*Article* **Learning-Aided Optimal Power Flow Based Fast Total Transfer Capability Calculation**

**Ji'ang Liu 1,\*, Youbo Liu 1, Gao Qiu 1,\* and Xiao Shao <sup>2</sup>**


**Abstract:** Total transfer capability (TTC) is a vital security indicator for power exchange among areas. It characterizes time-variants and transient stability dynamics, and thus is challenging to evaluate efficiently, which can jeopardize operational safety. A leaning-aided optimal power flow method is proposed to handle the above challenges. At the outset, deep learning (DL) is utilized to globally establish real-time transient stability estimators in parametric space, such that the dimensionality of dynamic simulators can be reduced. The computationally intensive transient stability constraints in TTC calculation and their sensitivities are therewith converted into fast forward and backward processes. The DL-aided constrained model is finally solved by nonlinear programming. The numerical results on the modified IEEE 39-bus system demonstrate that the proposed method outperforms several model-based methods in accuracy and efficiency.

**Keywords:** total transfer capability; surrogate assisted method; transient stability; deep learning; interior point method

**1. Introduction**

Power systems are currently operated near their stability boundary with the significant proliferation of interconnected grids and renewable penetration [1]. Therefore, online monitoring to transfer security margin of inter-area power transfer is in urgent demand. In the electric industry, total transfer capability (TTC), defined as maximum power exchange allowed to withstand multifarious security contingencies, is a widespread metric to quantify such a security margin. Limited by this issue, dispatchers generally use a conservative constant of offline TTC to decide online operations. Undoubtedly, such TTC values can incur the unwanted waste of line capacity and incorrect estimation to security margin. To untie these knots, the essence is to accelerate TTC calculation.

Thus far, several approaches have been proposed to model TTC calculation [2–4]. Among them, methods with only steady-state considered are inapplicable for TTC evaluation involving transient stability (TS) [5]. To enable TS assessment (TSA), TTC is preferred to be modeled as TS constrained (TSC) programming problem. As the models shown in [6–10], differential-algebraic equations (DAEs) representing system dynamics and TS constraints are discretized throughout the time domain simulation period. And the resulting differential equations are incorporated into the optimal power flow (OPF) model. Nevertheless, as mentioned before, solving such models is quite computationally expensive due to the high-dimensional and nonlinear DAEs involved. In light of this, under current time-varying power grids, inefficient physics-dominated methods can be problematic for fast TTC monitors.

Data-driven approaches have become mainstream to increase calculation speed for security assessment in large-scale power systems [11–13]. Reference [11] proposed an online measurement-based TTC estimator using the nonparametric estimation. Sun et al. developed an automatic learning technique based on the linear least-squares fitting method

**Citation:** Liu, J.; Liu, Y.; Qiu, G.; Shao, X. Learning-Aided Optimal Power Flow Based Fast Total Transfer Capability Calculation. *Energies* **2022**, *15*, 1320. https://doi.org/10.3390/ en15041320

Academic Editors: Luis Hernández-Callejo, Sergio Nesmachnow and Sara Gallardo Saavedra

Received: 30 December 2021 Accepted: 7 February 2022 Published: 11 February 2022

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to extract the TTC operating rules [12]. Unfortunately, these methods own two critical drawbacks. One is that they are hard to capture nonlinear patterns. An empirical and heuristic TTC calculation in the stage of prior sample production is the other problem. It cannot ensure finding the most extreme operating conditions, leading to low fidelity against true modes.

To overcome the first drawback, machine learning (ML) is a promising alternative thanks to its strong nonlinearity learning ability. Reference [13] introduced a hierarchical deep learning machine (HDLM) to successfully achieve real-time TSA, won over other physics-based methods with respect to speed, and beat linear data-driven methods on precision. But sustainable energy is under-investigated. In [14], a TSA framework based on a long short-term memory network was proposed; it improved assessment accuracy by learning from post-fault temporal PMU data dependencies. These applications manifest that ML is a better choice than linear learning methods in nonlinearity modeling tasks. A comparison table with the advantages and disadvantages of the above references is listed in Table 1.


**Table 1.** A comparison table with the advantages and disadvantages of each reference.

On the other hand, ML can substitute the most time-consuming TSA modules and partially participate in TTC calculation to deal with the second deficiency. This idea follows the classical roadmap of using optimal power flow (OPF) to approach extreme operations but tactfully bypasses high-dimensional modeling such that optimizers can quickly solve TTC. It is technically termed as a learning-aided (also known as surrogateassisted) method (LAM) [15–18], which utilizes ML algorithms to surrogate the most complex and computationally intensive parts in optimization problems. Reference [18] proposed a method that makes a fusion between surrogates and the evolutionary algorithm to improve the efficiency of optimizing high-dimensional expensive problems. In [1], LAM is also utilized to solve the TTC constrained operation planning problem. The above studies show that LAM can speed up solving optimization problems. At the same time, because it is a data-mechanism hybrid-driven method rather than an utterly data-driven method, it performs better in terms of fidelity.

By prioritizing both merits of physics- and data-driven modeling, this paper proposed a learning-aided optimal power flow based fast TTC calculation methods with the following features:

Deep belief network (DBN) is advocated to surrogate computationally intensive and high-dimensional time-domain based transient stability modelling. This learning-aided scheme allows us to significantly reduce complexity of TTC calculation.


IPM. This scheme is firstly used in OPF-based TTC calculation, and numerical studies justified its merits of compromising calculation efficiency and accuracy.

• A comprehensive comparative study is constructed. Numbers of traditional methods, such as the TSCOPF method [9], the sensitivity-based method [19], the repeated power flow (RPF) method [20], and the direct data-driven method [21], are used to demonstrate the superiority of our method.

The organization of this paper is as follows: Section 2 introduces TS constrained optimal power flow (TSCOPF), adopted to model TTC calculation. The learning-aided model for the TS constraints is introduced in Section 3. Section 4 details the proposed solving scheme method, where the Jacobin and Hessian matrices of the learning model are deduced to analytical form to enable combination with nonlinear programming. Section 5 illustrates the numerical study. Finally, the conclusion is presented in Section 6.
