as a load. That means that a negative power represents a power injection to the system. The same convention has been used for the external grid power flow. As constraints for the optimization problem, the thermal and electric demands must be supplied at any time. Equations (9) and (10) depict such constraints where ${p}_{th\_demand}$, ${p}_{elec\_demand}$, and ${p}_{pv}$ are fixed time series.$${p}_{th\_demand}\left(t\right)={p}_{th\_heat\_pump}\left(t\right)+{p}_{th\_boiler}\left(t\right)+{p}_{th\_chp}\left(t\right)-{p}_{th\_storage}\left(t\right)$$$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{p}_{elec\_demand}\left(t\right)={p}_{elec\_ext}+{p}_{pv}\left(t\right)+{p}_{elec\_chp}\left(t\right)-{p}_{battery}\left(t\right)-{p}_{ev}\left(t\right)-{p}_{elec\_heat\_pump}\left(t\right)$$As a result, the optimized power dispatch for each non-fixed source to supply the demand at each time step is obtained. However, up to this point, just the total installed capacities and demands have been considered as per the inequality (4). The actual topology of the electrical and thermal grid has been also disregarded as a common practice in the optimization of energy systems [19].
- Grid topology consideration: Before integrating these flows into the electrical grid in PowerFactory, the flow distribution must be considered. This applies only to the electric sources, sinks, and sector coupling technologies. The distribution of heat technologies is disregarded since only the electrical grid topology is being considered in this study.To represent a typical low voltage grid, one of the so-called “Merit Order Netz-Ausbau 2030” (MONA) reference grids is used. The ONT_8003 grid model is used as reference [26] and depicted in Figure 3. This topology represents a typical low voltage grid in Germany.The energy flows for each node per technology are derived from the optimization results and the capacity installed per building.
- Power flow calculation: Once the flow distributions at each node have been obtained, these energy flows must be added as time characteristics to the corresponding elements in PowerFactory.To determine if the optimized dispatch complies with the grid standards, a quasi-dynamic power flow (QDPF) study is performed for a year with 1 h time steps. From the results of the QDPF, lines exceeding 100% loading and voltage variation outside of the range of $\pm 10\%$ of the nominal voltage are considered as grid violations [8].The time steps containing such violations will be re-optimized by PowerFactory to avoid line congestion and bus voltage violations. For the system to converge, a dispatchable source is considered as slack, so it has enough power to cover the demand in case that the renewable sources are curtailed due to system violations.
- Constraints generation: Similar to the constraint generation for the linear optimization [27], the oemof-Solph model will be limited by constraints generated from the OPF as denoted in Equation (4). Power time-series are the link between the two tools. Therefore, PowerFactory and oemof-Solph will exchange information about the active power flows, being the reactive power flow after the last OPF calculation considered as optimal.To build the OPF problem, the following objective function and constraints are considered:$$\begin{array}{cc}\hfill min:& \sum _{t\in {T}_{opf}}({p}_{elec\_ext}^{opf}\left(t\right)\xb7{c}_{day\_ahead}\left(t\right)\hfill \\ & +{p}_{in\_chp}^{opf}\left(t\right)\xb7{c}_{gas})\xb7\tau \hfill \end{array}$$$$\begin{array}{cc}\hfill s.t.& {S}_{line}^{l}\left(t\right)\le {S}_{nom}^{l}\left(t\right),\forall l\in L\hfill \end{array}$$$$\begin{array}{cc}& {S}_{line}=\sqrt{{P}_{line}^{2}+{Q}_{line}^{2}}\hfill \end{array}$$$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0.9\xb7{V}_{nom}\le \left|{V}_{bus}^{b}\left(t\right)\right|\le 1.1\xb7{V}_{nom},\forall b\in B$$$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{-{e}_{battery}(t-\tau )}{\tau}\le {p}_{battery}^{opf}\left(t\right)\le \frac{{k}_{battery}-{e}_{battery}(t-\tau )}{\tau}$$$$0\le {p}_{pv}^{opf}\left(t\right)\le {p}_{pv}\left(t\right)$$$${S}^{n}\le {S}_{nom}^{n},\forall n\in N$$In Equation (11), the term for the thermal boiler is not considered. This is due to the fact that only electric units are taken into consideration in the OPF. The set ${T}_{opf}$ is a sub set of T comprising only the periods of time that led to grid violations from step 4. The variables ${p}_{elec\_ext}^{opf}$ and ${p}_{elec\_chp}^{opf}$ denote the optimized power flows to avoid such violations. The constraints (12) and (13) ensure that the nominal capacity ${S}_{nom}$ of any line of the system is not violated by the actual power flow ${S}_{line}$. The set of all lines in the energy system is denoted by L. The actual power flow in the line is given by the square root of the sum of squares of the real power ${P}_{line}$ and the reactive power ${Q}_{line}$. To ensure voltage compliance, the constraint (14) is included. This keeps the voltage magnitude $|{V}_{bus}|$ of all buses in the system, denoted by the set B, within $\pm 10\%$ of the nominal voltage ${V}_{nom}$. The constraint (15) limits the power that can be drawn by the batteries. The difference between the battery system capacity ${k}_{battery}$ and the battery system energy content ${e}_{battery}$ at the previous time-step $t-\tau $ provides the available energy to be drawn for charging the batteries. The energy content is provided from the linear optimization performed in step 2. The difference is divided by $\tau $ in order to get the charging power limit. A lower energy bound is given to allow the discharge of the batteries. This lower bound is equal to the negative of the energy content. In the OPF additional flexibility is provided with the PV system. The optimizer can reduce the power output ${p}_{pv}^{opf}$ as per constraint (16). Furthermore, to keep the thermal limits of all technologies within acceptable ranges, constraint (17) is included. Where the apparent power ${S}^{n}$ of each technology of the set N must not exceed the nominal apparent power ${S}_{nom}^{n}$. Similarly to the linear optimization, constraint (10) must be complied by the OPF, therefore:$$\begin{array}{cc}& {p}_{elec\_demand}\left(t\right)={p}_{elec\_ext}^{opf}\left(t\right)+{p}_{pv}^{opf}\left(t\right)+\hfill \\ & {p}_{elec\_chp}^{opf}\left(t\right)-{p}_{battery}^{opf}\left(t\right)-{p}_{ev}^{opf}\left(t\right)-{p}_{elec\_heat\_pump}^{opf}\left(t\right)\hfill \end{array}$$In Equation (18), the superscript $opf$ denotes the power flow obtained from the OPF from the respective source or sink. Considering that $CO{P}_{heat\_pump}=\frac{{p}_{th\_heat\_pump}}{{p}_{elec\_heat\_pump}}$ and ${p}_{th\_sources}={p}_{th\_heat\_pump}+{p}_{th\_boiler}+{p}_{th\_chp}$, Equations (18) and (9) can be combined to yield:$$\begin{array}{c}\hfill \frac{{p}_{th\_demand}-{p}_{th\_sources}-{p}_{th\_heat\_pump}-{p}_{th\_storage}}{CO{P}_{heat\_pump}}={p}_{elec\_ext}^{opf}+{p}_{elec\_gen}^{opf}-{p}_{elec\_loads}^{opf}\\ \hfill +{p}_{elec\_heat\_pump}^{opf}\end{array}$$Equation (19) shows how the changes in the OPF have to be compensated by the heat elements in the energy system. However, as previously indicated, the OPF only takes into consideration the electric components and the sector coupling technologies. Therefore, the changes in the OPF have to be reflected in the linear optimization. Equation (20) is employed to determine the deviation between the OPF and the initial power flow.$$\begin{array}{c}\hfill \Delta {p}_{n}=|{p}_{n}^{opf}-{p}_{n}|,\forall n\in N\end{array}$$The power flow per ${n}_{th}$ technology resulting from the OPF is represented by ${p}_{n}^{opf}$, whereas the initial power from the linear optimization is represented by ${p}_{n}$. Whenever $\Delta {p}_{n}\ne 0$, the value of the OPF is passed as an upper bound in (4) for the linear optimization in the next iteration.

#### 2.2. Evaluation Scenarios

- High generation scenario: The size of the PV installation is fixed to 1500 kW
_{p}and no storage or flexible loads are considered. The influence of line loading and voltage levels in the optimization is evaluated. - Heat pump and heat storage scenario: The influence of heat pump and heat storage is analyzed. Here, the size of the PV installation is reduced to 700 kW. Heat pumps and heat storage are added with 600 kW
_{th}and 150 m${}^{3}$ of capacity, respectively. The potential for flexibilization services from the heat sector is evaluated in this scenario through the implementation of the proposed method. - Electromobility scenario: A fleet of 62 EVs and 500 kWh of battery storage are added to the Heat pump and heat storage scenario. These are connected at eight different points within the network. It is assumed that the EVs are only connected from 6 p.m. to 7 a.m. of the next day [28]. Additionally, it is assumed that the daily required demand of the EVs is around 10 kWh, which is approximately the double required per EV per day [6]. Therefore, the state of charge is not the constraint for charging at the end of the charging period in the optimization, but to ensure enough daily coverage in a daily basis.

## 3. Results

#### 3.1. High Generation Scenario

#### 3.1.1. Line Loading Constraint

#### 3.1.2. Voltage Constraint

#### 3.2. Heat Pump and Heat Storage

_{p}for this scenario, the mismatch between the electric demand and the generation will still create energy flows back to the grid. Such energy flows are big enough to overload the main feeder. As the heat pumps and heat storage are considered in the energy system, the optimizer has the possibility to transfer the energy surplus from the PV system into another energy carrier. Figure 7 shows how the line loading constraint due to the high PV feed-in affects the power dispatch of the heat pumps and PV. The flexibilization of the heat pump activation avoids that the locally generated energy leaves the system. This diminishes the energy injected into the grid, avoiding in this manner the line overloading and diminishing the PV curtailment.

#### 3.3. Electromobility Scenario

## 4. Discussion

_{p}per household are prompted to lead to grid violations [8]. From the results, it may be inferred that a PV installed capacity of 5 kW

_{p}per household can be led to feed-in levels exceeding the nominal voltage in a typical low voltage grid [26]. Nevertheless, over-voltages exceeding $10\%$ of the nominal voltage are very unlikely to occur at this level of penetration. In the case of high share of PV without sector coupling or storage possibility, a curtailment up to ∼60 MWh per year is applied by the method to keep grid limits within acceptable ranges.

## 5. Summary and Outlook

- For scenarios with PV capacity above 5 kW
_{p}per household, the voltage and line loading constraints are violated, therefore affecting the optimization results. In the case study, curtailment of ∼60 MWh/year is needed to keep the grid within its limits. This due to the lack of storage or flexibilization options. - If flexible technologies are present, parameters as curtailment and self consumption are affected by the grid constraints. In the case study, the self-consumption can increase $7\%$ and the curtailment be reduced by $45\%$ as compared with the typical optimization without grid constraints. The method successfully integrated the additional load from the heat pumps and EV without grid violations.
- In comparison to other approaches, the decoupled constraint generation in the optimization problem determines the techno-economical power dispatch for all the different energy sectors and storage technologies.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Simplified example of an energy system structure representation in oemof-Solph and the energy flow between sectors. The blue lines represent the electrical part of the system, whereas the red and orange represent the heat and gas sectors, respectively.

**Figure 3.**Representation of the ONT_8003 MONA grid with the addition of the technologies mentioned in Table 1.

**Figure 4.**Main feeder loading with and without the consideration of the grid constraints into the optimization.

**Figure 5.**PV bus voltage at node number 4 with and without the consideration of the grid constraints into the optimization.

**Figure 6.**PV power dispatch optimization with and without the consideration of the grid constraints.

**Figure 7.**Flexibilization provided by heat pumps and heat storage. The blue area represents the energy otherwise curtailed without the grid constraints implementation in the dispatch optimization.

Source | Market | Storage | Coupling | Demand |
---|---|---|---|---|

Photovoltaic (PV) | Natural Gas | Battery | Heat Pump | Electricity |

Cogeneration (CHP) | Electricity | Hot Water Storage | CHP | Heat |

Gas Boiler | Electro-Vehicles |

**Table 2.**Summary of the influence of grid constraints in the optimization parameters. Values are shown in MWh/year.

Scenario | Self-Consumption | Curtailment | Energy Saved | ||
---|---|---|---|---|---|

Without Grid Constraints | With Grid Constraints | Without Grid Constraints | With Grid Constraints | ||

High PV Generation | 288.60 | 288.60 | 270.04 | 270.04 | 0 |

Heat Pump | 323.91 | 346.71 | 60.65 | 37.84 | 22.81 |

Electromobility | 434.87 | 462.16 | 49.55 | 27.64 | 21.91 |

Source | Considered Energy Carriers | Storage Technology | Grid Constraints | Approach |
---|---|---|---|---|

Cesena et al. [33] | Gas, heat, and power | Building heat inertia | Gas and power | MILP with heuristic penalization of grid constraints |

Huang et al. [34] | Electromobility and power | EV battery | Power | Optimal EV charging scheduling through a genetic algorithm |

Clegg et al. [35] | Gas, heat and power | — | Gas | Determination of flexibility according to gas availability and DC-OPF |

Nolden et al. [17] | Power | — | Power | Unit commitment and power flow solver to achieve a techno-economic dispatch |

This research | Electromobility, heat, gas, and power | Battery storage, EV battey, and heat storage | Power | MILP with iterative grid constraint generation |

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**MDPI and ACS Style**

Rubio, A.; Schuldt, F.; Klement, P.; von Maydell, K. Optimal Power Dispatch in Energy Systems Considering Grid Constraints. *Energies* **2022**, *15*, 192.
https://doi.org/10.3390/en15010192

**AMA Style**

Rubio A, Schuldt F, Klement P, von Maydell K. Optimal Power Dispatch in Energy Systems Considering Grid Constraints. *Energies*. 2022; 15(1):192.
https://doi.org/10.3390/en15010192

**Chicago/Turabian Style**

Rubio, Alejandro, Frank Schuldt, Peter Klement, and Karsten von Maydell. 2022. "Optimal Power Dispatch in Energy Systems Considering Grid Constraints" *Energies* 15, no. 1: 192.
https://doi.org/10.3390/en15010192