Optimal Power Flow

Author

Aleksandra Lekić, Haixiao Li

Published

February 20, 2026

Tip

AC/DC Optimal Power Flow (AC/DC OPF) is an optimization method used to determine the most economical and secure operating point of hybrid AC/DC power systems. It simultaneously considers AC network constraints, DC grid equations, and converter coupling relationships. By coordinating generators and converters, it minimizes operational cost while satisfying voltage, power flow, and equipment limits. AC/DC OPF is essential for modern grids with high renewable penetration and multi-terminal DC transmission systems.

Nomenclature

A. Indices and Sets

$\left(\overline{\cdot}\right)/\left(\underline{\cdot}\right)$: Upper/lower bound of corresponding optimization variables and parameters
$\textit{AC}$: Alternating current
$\textit{MTDC}$: Multi-terminal direct current
$\textit{VSC}$: Voltage source converter
$\textit{h, i, j}$: System node
$\textit{ij}$: System branch
$\textit{ref}$: Reference value
$\textit{droop}$: Droop parameters
$\textit{loss}$: Power loss
$\textit{cost}$: Generation cost
$\textit{gen}$: Generator
$\textit{load}$: Load
$\mathcal{L}$: Branch set
$\mathcal{N}$: Node set
$\lvert \cdot \rvert$: Magnitude

B. AC System Variables

$v_i^{AC}$: AC bus $i$ voltage
$\theta_{ij}^{AC}$: Phase difference of AC branch $ij$
$p_i^{AC}$ / $q_i^{AC}$: Real/Reactive power injection at AC bus $i$
$p_{ij}^{AC}$ / $q_{ij}^{AC}$: Real/Reactive power of AC branch $ij$
$p_{i,gen}^{AC}$ / $q_{i,gen}^{AC}$: Real/Reactive generator output at AC bus $i$
$p_{i,load}^{AC}$ / $q_{i,load}^{AC}$: Real/Reactive load at AC bus $i$
$p_{i,A2V}^{AC}$ / $q_{i,A2V}^{AC}$: Power delivered from AC grid to VSC at bus $i$
$c_{i,2}^{AC}$ / $c_{i,1}^{AC}$ / $c_{i,0}^{AC}$: Quadratic/Linear/Constant generation cost coefficients
$c_{ij}^{AC}$ / $s_{ij}^{AC}$: Optimization variables related to AC nodal voltages

C. MTDC System Variables

$v_j^{MTDC}$: MTDC bus $j$ voltage
$u_j^{MTDC}$: Squared MTDC bus voltage
$p_{jh}^{MTDC}$: Power of MTDC branch $jh$
$l_{jh}^{MTDC}$: Squared current of MTDC branch $jh$
$p_{i,ref}^{MTDC}$: DC power reference at MTDC bus $i$
$V_{i,ref}^{MTDC}$: DC voltage reference at MTDC bus $i$
$k_{i,droop}^{MTDC}$: Droop slope at MTDC bus $i$

D. VSC Variables

$v_s^{VSC}$ / $v_f^{VSC}$ / $v_c^{VSC}$: VSC bus $s/f/c$ voltage
$\theta_s^{VSC}$ / $\theta_f^{VSC}$ / $\theta_c^{VSC}$: VSC bus phase angle
$g_{sf}^{VSC}$ / $g_{fc}^{VSC}$: Conductance of VSC branches
$b_{sf}^{VSC}$ / $b_{fc}^{VSC}$: Susceptance of VSC branches
$b_f^{VSC}$: Shunt susceptance at VSC bus $f$
$p_{loss}^{VSC}$: VSC power loss
$a_{loss}^{VSC}$ / $b_{loss}^{VSC}$ / $c_{loss}^{VSC}$: Constant/Linear/Quadratic loss coefficients
$I_c^{VSC}$: Phase current at bus $c$
$l_c^{VSC}$: Squared phase current
$q_{s,ref}^{VSC}$: Reactive power reference
$v_{s,ref}^{VSC}$: AC voltage reference
$c_{sf}^{VSC}$ / $s_{sf}^{VSC}$ / $c_{cf}^{VSC}$ / $s_{cf}^{VSC}$: Optimization variables related to VSC nodal voltages

Second-Order Cone Programming (SOCP) Formulation

The speicifc AC/DC OPF model utilized in HARMONY is presented below.

Varables

\[\begin{align*} & p_i^{AC}, q_i^{AC}, p_{i, gen}^{A C}, q_{i, gen}^{A C}, p_{i, A2V}^{AC}, q_{i, A2V}^{AC}, s_{ij}^{AC}, c_{ij}^{AC}&& i,j\in\mathcal{N}^{AC} \\ & p_{ij}^{AC}, q_{ij}^{AC} && (i,j)\in\mathcal{L}^{AC} \\ & p_{j}^{MTDC}, v_{j}^{MTDC} && j\in\mathcal{N}^{M T D C} \\ & p_{jh}^{MTDC},l_{jh}^{MTDC}&& (j, h)\in\mathcal{L}^{M T D C} \\ & p_s^{VSC}, q_s^{VSC} && s\in\mathcal{N}^{MTDC}\\ & p_c^{VSC}, q_c^{VSC}, I_c^{VSC}, l_c^{VSC} && c\in\mathcal{N}^{MTDC}\\ & q_f^{VSC} && f\in\mathcal{N}^{MTDC}\\ & c_{mn}^{VSC}, c_{mn}^{VSC} && m, n \in \{s, f, c\} && s,f,c\in\mathcal{N}^{MTDC}\\ \end{align*}\]

AC Constraints

\[\begin{equation} p_{ij}^{AC} = G_{ij} \left( c_{ii}^{AC} - c_{ij}^{AC} \right) + B_{ij} s_{ij}^{AC}, \quad \forall i,j\in\mathcal{N}^{AC}, \quad \forall (i,j)\in\mathcal{L}^{AC} \tag{1a} \label{eq:1a} \end{equation}\]

\[\begin{equation} q_{ij}^{AC} = - B_{ij} \left( c_{ii}^{AC} - c_{ij}^{AC} \right) + G_{ij} s_{ij}^{AC}, \quad \forall i,j\in\mathcal{N}^{AC}, \quad \forall (i,j)\in\mathcal{L}^{AC} \tag{1b} \label{eq:1b} \end{equation}\]

\[\begin{equation} p_i^{AC} = c_{ii}^{AC} G_{ii} + \sum_{j} \left( c_{ij}^{AC} G_{ij} - s_{ij}^{AC} B_{ij} \right), \quad \forall i,j\in\mathcal{N}^{AC} \tag{1c} \label{eq:1c} \end{equation}\]

\[\begin{equation} q_i^{AC} = - c_{ii}^{AC} B_{ii} - \sum_{j} \left( c_{ij}^{AC} B_{ij} + s_{ij}^{AC} G_{ij} \right), \quad \forall i,j\in\mathcal{N}^{AC} \tag{1d} \label{eq:1d} \end{equation}\]

\[\begin{equation} c_{ij}^{AC} = c_{ji}^{AC}, \quad \forall i,j\in\mathcal{N}^{AC} \tag{1e} \label{eq:1e} \end{equation}\]

\[\begin{equation} s_{ij}^{AC} = - s_{ji}^{AC}, \quad \forall i,j\in\mathcal{N}^{AC} \tag{1f} \label{eq:1f} \end{equation}\]

\[\begin{equation} (c_{ij}^{AC})^2 + (s_{ij}^{AC})^2 \le c_{ii}^{AC} c_{jj}^{AC}, \quad \forall i,j\in\mathcal{N}^{AC} \tag{1g} \label{eq:1g} \end{equation}\]

\[\begin{equation} p_i^{AC} = p_{i,gen}^{AC} - p_{i,load}^{AC} - p_{i,A2V}^{AC}, \quad \forall i\in\mathcal{N}^{AC} \tag{2a} \label{eq:2a} \end{equation}\]

\[\begin{equation} q_i^{AC} = q_{i,gen}^{AC} - q_{i,load}^{AC} - q_{i,A2V}^{AC}, \quad \forall i\in\mathcal{N}^{AC} \tag{2b} \label{eq:2b} \end{equation}\]

\[\begin{equation} \underline{p}_{i,gen}^{AC} \le p_{i,gen}^{AC} \le \overline{p}_{i,gen}^{AC}, \quad \forall i\in\mathcal{N}^{AC} \tag{3a} \label{eq:3a} \end{equation}\]

\[\begin{equation} \underline{q}_{i,gen}^{AC} \le q_{i,gen}^{AC} \le \overline{q}_{i,gen}^{AC}, \quad \forall i\in\mathcal{N}^{AC} \tag{3b} \label{eq:3b} \end{equation}\]

\[\begin{equation} \underline{|v_i^{AC}|}^2 \le c_{ii}^{AC} \le \overline{|v_i^{AC}|}^2, \quad \forall i\in\mathcal{N}^{AC} \tag{4} \label{eq:4} \end{equation}\]

DC Constraints

\[\begin{equation} p_{j}^{MTDC} = \sum_{(j,h)} p_{jh}^{MTDC}, \quad \forall j\in\mathcal{N}^{MTDC}, \quad \forall(j,h)\in\mathcal{L}^{MTDC} \tag{5a} \label{eq:5a} \end{equation}\]

\[\begin{equation} p_{jh}^{MTDC} + p_{hj}^{MTDC} = r_{jh}^{MTDC} l_{jh}^{MTDC}, \quad \forall(j,h)\in\mathcal{L}^{MTDC} \tag{5b} \label{eq:5b} \end{equation}\]

\[\begin{equation} u_{j}^{MTDC} - u_{h}^{MTDC} = r_{jh}^{MTDC} \left( p_{jh}^{MTDC} - p_{hj}^{MTDC} \right), \quad \forall j,h\in\mathcal{N}^{MTDC}, \quad \forall(j,h)\in\mathcal{L}^{MTDC} \tag{5c} \label{eq:5c} \end{equation}\]

\[\begin{equation} (p_{jh}^{MTDC})^2 \le l_{jh}^{MTDC} u_j^{MTDC}, \quad \forall(j,h)\in\mathcal{L}^{MTDC} \tag{5d} \label{eq:5d} \end{equation}\]

\[\begin{equation} \underline{u}_j^{MTDC} \le u_j^{MTDC} \le \overline{u}_j^{MTDC}, \quad \forall j\in\mathcal{N}^{MTDC} \tag{6} \label{eq:6} \end{equation}\]

\[\begin{equation} p_j^{MTDC} = p_{j,ref}^{MTDC}, \quad \forall j\in\mathcal{N}^{MTDC} \tag{7a} \label{eq:7a} \end{equation}\]

\[\begin{equation} v_j^{MTDC} = v_{j,ref}^{MTDC}, \quad \forall j\in\mathcal{N}^{MTDC} \tag{7b} \label{eq:7b} \end{equation}\]

\[\begin{equation} p_j^{MTDC} = p_{j,ref}^{MTDC} - \frac{1}{k_{j,droop}^{MTDC}} \left( v_j^{MTDC} - v_{j,ref}^{MTDC} \right), \quad \forall j\in\mathcal{N}^{MTDC} \tag{7c} \label{eq:7c} \end{equation}\]

VSC Constraints

\[\begin{equation} p_s^{VSC}=p_{i,A2V}^{AC}, \quad s\in\mathcal{N}^{VSC} \tag{8a} \label{eq:8a} \end{equation}\]

\[\begin{equation} q_s^{VSC}=q_{i,A2V}^{AC}, \quad s\in\mathcal{N}^{VSC} \tag{8b} \label{eq:8b} \end{equation}\]

\[\begin{equation} p_{s}^{VSC} = c_{ss}^{VSC}g_{sf}^{VSC} - c_{sf}^{VSC}g_{sf}^{VSC} + s_{sf}^{VSC}b_{sf}^{VSC}, \quad s,f\in\mathcal{N}^{MTDC} \tag{9a} \label{eq:9a} \end{equation}\]

\[\begin{equation} q_{s}^{VSC} = - c_{ss}^{VSC}b_{sf}^{VSC} + c_{sf}^{VSC}b_{sf}^{VSC} + s_{sf}^{VSC}g_{sf}^{VSC}, \quad s,f\in\mathcal{N}^{MTDC} \tag{9b} \label{eq:9b} \end{equation}\]

\[\begin{equation} c_{sf}^{VSC} = c_{fs}^{VSC}, \quad s,f\in\mathcal{N}^{AC} \tag{9c} \label{eq:9c} \end{equation}\]

\[\begin{equation} s_{sf}^{VSC} = - s_{fs}^{VSC}, \quad s,f\in\mathcal{N}^{VSC} \tag{9d} \label{eq:9d} \end{equation}\]

\[\begin{equation} (c_{sf}^{VSC})^2+(s_{sf}^{VSC})^2 \le c_{ss}^{VSC}c_{ff}^{VSC}, \quad s,f\in\mathcal{N}^{AC} \tag{9e} \label{eq:9e} \end{equation}\]

\[\begin{equation} p_{c}^{VSC} = c_{cc}^{VSC}g_{cf}^{VSC} - c_{cf}^{VSC}g_{cf}^{VSC} + s_{cf}^{VSC}b_{cf}^{VSC}, \quad c,f\in\mathcal{N}^{MTDC} \tag{10a} \label{eq:10a} \end{equation}\]

\[\begin{equation} q_{c}^{VSC} = - c_{cc}^{VSC}b_{cf}^{VSC} + c_{cf}^{VSC}b_{cf}^{VSC} + s_{cf}^{VSC}g_{cf}^{VSC}, \quad c,f\in\mathcal{N}^{MTDC} \tag{10b} \label{eq:10b} \end{equation}\]

\[\begin{equation} c_{cf}^{VSC}=c_{fc}^{VSC}, \quad c,f\in\mathcal{N}^{VSC} \tag{10c} \label{10c} \end{equation}\]

\[\begin{equation} s_{cf}^{VSC}=-s_{fc}^{VSC}, \quad c,f\in\mathcal{N}^{VSC} \tag{10d} \label{eq:10d} \end{equation}\]

\[\begin{equation} (c_{cf}^{VSC})^2+(s_{cf}^{VSC})^2 \le c_{cc}^{VSC}c_{ff}^{VSC}, \quad c,f\in\mathcal{N}^{VSC} \tag{10e} \label{eq:10e} \end{equation}\]

\[\begin{equation} q_f^{VSC} = - c_{ff}^{VSC} b_f^{VSC}, \quad f\in\mathcal{N}^{VSC} \tag{11} \label{eq:11} \end{equation}\]

\[\begin{equation} p_c^{VSC} + p_{loss}^{VSC} + p_j^{MTDC} = 0, \quad j\in\mathcal{N}^{MTDC}, \quad c\in\mathcal{N}^{VSC} \tag{12} \label{eq:12} \end{equation}\]

\[\begin{equation} p_{loss}^{VSC} = a_{loss}^{VSC} + b_{loss}^{VSC}|I_c^{VSC}| + c_{loss}^{VSC}l_c^{VSC}, \quad c\in\mathcal{N}^{VSC} \tag{13a} \label{eq:13a} \end{equation}\]

\[\begin{equation} 0 \le |I_c^{VSC}| \le |\overline{I_c^{VSC}}|, \quad c\in\mathcal{N}^{VSC} \tag{13b} \label{eq:13b} \end{equation}\]

\[\begin{equation} 0 \le l_c^{VSC} \le |\overline{I_c^{VSC}}|^2, \quad c\in\mathcal{N}^{VSC} \tag{13c} \label{eq:13c} \end{equation}\]

\[\begin{equation} |I_c^{VSC}|^2 \le l_c^{VSC}, \quad c\in\mathcal{N}^{VSC} \tag{14a} \label{eq:14a} \end{equation}\]

\[\begin{equation} (p_c^{VSC})^2+(q_c^{VSC})^2 \le c_{cc}^{VSC}l_c^{VSC}, \quad c\in\mathcal{N}^{VSC} \tag{14b} \label{eq:14b} \end{equation}\]

\[\begin{equation} c_{ss}^{VSC} = (v_{s,ref}^{VSC})^2, \quad s\in\mathcal{N}^{VSC} \tag{15a} \label{eq:15a} \end{equation}\]

\[\begin{equation} q_s^{VSC} = q_{s,ref}^{VSC}, \quad s\in\mathcal{N}^{VSC} \tag{15b} \label{eq:15b} \end{equation}\]

Optimization Goal

\[\begin{equation} \min \sum_{i\in\mathcal{N}^{AC}} \left( c_{i,2}^{AC}\left(p_{i,gen}^{AC}\right)^2 + c_{i,1}^{AC}p_{i,gen}^{AC} + c_{i,0}^{AC} \right) \tag{16} \label{eq:16} \end{equation}\]

AC Constraints: ($\ref{eq:1a}$) to ($\ref{eq:1g}$) are SOCP AC power flow constraints Yasasvi et al. (2018). ($\ref{eq:2a}$) and ($\ref{eq:2b}$) are power balance constraints at each AC bus. ($\ref{eq:3a}$) and ($\ref{eq:3b}$) are generator output limits, while ($\ref{eq:4}$) is the voltage bound.
DC Constraints: ($\ref{eq:5a}$) is the power balance constraint at each DC bus, while ($\ref{eq:5b}$) and ($\ref{eq:5d}$) are SOCP DC power flow constratins Gan and Low (2014). ($\ref{eq:6}$) is the voltage bound, while ($\ref{eq:7a}$), ($\ref{eq:7b}$), and ($\ref{eq:7c}$) are the power/voltage reference and droop control constraints.
VSC Constraints: ($\ref{eq:8a}$) and ($\ref{eq:8b}$) are the power coupling constraints between AC grid and VSC, while ($\ref{eq:9a}$) to ($\ref{eq:9e}$) and ($\ref{eq:10a}$) to ($\ref{eq:10e}$) are the power flow constraints at VSC AC and DC sides, respectively. ($\ref{eq:11}$) is the shunt reactive power constraint, while ($\ref{eq:12}$) is the power balance constraint at VSC DC side. ($\ref{eq:13a}$) to ($\ref{eq:13c}$) are the loss model and current limits, while ($\ref{eq:14a}$) and ($\ref{eq:14b}$) are the second-order cone relaxations of current and apparent power limits, respectively. Finally, ($\ref{eq:15a}$) and ($\ref{eq:15b}$) are the voltage and reactive power reference constraints for VSCs operating in grid-following mode.
Optimization Goal: ($\ref{eq:16}$) is minimization of total generation cost.

Voltage phase recovery

For second-order cone relaxations in AC grid power flow or VSC AC-side power flow, the bus voltage phase is eliminated. However, the voltage phase is important in harmonic stability analysis of AC/DC hybrid power systems. Therefore, it is necessary to recover the system-wide voltage phase from the obtained AC/DC OPF results.

Considering the complex AC nodal voltage is $\dot{v}_i^{A C}=v_i^{A C} e^{\mathrm{j} \theta_i^{A C}}$, the nonlinear coupling appears in power flow equations, such that:

\[\begin{equation} \dot{v}_i^{AC}\bigl(\dot{v}_j^{AC}\bigr)^{*} = v_i^{AC} v_j^{AC}\left(\cos\delta_{ij}^{AC} + \mathrm{j}\sin\delta_{ij}^{AC}\right). \tag{17} \label{eq:17} \end{equation}\]

where $\delta_{i j}^{A C}=\theta_i^{A C}-\theta_j^{A C}$.

To convexify the OPF model, SOC relaxation substitutes that $c_{i j}^{A C}=v_i^{A C} v_j^{A C} \cos \delta_{i j}^{A C}$ and $s_{i j}^{A C}=v_i v_j \sin \delta_{i j}^{A C}$. Thus avoiding explicit use of $\theta_i^{A C}$. According to $\ref{eq:17}$, we can see that the optimized variables satisfy:

\[\begin{equation} \delta_{i j}^{A C}=\arctan 2\left(s_{i j}^{A C}, c_{i j}^{A C}\right), \tag{18} \label{eq:18} \end{equation}\] thus $\delta_{i j}^{A C}$, the AC branch phase difference, can be directly obtained based on $s_{i j}^{A C}$ and $c_{i j}^{A C}$.

Let $\mathcal{G}^{A C}:=\left(\mathcal{N}^{A C}, \varepsilon^{A C}\right)$ denote the AC network graph. If we choose a slack bus with reference phase 0, then all bus phases can be recovered, along any spanning tree of $\mathcal{G}^{A C}$, such that:

\[\begin{equation} \theta_j^{A C}=\theta_i^{A C}+\delta_{i j}^{A C}, \tag{19} \label{eq:19} \end{equation}\]

Then, we continue to consider that a VSC is connected to AC grid bus $j$. In this case, the PCC node s voltage phase can be directly obtained and regarded as reference voltage. Following the approach same to ($\ref{eq:17}$)-($\ref{eq:19}$), the VSC AC-side voltage phase can be calculated.

References

Gan, Lingwen, and Steven H Low. 2014. “Optimal Power Flow in Direct Current Networks.” IEEE Transactions on Power Systems 29 (6): 2892–2904.

Yasasvi, P Naga, Snehil Chandra, A Mohapatra, and SC Srivastava. 2018. “An Exact SOCP Formulation for AC Optimal Power Flow.” In 2018 20th National Power Systems Conference (NPSC), 1–6. IEEE.

--- layout: page title: Optimal Power Flow # be as specific as possible tags: [Opensource, Optimal Power Flow Analysis, Power Flow Analysis] date: 2026-02-20 last-updated: 3/9/2026 author: "Aleksandra Lekić, Haixiao Li" reviewers: "Aleksandra Lekić" bibliography: references.bib order: 1 # MathJax format: html: standalone: false toc: false page-layout: full html-math-method: mathjax code-fold: true code-summary: "Show/Hide Code" code-tools: true highlight-style: github include-in-header: text: | <script> MathJax = { tex: { tags: 'ams', tagSide: 'right', tagIndent: '0em' } }; </script> ---  :::{.callout-tip} *AC/DC Optimal Power Flow (AC/DC OPF) is an optimization method used to determine the most economical and secure operating point of hybrid AC/DC power systems. It simultaneously considers AC network constraints, DC grid equations, and converter coupling relationships. By coordinating generators and converters, it minimizes operational cost while satisfying voltage, power flow, and equipment limits. AC/DC OPF is essential for modern grids with high renewable penetration and multi-terminal DC transmission systems.* ::: ## Nomenclature ### A. Indices and Sets $\left(\overline{\cdot}\right)/\left(\underline{\cdot}\right)$ : Upper/lower bound of corresponding optimization variables and parameters $\textit{AC}$ : Alternating current $\textit{MTDC}$ : Multi-terminal direct current $\textit{VSC}$ : Voltage source converter $\textit{h, i, j}$ : System node $\textit{ij}$ : System branch $\textit{ref}$ : Reference value $\textit{droop}$ : Droop parameters $\textit{loss}$ : Power loss $\textit{cost}$ : Generation cost $\textit{gen}$ : Generator $\textit{load}$ : Load $\mathcal{L}$ : Branch set $\mathcal{N}$ : Node set $\lvert \cdot \rvert$ : Magnitude --- ### B. AC System Variables $v_i^{AC}$ : AC bus $i$ voltage $\theta_{ij}^{AC}$ : Phase difference of AC branch $ij$ $p_i^{AC}$ / $q_i^{AC}$ : Real/Reactive power injection at AC bus $i$ $p_{ij}^{AC}$ / $q_{ij}^{AC}$ : Real/Reactive power of AC branch $ij$ $p_{i,gen}^{AC}$ / $q_{i,gen}^{AC}$ : Real/Reactive generator output at AC bus $i$ $p_{i,load}^{AC}$ / $q_{i,load}^{AC}$ : Real/Reactive load at AC bus $i$ $p_{i,A2V}^{AC}$ / $q_{i,A2V}^{AC}$ : Power delivered from AC grid to VSC at bus $i$ $c_{i,2}^{AC}$ / $c_{i,1}^{AC}$ / $c_{i,0}^{AC}$ : Quadratic/Linear/Constant generation cost coefficients $c_{ij}^{AC}$ / $s_{ij}^{AC}$ : Optimization variables related to AC nodal voltages --- ### C. MTDC System Variables $v_j^{MTDC}$ : MTDC bus $j$ voltage $u_j^{MTDC}$ : Squared MTDC bus voltage $p_{jh}^{MTDC}$ : Power of MTDC branch $jh$ $l_{jh}^{MTDC}$ : Squared current of MTDC branch $jh$ $p_{i,ref}^{MTDC}$ : DC power reference at MTDC bus $i$ $V_{i,ref}^{MTDC}$ : DC voltage reference at MTDC bus $i$ $k_{i,droop}^{MTDC}$ : Droop slope at MTDC bus $i$ --- ### D. VSC Variables $v_s^{VSC}$ / $v_f^{VSC}$ / $v_c^{VSC}$ : VSC bus $s/f/c$ voltage $\theta_s^{VSC}$ / $\theta_f^{VSC}$ / $\theta_c^{VSC}$ : VSC bus phase angle $g_{sf}^{VSC}$ / $g_{fc}^{VSC}$ : Conductance of VSC branches $b_{sf}^{VSC}$ / $b_{fc}^{VSC}$ : Susceptance of VSC branches $b_f^{VSC}$ : Shunt susceptance at VSC bus $f$ $p_{loss}^{VSC}$ : VSC power loss $a_{loss}^{VSC}$ / $b_{loss}^{VSC}$ / $c_{loss}^{VSC}$ : Constant/Linear/Quadratic loss coefficients $I_c^{VSC}$ : Phase current at bus $c$ $l_c^{VSC}$ : Squared phase current $q_{s,ref}^{VSC}$ : Reactive power reference $v_{s,ref}^{VSC}$ : AC voltage reference $c_{sf}^{VSC}$ / $s_{sf}^{VSC}$ / $c_{cf}^{VSC}$ / $s_{cf}^{VSC}$ : Optimization variables related to VSC nodal voltages ## Second-Order Cone Programming (SOCP) Formulation The speicifc AC/DC OPF model utilized in HARMONY is presented below. ### Varables \begin{align*} & p_i^{AC}, q_i^{AC}, p_{i, gen}^{A C}, q_{i, gen}^{A C}, p_{i, A2V}^{AC}, q_{i, A2V}^{AC}, s_{ij}^{AC}, c_{ij}^{AC}&& i,j\in\mathcal{N}^{AC} \\ & p_{ij}^{AC}, q_{ij}^{AC} && (i,j)\in\mathcal{L}^{AC} \\ & p_{j}^{MTDC}, v_{j}^{MTDC} && j\in\mathcal{N}^{M T D C} \\ & p_{jh}^{MTDC},l_{jh}^{MTDC}&& (j, h)\in\mathcal{L}^{M T D C} \\ & p_s^{VSC}, q_s^{VSC} && s\in\mathcal{N}^{MTDC}\\ & p_c^{VSC}, q_c^{VSC}, I_c^{VSC}, l_c^{VSC} && c\in\mathcal{N}^{MTDC}\\ & q_f^{VSC} && f\in\mathcal{N}^{MTDC}\\ & c_{mn}^{VSC}, c_{mn}^{VSC} && m, n \in \{s, f, c\} && s,f,c\in\mathcal{N}^{MTDC}\\ \end{align*} ### AC Constraints \begin{equation} p_{ij}^{AC} = G_{ij} \left( c_{ii}^{AC} - c_{ij}^{AC} \right) + B_{ij} s_{ij}^{AC}, \quad \forall i,j\in\mathcal{N}^{AC}, \quad \forall (i,j)\in\mathcal{L}^{AC} \tag{1a} \label{eq:1a} \end{equation} \begin{equation} q_{ij}^{AC} = - B_{ij} \left( c_{ii}^{AC} - c_{ij}^{AC} \right) + G_{ij} s_{ij}^{AC}, \quad \forall i,j\in\mathcal{N}^{AC}, \quad \forall (i,j)\in\mathcal{L}^{AC} \tag{1b} \label{eq:1b} \end{equation} \begin{equation} p_i^{AC} = c_{ii}^{AC} G_{ii} + \sum_{j} \left( c_{ij}^{AC} G_{ij} - s_{ij}^{AC} B_{ij} \right), \quad \forall i,j\in\mathcal{N}^{AC} \tag{1c} \label{eq:1c} \end{equation} \begin{equation} q_i^{AC} = - c_{ii}^{AC} B_{ii} - \sum_{j} \left( c_{ij}^{AC} B_{ij} + s_{ij}^{AC} G_{ij} \right), \quad \forall i,j\in\mathcal{N}^{AC} \tag{1d} \label{eq:1d} \end{equation} \begin{equation} c_{ij}^{AC} = c_{ji}^{AC}, \quad \forall i,j\in\mathcal{N}^{AC} \tag{1e} \label{eq:1e} \end{equation} \begin{equation} s_{ij}^{AC} = - s_{ji}^{AC}, \quad \forall i,j\in\mathcal{N}^{AC} \tag{1f} \label{eq:1f} \end{equation} \begin{equation} (c_{ij}^{AC})^2 + (s_{ij}^{AC})^2 \le c_{ii}^{AC} c_{jj}^{AC}, \quad \forall i,j\in\mathcal{N}^{AC} \tag{1g} \label{eq:1g} \end{equation} \begin{equation} p_i^{AC} = p_{i,gen}^{AC} - p_{i,load}^{AC} - p_{i,A2V}^{AC}, \quad \forall i\in\mathcal{N}^{AC} \tag{2a} \label{eq:2a} \end{equation} \begin{equation} q_i^{AC} = q_{i,gen}^{AC} - q_{i,load}^{AC} - q_{i,A2V}^{AC}, \quad \forall i\in\mathcal{N}^{AC} \tag{2b} \label{eq:2b} \end{equation} \begin{equation} \underline{p}_{i,gen}^{AC} \le p_{i,gen}^{AC} \le \overline{p}_{i,gen}^{AC}, \quad \forall i\in\mathcal{N}^{AC} \tag{3a} \label{eq:3a} \end{equation} \begin{equation} \underline{q}_{i,gen}^{AC} \le q_{i,gen}^{AC} \le \overline{q}_{i,gen}^{AC}, \quad \forall i\in\mathcal{N}^{AC} \tag{3b} \label{eq:3b} \end{equation} \begin{equation} \underline{|v_i^{AC}|}^2 \le c_{ii}^{AC} \le \overline{|v_i^{AC}|}^2, \quad \forall i\in\mathcal{N}^{AC} \tag{4} \label{eq:4} \end{equation} ### DC Constraints \begin{equation} p_{j}^{MTDC} = \sum_{(j,h)} p_{jh}^{MTDC}, \quad \forall j\in\mathcal{N}^{MTDC}, \quad \forall(j,h)\in\mathcal{L}^{MTDC} \tag{5a} \label{eq:5a} \end{equation} \begin{equation} p_{jh}^{MTDC} + p_{hj}^{MTDC} = r_{jh}^{MTDC} l_{jh}^{MTDC}, \quad \forall(j,h)\in\mathcal{L}^{MTDC} \tag{5b} \label{eq:5b} \end{equation} \begin{equation} u_{j}^{MTDC} - u_{h}^{MTDC} = r_{jh}^{MTDC} \left( p_{jh}^{MTDC} - p_{hj}^{MTDC} \right), \quad \forall j,h\in\mathcal{N}^{MTDC}, \quad \forall(j,h)\in\mathcal{L}^{MTDC} \tag{5c} \label{eq:5c} \end{equation} \begin{equation} (p_{jh}^{MTDC})^2 \le l_{jh}^{MTDC} u_j^{MTDC}, \quad \forall(j,h)\in\mathcal{L}^{MTDC} \tag{5d} \label{eq:5d} \end{equation} \begin{equation} \underline{u}_j^{MTDC} \le u_j^{MTDC} \le \overline{u}_j^{MTDC}, \quad \forall j\in\mathcal{N}^{MTDC} \tag{6} \label{eq:6} \end{equation} \begin{equation} p_j^{MTDC} = p_{j,ref}^{MTDC}, \quad \forall j\in\mathcal{N}^{MTDC} \tag{7a} \label{eq:7a} \end{equation} \begin{equation} v_j^{MTDC} = v_{j,ref}^{MTDC}, \quad \forall j\in\mathcal{N}^{MTDC} \tag{7b} \label{eq:7b} \end{equation} \begin{equation} p_j^{MTDC} = p_{j,ref}^{MTDC} - \frac{1}{k_{j,droop}^{MTDC}} \left( v_j^{MTDC} - v_{j,ref}^{MTDC} \right), \quad \forall j\in\mathcal{N}^{MTDC} \tag{7c} \label{eq:7c} \end{equation} ### VSC Constraints \begin{equation} p_s^{VSC}=p_{i,A2V}^{AC}, \quad s\in\mathcal{N}^{VSC} \tag{8a} \label{eq:8a} \end{equation} \begin{equation} q_s^{VSC}=q_{i,A2V}^{AC}, \quad s\in\mathcal{N}^{VSC} \tag{8b} \label{eq:8b} \end{equation} \begin{equation} p_{s}^{VSC} = c_{ss}^{VSC}g_{sf}^{VSC} - c_{sf}^{VSC}g_{sf}^{VSC} + s_{sf}^{VSC}b_{sf}^{VSC}, \quad s,f\in\mathcal{N}^{MTDC} \tag{9a} \label{eq:9a} \end{equation} \begin{equation} q_{s}^{VSC} = - c_{ss}^{VSC}b_{sf}^{VSC} + c_{sf}^{VSC}b_{sf}^{VSC} + s_{sf}^{VSC}g_{sf}^{VSC}, \quad s,f\in\mathcal{N}^{MTDC} \tag{9b} \label{eq:9b} \end{equation} \begin{equation} c_{sf}^{VSC} = c_{fs}^{VSC}, \quad s,f\in\mathcal{N}^{AC} \tag{9c} \label{eq:9c} \end{equation} \begin{equation} s_{sf}^{VSC} = - s_{fs}^{VSC}, \quad s,f\in\mathcal{N}^{VSC} \tag{9d} \label{eq:9d} \end{equation} \hypertarget{eq:9e}{} \begin{equation} (c_{sf}^{VSC})^2+(s_{sf}^{VSC})^2 \le c_{ss}^{VSC}c_{ff}^{VSC}, \quad s,f\in\mathcal{N}^{AC} \tag{9e} \label{eq:9e} \end{equation} \begin{equation} p_{c}^{VSC} = c_{cc}^{VSC}g_{cf}^{VSC} - c_{cf}^{VSC}g_{cf}^{VSC} + s_{cf}^{VSC}b_{cf}^{VSC}, \quad c,f\in\mathcal{N}^{MTDC} \tag{10a} \label{eq:10a} \end{equation} \begin{equation} q_{c}^{VSC} = - c_{cc}^{VSC}b_{cf}^{VSC} + c_{cf}^{VSC}b_{cf}^{VSC} + s_{cf}^{VSC}g_{cf}^{VSC}, \quad c,f\in\mathcal{N}^{MTDC} \tag{10b} \label{eq:10b} \end{equation} \begin{equation} c_{cf}^{VSC}=c_{fc}^{VSC}, \quad c,f\in\mathcal{N}^{VSC} \tag{10c} \label{10c} \end{equation} \begin{equation} s_{cf}^{VSC}=-s_{fc}^{VSC}, \quad c,f\in\mathcal{N}^{VSC} \tag{10d} \label{eq:10d} \end{equation} \begin{equation} (c_{cf}^{VSC})^2+(s_{cf}^{VSC})^2 \le c_{cc}^{VSC}c_{ff}^{VSC}, \quad c,f\in\mathcal{N}^{VSC} \tag{10e} \label{eq:10e} \end{equation} \begin{equation} q_f^{VSC} = - c_{ff}^{VSC} b_f^{VSC}, \quad f\in\mathcal{N}^{VSC} \tag{11} \label{eq:11} \end{equation} \begin{equation} p_c^{VSC} + p_{loss}^{VSC} + p_j^{MTDC} = 0, \quad j\in\mathcal{N}^{MTDC}, \quad c\in\mathcal{N}^{VSC} \tag{12} \label{eq:12} \end{equation} \begin{equation} p_{loss}^{VSC} = a_{loss}^{VSC} + b_{loss}^{VSC}|I_c^{VSC}| + c_{loss}^{VSC}l_c^{VSC}, \quad c\in\mathcal{N}^{VSC} \tag{13a} \label{eq:13a} \end{equation} \begin{equation} 0 \le |I_c^{VSC}| \le |\overline{I_c^{VSC}}|, \quad c\in\mathcal{N}^{VSC} \tag{13b} \label{eq:13b} \end{equation} \begin{equation} 0 \le l_c^{VSC} \le |\overline{I_c^{VSC}}|^2, \quad c\in\mathcal{N}^{VSC} \tag{13c} \label{eq:13c} \end{equation} \begin{equation} |I_c^{VSC}|^2 \le l_c^{VSC}, \quad c\in\mathcal{N}^{VSC} \tag{14a} \label{eq:14a} \end{equation} \begin{equation} (p_c^{VSC})^2+(q_c^{VSC})^2 \le c_{cc}^{VSC}l_c^{VSC}, \quad c\in\mathcal{N}^{VSC} \tag{14b} \label{eq:14b} \end{equation} \begin{equation} c_{ss}^{VSC} = (v_{s,ref}^{VSC})^2, \quad s\in\mathcal{N}^{VSC} \tag{15a} \label{eq:15a} \end{equation} \begin{equation} q_s^{VSC} = q_{s,ref}^{VSC}, \quad s\in\mathcal{N}^{VSC} \tag{15b} \label{eq:15b} \end{equation} ### Optimization Goal \begin{equation} \min \sum_{i\in\mathcal{N}^{AC}} \left( c_{i,2}^{AC}\left(p_{i,gen}^{AC}\right)^2 + c_{i,1}^{AC}p_{i,gen}^{AC} + c_{i,0}^{AC} \right) \tag{16} \label{eq:16} \end{equation} - AC Constraints: (\ref{eq:1a}) to (\ref{eq:1g}) are SOCP AC power flow constraints @yasasvi2018exact. (\ref{eq:2a}) and (\ref{eq:2b}) are power balance constraints at each AC bus. (\ref{eq:3a}) and (\ref{eq:3b}) are generator output limits, while (\ref{eq:4}) is the voltage bound. - DC Constraints: (\ref{eq:5a}) is the power balance constraint at each DC bus, while (\ref{eq:5b}) and (\ref{eq:5d}) are SOCP DC power flow constratins @gan2014optimal. (\ref{eq:6}) is the voltage bound, while (\ref{eq:7a}), (\ref{eq:7b}), and (\ref{eq:7c}) are the power/voltage reference and droop control constraints. - VSC Constraints: (\ref{eq:8a}) and (\ref{eq:8b}) are the power coupling constraints between AC grid and VSC, while (\ref{eq:9a}) to (\ref{eq:9e}) and (\ref{eq:10a}) to (\ref{eq:10e}) are the power flow constraints at VSC AC and DC sides, respectively. (\ref{eq:11}) is the shunt reactive power constraint, while (\ref{eq:12}) is the power balance constraint at VSC DC side. (\ref{eq:13a}) to (\ref{eq:13c}) are the loss model and current limits, while (\ref{eq:14a}) and (\ref{eq:14b}) are the second-order cone relaxations of current and apparent power limits, respectively. Finally, (\ref{eq:15a}) and (\ref{eq:15b}) are the voltage and reactive power reference constraints for VSCs operating in grid-following mode. - Optimization Goal: (\ref{eq:16}) is minimization of total generation cost. ## Voltage phase recovery For second-order cone relaxations in AC grid power flow or VSC AC-side power flow, the bus voltage phase is eliminated. However, the voltage phase is important in harmonic stability analysis of AC/DC hybrid power systems. Therefore, it is necessary to recover the system-wide voltage phase from the obtained AC/DC OPF results. Considering the complex AC nodal voltage is $\dot{v}_i^{A C}=v_i^{A C} e^{\mathrm{j} \theta_i^{A C}}$, the nonlinear coupling appears in power flow equations, such that: \begin{equation} \dot{v}_i^{AC}\bigl(\dot{v}_j^{AC}\bigr)^{*} = v_i^{AC} v_j^{AC}\left(\cos\delta_{ij}^{AC} + \mathrm{j}\sin\delta_{ij}^{AC}\right). \tag{17} \label{eq:17} \end{equation} where $\delta_{i j}^{A C}=\theta_i^{A C}-\theta_j^{A C}$. To convexify the OPF model, SOC relaxation substitutes that $c_{i j}^{A C}=v_i^{A C} v_j^{A C} \cos \delta_{i j}^{A C}$ and $s_{i j}^{A C}=v_i v_j \sin \delta_{i j}^{A C}$. Thus avoiding explicit use of $\theta_i^{A C}$. According to \ref{eq:17}, we can see that the optimized variables satisfy: \begin{equation} \delta_{i j}^{A C}=\arctan 2\left(s_{i j}^{A C}, c_{i j}^{A C}\right), \tag{18} \label{eq:18} \end{equation} thus $\delta_{i j}^{A C}$, the AC branch phase difference, can be directly obtained based on $s_{i j}^{A C}$ and $c_{i j}^{A C}$. Let $\mathcal{G}^{A C}:=\left(\mathcal{N}^{A C}, \varepsilon^{A C}\right)$ denote the AC network graph. If we choose a slack bus with reference phase 0, then all bus phases can be recovered, along any spanning tree of $\mathcal{G}^{A C}$, such that: \begin{equation} \theta_j^{A C}=\theta_i^{A C}+\delta_{i j}^{A C}, \tag{19} \label{eq:19} \end{equation} Then, we continue to consider that a VSC is connected to AC grid bus $j$. In this case, the PCC node s voltage phase can be directly obtained and regarded as reference voltage. Following the approach same to (\ref{eq:17})-(\ref{eq:19}), the VSC AC-side voltage phase can be calculated. ## References ::: {#refs} :::