# A Survey of Real-Time Optimal Power Flow

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**x**represents the vector of state variables,

**u**is the vector of continuous decision variables,

**l**denotes the vector of integer decision variables,

**y**represents the vector of binary decision variables, and

**ξ**is the vector of random variables. Here,

**g**denotes dynamic model equations with initial states of

**x**

_{0}at t

_{0},

**x**represents the lower/upper limits of state variables,

_{min/max}**u**stands for the lower/upper limits of continuous decision variables, and t

_{min/max}_{f}denotes the final time.

**x**could contain voltages of the PQ buses [40,41,42,43,44], active–reactive power at swing bus, and power flows in feeders [45]. The vector of continuous control variables

**u**could include active and reactive power generation [40] (conventional and/or renewable generations) and charge and discharge power of the storage units [25,26]. The vector of discrete control variables

**l**could consist of reference values of slack bus voltage [46,47] (for the sake of simplicity, some studies consider the slack bus voltage as a continuous variable [48,49]), and the vector of binary control variables could consist of the charge/discharge status of BSSs. The vector of random variables

**ξ**could consist of REGs and/or load demand [35,50,51]. The random variables are the generated amounts of REGs usually considered as being stochastically distributed with a known probability density function ρ(ξ). The time step t could also be considered as an integer variable to be optimized in the optimization problem; for instance, it could be the number of daily charge/discharge hours of BSSs [26]. The dynamic model equations

**g**generally consists of power flow equations at the buses [46] and the energy equations for batteries [25]. The inequality constraints could include lower and upper limits of state and decision variables.

## 3. Offline EMSs

- (1)
- Deterministic EMSs, by which the outputs are determined using forecasted parameter values. In other words, uncertainties are not considered when computing the solutions.
- (2)
- Stochastic EMSs, which consider the uncertainties and inaccuracies of the forecasted values when computing the solutions. It means the control strategies obtained in this way are more likely to be functional in practical applications under uncertainty.

#### 3.1. Deterministic EMSs

**ξ**(t) has been removed from the objective function f and model equations

**g**(usually, they are replaced by their nominal or expected values), meaning that there is no randomness in the problem formulation. Focusing on different aspects of Equation (2), a vast number of studies have been made on OPF for offline operation planning of energy networks, since it was proposed by Carpentier in 1962 [54]. Indeed, from 1962 to late 1990s, most of the studies focused on OPF without REGs. For instance, a reactive volt-ampere control method was proposed in reference [62] to minimize the losses of a power transmission network. Based on reference [54], the general problem of minimizing the total operation cost of a power system was formulated [63] as an NLP problem. This study [63] was later extended [37] and proposed a method to solve OPF considering active–reactive power and the tap ratios of transformers as decision variables to minimize losses or costs in a network. Besides, a unified approach based on the Carpentier’s formulation [54] was proposed in another work [64] to solve the OPF problem.

#### 3.2. Stochastic EMSs

## 4. Real-Time EMSs

- (1)
- Constraint satisfaction-based RT-EMSs, which provide solutions to satisfy technical constraints. The solutions obtained in this way may not be optimal.
- (2)
- OPF-based RT-EMSs, which provide ‘(sub)optimal’ solutions in real time, while satisfying technical constraints.

#### 4.1. Constraint Satisfaction-Based RT-EMSs

#### 4.2. OPF-Based RT-EMSs

**ξ**in the energy network at every sampling time T

_{S}. These reactions could be either optimal or suboptimal. This is in contrast to offline OPF (deterministic and stochastic), where the optimal operation strategies are calculated only once for each prediction horizon T

_{P}. A general RT-OPF method is illustrated in Figure 2. The optimal set points of the controller are obtained by OPF in each T

_{P}, which are later corrected in real time on the basis of actual measurements. A feedback control system measures, monitors, and controls the variables to ensure the desired response of the network.

## 5. Conclusions and Future Challenges

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

f | Objective function |

g | Dynamic model equations |

l | Vector of discrete decision variables |

t | Time |

t_{0} | Initial time |

t_{f} | Final time |

T_{P} | Prediction horizon |

T_{S} | Sampling interval |

u | Vector of continuous decision variables |

u_{max} | Upper boundaries of continuous decision variables |

u_{min} | Lower boundaries of continuous decision variables |

x | Vector of state variables |

x_{0} | Initial states |

x_{max} | Upper boundaries of state variables |

x_{min} | Lower boundaries of state variables |

y | Vector of binary decision variables |

ξ | Vector of uncertain variables |

Ω | Set of random variables |

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**MDPI and ACS Style**

Mohagheghi, E.; Alramlawi, M.; Gabash, A.; Li, P.
A Survey of Real-Time Optimal Power Flow. *Energies* **2018**, *11*, 3142.
https://doi.org/10.3390/en11113142

**AMA Style**

Mohagheghi E, Alramlawi M, Gabash A, Li P.
A Survey of Real-Time Optimal Power Flow. *Energies*. 2018; 11(11):3142.
https://doi.org/10.3390/en11113142

**Chicago/Turabian Style**

Mohagheghi, Erfan, Mansour Alramlawi, Aouss Gabash, and Pu Li.
2018. "A Survey of Real-Time Optimal Power Flow" *Energies* 11, no. 11: 3142.
https://doi.org/10.3390/en11113142