## 1 Introduction
In addition to the challenges brought by the large-scale system modeling of power grids, the destructive impacts caused by cyber attacks have brought new challenges to the design of power grid control systems following the rapid advancement of power grid digitization and informatization [15]. Denial of service (DoS) attacks are common network attacks in information attacks. DoS attacks damage the integrity and timeliness of communication data by blocking communication channels [27; 28]. In addition, these attacks also reduce the performance of power grid control systems and cause transient instability of power grid phase angles in severe cases or even system collapse. Thus far, there have been many power failures caused by DoS attacks [29; 30]. Therefore, it is significant to conduct related research regarding power grid frequency control under DoS attacks. The existing research has proposed certain solutions for power system frequency control under network attacks, including resilient model predictive control [12], event triggering control [13], etc. The methods used to resist DoS attacks include data dimensionality reduction [31], data filtering [32], and event-triggered control structure [33]. In particular, actor critic structure algorithms in reinforcement learning [34; 35; 36] can be used to defend against DoS attacks. In actor-critic structure reinforcement-learning algorithms, the actor and critic play different functions. The actor is used to output control strategies. The critic is used to evaluate the rationality of the current control strategy formulated by the actor. Actor-critic structure algorithms focus on obtaining parameter vectors or network weights that can maximize the return by using the gradient descent update method. Actor-critic structure algorithms can be implemented by neural networks, such as back-propagation neural networks (BPNN) [37] and radial basis neural networks (RBFNN) [38]. BPNN was used to implement both actor and critic networks in the actor-critic structure algorithms [34; 35; 36]. However, BPNN may cause the optimization process of reinforcement learning to only reach the local optimum during the implementation of critic structure, resulting in suboptimal control output [37; 39]. The traditional model-based frequency control strategy cannot solve the problem of large-scale analytical modeling of control strategy in modern power systems. Although data-driven approaches (reinforcement learning etc.) have been proposed to solve this problem, the rapid response of control strategy optimization in reinforcement learning has not been well considered. There may be a local optimization problem of any control strategy. Therefore, in order to improve the performance of load-frequency control under DoS attacks, this paper introduces convex neural networks [40; 41; 42; 43] to realize the actor-critic structure. Convex neural networks are used to approximate the long-term goal of reinforcement learning and the output of the controller. The optimization process of reinforcement learning is transformed into a convex optimization process. When partial weights of the convex neural networks meet certain constraints, the output of the convex neural networks is a convex function of the input. This ensures the existence of the global optimum and improves the efficiency of the optimization process. This paper has two main contributions as follows. * In this paper, we propose a load-frequency control strategy of convex neural network-based reinforcement learning that can resist DoS attacks and analyze the sufficient conditions for the stability of power grids as well as the convergence of convex neural network parameters during online learning. * A long-term utility model of load-frequency control based on convex neural network approximation is proposed. Thus, the control output can be improved by the near global optimum obtained from the convex approximation. Additionally, the optimization speed is accelerated, and the efficiency of controllers is improved.