# Optimal Load Shedding for Smart Power Grid Resilience Enhancement Considering Cyber-Physical Constraints

Sonia Hosseinpour, João Martins

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Sonia Hosseinpour, João Martins. Optimal Load Shedding for Smart Power Grid Resilience Enhancement Considering Cyber-Physical Constraints. 14th Doctoral Conference on Computing, Electrical and Industrial Systems (DoCEIS), Jul 2023, Monte Da Caparica, Portugal. pp.67-81, 10.1007/978-3-031-36007-7_5 . hal-05473071

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# Optimal Load Shedding for Smart Power Grid Resilience Enhancement Considering Cyber-Physical Constraints

Sonia Hosseinpour and João Martins

NOVA School of Science and Technology, Center of Technology and Systems (UNINOVA-CTS) and Associated Lab of Intelligent Systems (LASI), NOVA University Lisbon, 2829-516 Lisbon, Portugal

s.hosseinpour@campus.fct.unl.pt, jf.martins@fct.unl.pt

**Abstract.** The increasing use of measurement and telecommunication equipment has transformed power systems into smart networks, which have created new challenges in modeling and operation studies. One critical challenge is the transformation of power networks into a vast cyber-physical system, which has made mathematical modeling difficult. Any fault in cyber or physical systems can affect the performance of another part, so the interaction of cyber and physical sectors should be considered in power system studies. Load shedding studies have become important for improving the resilience of power systems, but the interaction of cyber and physical parts in load shedding is rarely studied. This paper proposes a new mathematical model for optimal load shedding in power systems by considering the interaction of cyber and physical parts to improve system resilience. The proposed model is applied to an IEEE RTS-79 power grid and demonstrates its effectiveness in improving resilience by considering cyber-physical constraints.

**Keywords:** Cyber-Physical Systems · Optimal Load Shedding · Resilience Enhancement · Grey Wolf Optimizer

## 1 Introduction

A severe disturbance in the power grid, such as a line failure or generator outage, results in an imbalance between production and demand, which can cause instability in the frequency and voltage of the power grid. Modern power grids operate with low operating reserves and stability margins [1], exacerbating the risk of imbalance. When an imbalance occurs, power grid operators take several measures to prevent instability. Optimal load shedding is the last step in maintaining power system stability following a severe disturbance [2–4]. Load shedding is a mechanism used to balance production and demand during energy shortages. However, load shedding faces numerous challenges, particularly in modern power grids. One of the most critical challenges is the impact of cyber-physical constraints on the power grid, which relates to the interdependence of cyber and physical systems.
With recent developments in the field of smart metering and information and telecom-
munication technologies, the use of these devices in power systems has increased, turning
the power system into an extensive cyber-physical system [5]. While the use of intelli-
gent measurement and telecommunication equipment provides many advantages for the
power system, it also faces significant challenges [6]. This equipment offers the possibil-
ity of understanding more about the current conditions of the power system and enabling
optimal control and load shedding in emergencies. However, the power grid's depen-
dence on the cyber system increases its vulnerability and creates new risks. In recent
years, numerous reports have emerged of power grid blackouts in various countries due
to cyber-attacks [7, 8]. As a result, studies on improving the resilience of modern power
networks by considering the cyber-physical power system have become essential.

In power grids, optimal load shedding requires coordination between the generation, transmission, and distribution systems, which is highly dependent on the cyber system. Therefore, cyber-physical constraints can affect the load-shedding program, and its failure leads to severe disruption in the network. For this reason, the study of load shedding in modern power networks should be done considering cyber-physical constraints. The interaction of cyber and physical parts in power systems can be categorized by methods based on graph theory [9], complex network [10], finite estimation machine [11], and Petri net [12]. In [13] and [14], the graph model is used to model cyber-attacks in cyber-physical systems.

Most studies on the resilience of cyber-physical power systems have focused on only one specific aspect of the system. In early works, power systems have been studied only from the aspect of a physical system or cyber system. In [15], the impact of generation loss on the telecommunication network has been investigated. In some studies, only the cyber side is considered. For example, in [16], the use of an information masking strategy to protect data security in cloud-based energy management systems has been investigated. Many studies have been done on the impact of the cyber network on the power network. The impact of cyber events on the power network [17], the impact of the loss of telecommunications on the power grid [18], and dynamic state estimation based on the risk mitigation strategy to improve the resilience of the network against cyber-attacks [19] have been studied.

Although the above works have tried to study the interactions between physical and cyber networks, the physical and cyber section's mutual dependence has not been considered. It has been tried to model the mutual influence of cyber and physical networks [20–23]. In [22], the impact of cascading failures on the power network coupled with three different types of cyber networks has been evaluated. However, in these works, the physical laws of the power network are not considered [23–26]. In [24] and [26], by adding Kirchhoff's laws, it has been tried to overcome the mentioned weaknesses. However, in these works, only power networks have been modeled, and the proposed models were not enough to cover all the features of the cyber-physical system. The reliability of the cyber-physical power system was evaluated in [27] and [28], but the focus was solely on the impact of cyber-attacks on the power system.

Optimal load shedding with the aim of improving resilience and considering the
mutual influence of physical and cyber networks has rarely been studied.
The study in [29] investigates the damage resulting from cyber-attacks on cyber-physical systems while considering the interaction between the cyber and physical sections. The study assumes that the cyber system is backed up by an uninterruptible power supply (UPS) and neglects the effect of the power network on the cyber network. In [30], vulnerability analysis of the coupled power/communication system is performed, considering optimal load shedding to improve resilience. This study also neglects the effect of power loss in the cyber system due to load shedding. In [31], the mutual influence of cyber and physical parts on the load-shedding program has been studied and evaluated to prevent the propagation of faults in the system. The results of this study have shown that the efficiency of the load-shedding program is sensitive to the performance of the cyber domain. Some studies have also focused on the design of the degree of dependence of the cyber and physical sectors [32]. In [33], optimal power flow (OPF) has been studied by considering the mutual influence of cyber and physical networks, however, this model was based on DC OPF and does not provide a suitable criterion for evaluating network voltage stability. In [34], an adaptive-dynamic load-shedding method is presented to improve the frequency resilience of the power grid, but only the physical part of the system is considered. In some works, optimal OPF has been studied by considering cyber-physical constraints and their mutual influence, but it has not been focused on improving resilience [35].

According to the stated content, in this article, an optimal load-shedding program is proposed considering the cyber-physical constraints, in which the mutual influence of the power grid and the cyber network is accurately modeled. In this model, the dependence of the power network on the cyber network and the dependence of the cyber network on the power network are considered. Our goal is to reduce the catastrophic impact of cascading effects on cyber-physical systems, leading to improved power grid resilience. In the proposed model, AC OPF is used so that the physical laws of the power grid are fully applied. For this reason, the final model is a problem with highly nonlinear constraints. To solve this model, the gray wolf optimizer (GWO) algorithm is used along with the Matpower toolbox in Matlab. The proposed model is applied to a standard IEEE system and its efficiency is proved. It is shown by the simulation results that the proposed model can prevent the cascading shutdown of the power network by removing the minimum load. Therefore, this model can improve the resilience of the cyber-physical power system.

## 2 The Mutual Influence of Cyber and Physical Networks

Figure 1 depicts the cyber-physical structure of a power system, where the lower grid represents the power grid with generator buses, demand buses, and intermediate buses, while the upper grid represents the cyber network with central control and telecommunication nodes. The power grid’s objective is to deliver power production from generators to consumers, and the cyber network’s objective is to support this process. For stable and optimal operation of the power grid, continuous monitoring and control are necessary, requiring the collection of information from production and load buses and sending it to the central control unit. This process necessitates an interconnected cyber system. In turn, the cyber system requires electrical energy supplied through the power network
to carry out telecommunication processes [24]. The power grid and cyber network are interdependent and have a mutual influence on each other.

<figure><img src="image_4.png" /><figcaption>Fig. 1. Cyber-physical structure of a power grid</figcaption></figure>

In traditional power networks, the grid operator does not know the state of the cyber system, and when a failure occurs in the network, he seeks to minimize the load only by considering the state of the power network. In this structure, the power network is operated independently from the cyber network [33]. However, any damage to the cyber network affects the operation of the power network, and the state of the power grid affects the state of the cyber network. On the other hand, in modern power networks, the network operator is fully aware of the state of the cyber network and makes decisions based on measured data in the entire cyber-physical system, considering the mutual influence of cyber and physical networks on each other [33]. This article focuses on the second structure where load shedding during system failure is solved by considering the instantaneous conditions of both cyber and physical networks and their mutual influence.

## 3 Mathematical Formulation

To study the resilience of the power system, events with low probability but high impacts, such as storms or human attacks, must be considered. These events can cause the failure of one or more transmission lines and disrupt the balance of production and consumption, thus posing a significant threat to the security of the power system. In such situations, the grid operator must ensure the stability of the power grid by implementing optimal load shedding. To achieve this, the problem of optimal load shedding is modeled with the following objective function:

$$ \min \sum_{i \in I} C^{sh} P_{i}^{sh} + \sum_{g \in G} (a_{g} P_{g}^{2} + b_{g} P_{g} + c_{g}) \quad (1) $$
Where $P_i^{sh}$ is the shedding quantity of load, $P_g$ is the active power output of generator g, and $C_i^{sh}$ is the shedding price of load. Also, $a_g$, $b_g$, and $c_g$ are the cost coefficients of generators. The first part of (1) is related to the cost of load shedding and the second part is the cost of generating power by generators. Based on the AC power flow model, the power grid constraints are presented in the form of the following relations.

$$ \sum_{g \in G(i)} P_g - P_i^d + P_i^{sh} = \sum_{j} P_{i,j}^L, \quad i \in I \qquad (2) $$

$$ \sum_{g \in G(i)} Q_g - Q_i^d + Q_i^{sh} = \sum_{j} Q_{i,j}^L, \quad i \in I \qquad (3) $$

$$ P_{i,j}^{L} = V_{i} \sum_{j} Y_{i,j} V_{j} \cos(\delta_{i} - \delta_{j} - \theta_{i,j}), \quad (i,j) \in \ell \qquad (4) $$

$$ Q_{i,j}^{L} = V_{i} \sum_{j} Y_{i,j} V_{j} \sin(\delta_{i} - \delta_{j} - \theta_{i,j}), \quad (i,j) \in \ell \qquad (5) $$

$$ (P_{i,j}^{L})^{2} + (Q_{i,j}^{L})^{2} \leq (S_{i,j}^{L, \max})^{2}, \quad (i,j) \in \ell \qquad (6) $$

$$ P_{g}^{\min} U_{g} \leq P_{g} \leq P_{g}^{\max} U_{g}, \quad g \in G \qquad (7) $$

$$ 0 \leq P_i^{sh} \leq P_i^d, \quad i \in I \qquad (8) $$

The variable $U_g$ is a binary representation of the state of generator g, denoting a value of 1 for those generators that are capable of producing power. In the event that a generator becomes disconnected from the grid, either due to physical or cyber-related issues, the corresponding $U_g$ value becomes zero. $Q_g$ is the reactive power output of generator g. $P_{i,j}^L$, $Q_{i,j}^L$ and $S_{i,j}^{L, \max}$ represent the active power flow, reactive power flow, and nominal capacity for transmission line (i,j), respectively. $P_i^d$ and $Q_i^d$ represent the active and reactive power demand of bus i, respectively. $V_i$ and $\delta_i$ represent the voltage amplitude and phase angle, while $Y_{i,j}$ and $\theta_{i,j}$ representing the admittance amplitude and angle, respectively. Additionally, $P_g^{\min}$ and $P_g^{\max}$ represent the minimum and maximum active power output limits of the generator g, respectively.

The balance of active and reactive power for each bus is described in Eqs. (2) and (3), respectively. Equations (4) and (5) describe the active and reactive power flowing through the transmission lines, based on the AC power flow model. The thermal limit of the transmission lines, which is determined by their capacity, is described in (6). The output power of each generator is limited by its capacity, as stated in (7). Additionally, the maximum amount of load shedding at each bus is limited, as described in (8), to ensure that it does not exceed the bus's demand.

In the proposed method, the power grid and cyber network are considered separate graphs that depend on each other. If a cyber-node fails, its corresponding buses in the power grid become invisible, and the central control will not be able to monitor or control them. If this happens to buses with generators, the generator will be out of reach and
will not produce power [24]. Generator unavailability is the main reason for most power outages [36]. Therefore, the status of the generators depends on the state of the cyber network and must be modeled based on its conditions. To mathematically describe these conditions, a binary variable $\gamma_v$ is defined that specifies the status of node v in the cyber network. Thus, the generators in the i-th bus of the physical system will be able to produce power only if the corresponding node in the cyber system is in normal conditions. This constraint for the power network is modeled as follows.

$$U_g \le \gamma_v, g \in G(v) \quad (9)$$

Where $G(v)$ represents the set of generators installed in the physical bus that corresponds to the cyber node v. As per Eq. (9), in case the cyber node v, which is connected to the physical bus of generator g, suffers damage, the binary value $U_g$ assumes zero, consequently leading to the generator's incapacity to generate power.

As mentioned earlier, cyber nodes also require energy to function, which must be provided by the physical network. Therefore, the functionality of cyber nodes depends on the working status and connectivity of power buses. If the power supply of a physical bus is interrupted, the corresponding cyber node may become unserviceable due to the lack of sufficient power for the telecommunication system. Hence, it is necessary to determine the value of the variable $\gamma_v$ based on the status of its corresponding bus in the power grid. To model these conditions, we have used the modified method [15], which is described as follows:

$$\left\{ \begin{array}{ll} \gamma_v = 1 & \text{if } P_i^{sh} \le (1-\alpha)P_i^d \\ & \nu \in \nu_i \\ \gamma_v = 0 & \text{else} \end{array} \right. \quad (10)$$

Where $\nu_i$ is the set of all cyber nodes connected to the i-th bus in the power network, and $\alpha$ is a parameter in the range [0,1], that indicates the degree of dependence on the cyber network and the power network [15]. Thus, a larger $\alpha$ indicates a greater dependence of the cyber network on the power network.

Unlike the power grid, which must obey Kirchhoff's laws, the cyber grid is much more flexible in operation. In general, if a cyber node maintains its connection with the control center, it has the potential to perform well. However, as previously stated, if the cyber node's energy needs are not supplied by the power grid, it cannot function. In other words, the working state of a cyber node depends not only on its connection to the control center but also on the state of energy support. To apply these conditions, an iterative process is used according to the flowchart presented in Fig. 2.

According to Fig. 2, firstly, the amount of load shedding in each bus and the parameter $\alpha$ are received as input. Based on these parameters, the status of cyber system nodes is determined based on (10). In the next step, the status of cyber nodes is checked in terms of connection to the central control, and the status of isolated cyber nodes is changed to unusable. After the state of all the nodes of the cyber system is determined, the status of the generators is updated based on the status of the cyber nodes and corresponding physical buses. At this stage, the generators whose corresponding nodes are damaged in the cyber system go to the unreachable state, and their $U_g$ value is set to zero. After updating the status of cyber nodes and the status of the generators, if there is a change in
<figure><img src="image_5.png" /><figcaption>Fig. 2. Flowchart of determining the status of cyber nodes and generators considering cyber-physical systems interaction</figcaption></figure>

the status of cyber nodes or physical buses, the process is repeated. This process continues until there is no change in the value of $U_g$ and $\gamma_v$ in two consecutive iterations.

## 4 Solution Methodology

To solve the problem, the GWO is used, which is an algorithm based on artificial intelligence and has a better performance compared to many other algorithms [37]. One of the main advantages of this algorithm is that there is no need for users to set any of its parameters. This algorithm is based on two functions, encircling prey and hunting.

In the prey encircling step, the distance between each wolf and the prey is calculated according to (11), where $\vec{X}_p$ is the prey position vector, $\vec{X}$ is the wolf position vector, t is the number of iterations, and $r_1$ is a random vector in the range [0,1].

$$ \vec{D} = |\vec{C} \cdot \vec{X}_p(t) - \vec{X}(t)|, \vec{C} = 2\vec{r}_1 \quad (11) $$

The hunting stage includes approaching the prey based on the information obtained from the encircling prey. This stage is modeled as follows:

$$ \vec{X}(t+1) = \vec{X}_p(t) - \vec{A} \cdot \vec{D} \quad (12) $$
$$ \vec{A} = 2\vec{a} \cdot \vec{r}_2 - \vec{a} \qquad (13) $$

Where parameter a decrease linearly from 2 to zero during iterations, and $r_2$ is a vector of random numbers in the range [0,1]. The position of the prey, or in other words, the optimal response in the solution space, is unknown. It is assumed that the wolves $\alpha$, $\beta$, and $\delta$ have the best information about the position of the prey [37]. Therefore, the position of these wolves is used to update the position of the remaining wolves. Using these three positions, which are the best answers in each iteration, the distance between each wolf and these three positions is calculated as:

$$ \begin{aligned} \vec{D}_{\alpha} &= |\vec{C}_1 \cdot \vec{X}_{\alpha} - \vec{X}| \\ \vec{D}_{\beta} &= |\vec{C}_2 \cdot \vec{X}_{\beta} - \vec{X}| \\ \vec{D}_{\delta} &= |\vec{C}_3 \cdot \vec{X}_{\delta} - \vec{X}| \end{aligned} \qquad (14) $$

These distances can be used to update the positions of the wolves in the next iteration. The position of the wolves in the next iteration is calculated using the following relations:

$$ \begin{aligned} \vec{X}_1 &= \vec{X}_{\alpha} - \vec{A}_1 \cdot \vec{D}_{\alpha} \\ \vec{X}_2 &= \vec{X}_{\beta} - \vec{A}_2 \cdot \vec{D}_{\beta} \\ \vec{X}_3 &= \vec{X}_{\delta} - \vec{A}_3 \cdot \vec{D}_{\delta} \end{aligned} \qquad (15) $$

$$ \vec{X}(t+1) = \frac{\vec{X}_1 + \vec{X}_2 + \vec{X}_3}{3} \qquad (16) $$

By repeatedly applying the encircling prey and hunting operators, the position of the prey or the best answer is obtained. In the proposed method, for each candidate response of the GWO, which includes the amount of load shedding in each bus, the algorithm in Fig. 2 is applied to determine the status of cyber nodes and generators. Then, considering the state of the cyber-physical system, the OPF is solved. If the total constraints of the problem are met, the results of the OPF and the amount of load shedding are recorded. If the OPF does not converge, a large penalty is sent to the GWO as a cost function. The flowchart of the proposed algorithm for solving the problem is presented in Fig. 3.

# 5 Simulation Results

The proposed model is applied to the IEEE RTS-79 system for evaluation. Based on [24], it is assumed that each bus of the power network is equipped with a telecommunication node. Additionally, the control center is placed next to the node that has the most connections to other nodes. The simulation is performed on a 3.3 GHz Ryzen 9 5900 hs-based laptop. To solve the problem, the GWO algorithm is implemented in MATLAB, and the Matpower toolbox is used for OPF.

The IEEE RTS-79 system has 12 generators, 36 transmission lines, and 17 load centers. Its complete information is available in [38]. The coupling parameter of the physical and cyber network, $\alpha$, is set to 0.75 [15]. Figure 4 shows the single-line topology of the power grid along with its cyber network.
To evaluate the resilience of the studied system, various scenarios with different levels of failure have been considered. To this end, ten different levels of severity of failure in the system have been evaluated. For instance, a failure level of “N” implies that N number of system lines have faulted at the same time and are out of service. For each level of failure, ten scenarios are generated. Consequently, by considering ten failure levels and ten scenarios for each level, 100 different modes are obtained, and the problem is solved for each of these modes.

<figure><img src="image_6.png" /><figcaption>Fig. 3. Proposed strategy for load shedding in cyber-physical system</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 4. Topology of the case study system</figcaption></figure>

Considering that the study is related to resilience, to generate line failure scenarios, the failure probability of all lines is assumed to be the same at first, and the first failed line is randomly determined. Next, the probability of failure of the lines that are closer to the failed line is increased, and a second failed line is randomly selected. This process is repeated until ten scenarios are reached. By applying this method, line failures occur in a specific region of the system that can be considered caused by storms or floods.

For the GWO algorithm, the number of wolf populations and the maximum iteration of the algorithm is set to 50 and 50, respectively. The time required to solve the problem for one mode is approximately 80 s. Thus, it takes 8000 s to solve the mentioned 100 situations. Since the outcome of meta-heuristic algorithms relies on the initial population, which is generated randomly, the problem is solved ten times using the GWO algorithm. The best solution obtained from the ten runs is chosen as the final solution.
The problem of optimal load shedding following a disturbance has been solved twice: once by considering the proposed strategy and once without considering the interaction between the physical grid and the cyber network, which is commonly referred to as the traditional strategy. By comparing these two strategies, the impact of considering the interdependence between the cyber and physical networks is determined. Figures 5, 6 and 7 illustrate the simulation results for both strategies.

Figure 5 displays the average total cost value for each fault severity based on (1). This figure is generated by solving the problem for the 100 specified modes and computing the average over the scenarios for each damage severity.

<figure><img src="image_8.png" /><figcaption>Fig. 5. Average total cost for each failure level</figcaption></figure>

Based on Fig. 5, it is evident that the system cost increases with the increase in the failure level for both methods. As more lines go out of service with increasing failure levels, more loads are isolated or cut off due to the unavailability of generators. However, the proposed strategy shows significantly lower overall cost than the traditional strategy for failure levels 2 and above. The proposed strategy considers load shedding in a way that minimizes the impact on both the physical and cyber networks. In contrast, the traditional method only considers the physical network's conditions, resulting in some load interruptions caused by cyber line failures, reducing power production.

Figure 6 shows the average cost reduction of the proposed strategy compared to the traditional strategy. As shown, for fault level 1, there is no significant cost reduction because this level of fault does not cause widespread failure in the cyber network. It can also be seen that for fault level 2, the overall cost is reduced by nearly 90%. The reason for this is that in the proposed strategy, load shedding is done in such a way that it does not damage the cyber network. For higher fault levels, the cost reduction of the proposed strategy is evident and significant. Figure 7 and 8 display the average number of online cyber nodes and the average interrupted load, respectively, for each failure level.
<figure><img src="image_9.png" /><figcaption>Fig. 6. Average cost reduction for each fault level</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. The average number of cyber nodes in service for each failure level</figcaption></figure>

Figure 7 further supports the benefits of the proposed strategy by showing the number of healthy cyber lines for each fault level. The proposed strategy results in fewer out-of-service cyber lines, indicating that load shedding is performed in a way that preserves the health of the cyber network.

Finally, Fig. 8 displays the average load interruption for each failure level. It is evident that, for failure level 10, the traditional strategy results in over 90% of the system load being cut off (2580 MW), nearly leading to a complete blackout. In contrast, the proposed strategy results in much less load shedding than the traditional method.
<figure><img src="image_11.png" /><figcaption>Fig. 8. Amount of interrupted load for each failure level</figcaption></figure>

## 6 Conclusion

In this study, a novel approach for optimal load shedding was proposed to enhance resilience in cyber-physical power systems. The proposed strategy involved modeling the interaction between the physical and cyber networks and employing AC optimal power flow to accurately represent the physical network rules. The standard IEEE system was used as a case study to evaluate the effectiveness of the proposed strategy. The problem of optimal load shedding was addressed by considering various levels of system failure and scenarios of failed lines in the physical network. The simulation results demonstrated that the proposed strategy can reduce the amount of load shedding during severe disturbances, thereby improving the system's resilience. The results also showed that the proposed strategy can decrease the cost of load shedding compared to conventional methods. This advantage can be attributed to the proposed strategy's simultaneous consideration of both the physical and cyber network conditions during load shedding. Consequently, load shedding is performed with a minimal negative impact on the cyber network, leading to minimal damage to the cyber lines.

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