The main goal of the OPF model is to explore how flexibility in the distribution grid can benefit DSO and consumers and for TSO who operate the transmission grid. Therefore, including the transmission grid in the optimization model will be necessary to analyze the impact resource activation in the distribution grid will have on the transmission grid. A convex conic program like the SOC-ACOPF is not suited for a meshed grid, which often is how a transmission grid is structured. The focus of the transmission grid is to analyze production and congestion management, which are information that does not require a detailed grid
description like voltage magnitude and reactive power. A common method used for such application in the transmission grid and market operation is the DC-OPF method [62]. Thus, this method will be integrated within the optimization model to solve the power flow in the transmission grid.
The mathematical formulation of the DC-OPF is less comprehensive than the SOC-ACOPF due to the possibility of neglecting voltage magnitudes and reactive power. The optimization objective will be the same as in the SOC-ACOPF, which minimizes production cost while still respecting the grid constraints.
The first constraint is a balancing equation (12), where the sum of susceptance (B) multiplied with voltage angle (θ) dictate the power flow in and out of the node. This sum should then be equal to the difference between production (PnG) and the load (PnL) on that said node.
PnG−PnL=− X
j∈k(n)
Bnjθj (12)
Equation (13) dictate the flow of power for each line. Variable representing the power flow for each line (Pnjf l) is decided by the voltage angle between each node and the susceptance of the line connecting these two nodes.
Pnjf l=Bnj(θn−θj) (13)
Each generator can not produce infinite amount of power, so each generator requires a maximum and min-imum constraint which limits the production as shown in equation (14).
PnG,min≤PnG ≤PnG,max (14)
In the same way there are limitation on generators, power flow can also be limited. Equation (15) dictates the upper and lower boundary for power flow for each line.
Pnjmin≤Pnj≤Pnjmax (15)
Lastly, one node need to be defined as the slack node, which is a node used to provide balance to the whole grid [63]. This is done by defining the voltage angle on this node to be equal to 0, as shown with equation (16).
θslackn = 0 (16)
The combined model for the distribution and transmission grid will come in the form of a hybrid DC/AC-OPF model where the DC-DC/AC-OPF will apply for the transmission grid and SOC-ACDC/AC-OPF to the distribution grid. This model gives rise to the formulated optimization problem as shown on the next page.
Hybrid AC/DC Optimization Model Formulation
Subject to SOC-ACOPF to DC-OPF Connection Constraint:
PnG,DC−PmG,AC =−X
The combination of all the variables from the SOC-ACOPF and DC-OPF leads to a system with eight variables. These consist of auxiliary variables, active and reactive production, power flow, and voltage angle. The optimization objective will remain the same as for the standalone SOC-ACOPF and DC-OPF methods. The two objectives will be combined to minimize operational costs for both the distribution and transmission grid simultaneously. Nothing has changed for SOC-ACOPF constraint, besides marking the power production variable and load parameters with ”AC” to distinguish between the two power flow methods.
One new constraint is the AC to DC connection constraint, which connects the distribution grid to the transmission grid. This constraint requires one distribution node to be defined as a feeder/slack node that supplies the power to the grid. While the distribution grid sees this node as a production node, it appears as a single load from the transmission grid’s perspective. The criteria for activation of this constraint is that the production for feeder node (PmG,AC) needs to be equal to the transmission system’s load (PnL,DC) if a connection between these nodes is established. This constraint resembles a DC power-balance constrain, where the DC load parameter (PnL,DC) has been replaced by the AC production variable (PmG,AC) in the distribution grid.
In the same way, the SOC-ACOPF parameters and variables have been marked with ”AC”, so will the DC-ACOPF parameters and variables be marked with ”DC.” Another modification is the reformulated constraint criteria for the power balance equation and slack bus voltage angle. The new criteria for the power balance constraint are that activating this constraint will only happen as long as the transmission node is not connected to a distribution node. Moreover, the slack voltage angle can only be applied to a node not connected to the distribution feeder/slack node.
4.4.1 Hybrid AC/DC-OPF Model Verification
This chapter will provide a comparison between power flow models to prove that the hybrid AC/DC-OPF model will converge towards a true power flow solution. This comparison will analyze if the combined hybrid AC/DC-OPF model will yield the same power flow and production results as a SOC-AC and DCOPF for the same distribution and transmission grid, run separately. This approach is as follows:
• Perform power flow simulation for the distribution grid with the SOC-ACOPF method.
• Utilize the feeder node production found with the SOC-ACOPF as load in the transmission grid and run DC-OPF simulation.
• Conduct simulation for the combined system with the hybrid AC/DC-OPF model.
• Compare the power flow results found in both the distribution and transmission grid with the result found with the SOC-AC and DC-OPF.
The test system in use, is a 33 node distribution grid [64], and a 9 node transmission grid system [65]. These systems will also be utilized further in this thesis and will be given a more thorough explanation in chapter
Table 2: Result comparison between the combined hybrid AC/DC-OPF model and the standalone DC-OPF and SOC-ACOPF models.
Active power production [pu]
Reactive power Production [pu]
Voltage angle [deg]
Voltage magnitude [pu]
Deviation from DC-OPF
4,87E-05 - 6,30E-04
-Deviation from SOC-ACOPF
7,00E-05 4,00E-05 1,35E-04 5,49E-05
Table 2 presents the most significant deviations for different values between the hybrid AC/DC-OPF model and the SOC-ACOPF and DC-OPF model. As it appears, there are some minor deviations between the values produced by the different models. The SOC-ACOPF and DC-OPF method are already confirmed to be correct. The only new concept is the connection constraint. Thus, the comparison demonstrates that the connection between the two methods by setting the feeder node’s production equal to the transmission node’s load is correct.