Power source-power grid coordinated typhoon defense strategy based on multiagent dynamic game theory

0 Introduction

Large-scale power outages caused by unavoidable natural disasters,e.g.,typhoons,occur frequently [1],resulting in serious economic losses and political impacts[2].These events are vital threats and lead to huge load losses to the energy internet [3] with a high renewableenergy penetration rate [4-6].Minimizing the load loss [7]through the coordination of the power source and grid [8-10]has become a problem that needs to be solved urgently.

Preventive measures against these natural disasters have been taken to address this issue,which involve strengthening or building additional grid infrastructure [11,12].The black start unit (BSU)[13] was added at the power source side,overhead lines were replaced by underground cables,and poles and towers were strengthened on the grid side.These measures have a significant positive effect on typhoon resistance and enhanced the resilience of the power grid[14],but they are extremely costly.Achieving maximum grid protection with minimum investments is essential.Considering limited budgets,optimal allocation strategies of preventive measures and other resources [15] have been intensively studied [16],optimal allocation of various power sources,such as electric vehicles,diesel generators,and energy storage devices,helps to minimize the grid loss caused by hurricanes [17,18].In these studies,natural disasters were modeled as a probabilistic line failure that can be classified as a stochastic programming problem [18].However,the worst scenario for the power grid was ignored.

Transmission line failure caused by typhoons not only leads to grid operation performance deterioration but also tests the reasonability of grid planning [19],which requires improvement from both power grid side and power source side.From power grid side,line reinforcement measures should be considered by the power grid planner; from the power source side,the BSU location and capacity should be confirmed by the BSU planner.However,aforementioned measures that have been undertaken are unilateral preventive measures; bilateral measures,e.g.,a coordinated defense strategy [20,21],have not been considered.A twolevel regional integrated energy system scheduling strategy was proposed [22] to realize a balance of constraints and intraday joint optimization of three parties.A coordinated planning method considering multiagent gaming and power grid uncertainties was proposed for an incremental distribution electric network [23].Additionally,a marketoriented multiagent game model was proposed as an optimal scheduling strategy [24,25],including power generation companies,electricity sales companies,and end users.Each party’s revenue was set as the optimization target.However,these studies mainly focused on the power market and dispatching decision making.Typhoon resistance was not considered.

The three parties (power grid planner,BSU planner,and typhoon)can be considered to constitute a multiagent game.In this study,typhoons were regarded as virtual rational gamers,and maximum grid load loss was assumed when a typhoon attacks the power system,causing the worst grid failure scenario.A multiagent defender-attackerdefender (DAD)dynamic game model [24] is proposed.It includes defense measures for power grid planners and BSU planners.An attempt was made to determine the optimal line reinforcement,calculate the placement and capacity of the BSU,and minimize power grid losses.

1 Multiagent DAD Dynamic Game

In this study,it was assumed that typhoons are sufficiently rational,any defender can implement a defense strategy,and the attack strategy is updated immediately.The information exchange between the BSU planner and the power grid planner in the same round of the game is considered.The multiagent DAD dynamic game model can be divided into three stages to simulate the gaming procedure.In the first stage,it was assumed that typhoons are sufficiently rational,any defender can implement a defense strategy,and the attack strategy is updated immediately.The information exchange between the BSU planner and the power grid planner in the BSU planner acts as a defender,and a defense strategy is formulated to minimize the load loss through deployment of the BSU with a fixed capacity and location.The power grid topology information is sent to the typhoon.In the second stage,based on the received information,the typhoon starts attacking to maximize the grid load loss,stop transmission lines from operating,and cause power grid failure.Then,the grid damage and load losses are transmitted to the grid planner.In the third stage,the grid planner serves as the defender as well,reads damages and load loss information,and takes measures,e.g.,line reinforcements,to minimize load losses.The static stages are shown in Fig.1,and the iterative process is illustrated in Fig.2.

Fig.1 DAD static game model

Fig.2 DAD game-based natural disaster resisting planning

The game can be described by a tri-layer DAD mathematical model [26],please find details in reference:

Here,u is the 0-1 decision variable of the upper-level planning,which indicates whether the node is connected to the BSU.PBSU represents the output of the connected BSU.v is the 0-1 decision variable of the middle-level planning model,which indicates whether the power line is out of service.w is the 0-1 decision variable of the lower-level planning model,which indicates whether the power line is being reinforced.f is the expected value of the grid load loss. hu,hm,and hl represent the upper,middle,and lower planning constraints,respectively.

2 Decoupled Model

The trilevel planning model described by Eqs.(1)-(6)has the following characteristics.

(1)The upper-,middle-,and lower-level planning correspond to the BSU planner,typhoon,and power grid planner,respectively.

(2)Trilevel planning has a clear solving sequence.

(3)All decision variables of the trilevel planning model are independent of each other.

The tri-layer mathematical model can be decoupled to obtain independent mathematical models of the BSU planner,typhoon,and power grid planner [26,27].

2.1 BSU planner model

2.1.1 Objective function

The goal of the BSU planner is to minimize the load loss; the objective function in Eq.(7)can be obtained by decoupling Eqs.(1)-(6):

2.1.2 Constraints

The constraints of the BSU planner model include constraints on BSU investment,branch power flow,power balance,conventional unit output,BSU output,node voltage,branch transmission capacity,line failure,and line reinforcement.

(1)BSU investment constraints

The cost of building a new BSU or modifying an original BSU is extremely high,and investment constraints should be considered when planning placement and capacity.

Here,un indicates whether node n is connected to the BSU.un = 1 indicates that node n is connected to the BSU,whereas un = 0 indicates that node n is not connected to the BSU.In addition,S indicates the value of the BSU investment budget; the total expense of the nodes connected to the BSU is not more than S.

(2)Branch flow constraints

where l is the line between nodes n and m. Pl and Ql are the active and reactive powers of branch l,respectively.Vn and Vm are the voltage amplitudes of nodes n and m,respectively.gl and bl are the real and imaginary parts of the reciprocal impedance of line l,respectively.θl is the phase angle difference between nodes n and m. blc is the susceptance of the ground branch of the π-type equivalent circuit of branch l.

(3)Power balance constraint

Here,Png and Qng are the actual active and reactive power outputs of the unit at node n,respectively,PnBSU and QnBSU are the actual active and reactive power outputs of the BSU at node n,respectively,and Pnd and Qnd are the active and reactive power outputs of node n,respectively.

(4)Conventional unit output constraints

where and are the minimum and maximum active output limits of unit n,respectively,and and are the minimum and maximum reactive output limits of unit n,respectively.

(5)BSU output constraints

Here,are the lower and upper limits of the BSU output,respectively.

(6)Node voltage constraints

where Vn,min and Vn,max are the minimum and maximum voltage amplitude limits of node n,respectively.

(7)Branch transmission capacity constraint

Here,Pl,max is the maximum transmission power of branch l.

2.2 Attack model of typhoon

Natural disasters are highly uncertain [27,28].Fig.3 shows the time sequence of typhoon Hato passing through the southern coast of China at the end of August 2017.Hato formed at 00:00 hours and landed at 48:00 hours The wind speed-time is shown in Fig.3.The wind speed gradually increased from formation to landing; it reached a maximum wind speed of 48 m/s at the time of landing.Then,the wind speed began to decrease until the typhoon disappeared.

A regional power grid in the Guangdong province coastal area was taken as an example; the geographical distribution of its nodes is shown in Fig.3.The path of typhoon Hato is shown by the dotted line in Fig.4.The components affected by Hato at t1 are the power equipment contained in the circle with t1 as 4; the same happened at t2 and t3.As Hato gradually developed inland,its influence decreased.

Fig.3 Path of typhoon Hato

Fig.4 Typhoon Hato wind speed-time diagram

Fig.5 Hato transiting in Guangdong power grid

2.2.1 Objective function

As stated earlier,Hato is a virtual rational player,launching an attack on the power system under a specific grid topology,forcing transmission lines to stop,and maximizing the load loss.The objective function of the typhoon attack model can be obtained as

2.2.2 Constraints

The constraints are mainly the power grid reliability,branch power flow,power balance,conventional unit output,BSU output,node voltage,and branch transmission capacity constraints.

(1)Grid reliability constraints

Here,vl,T indicates whether power line l is damaged by a typhoon at time step T,resulting in operation,vl,T = 1 indicates that line l is not affected,and vl,T = 0 indicates that line l is out of service.Furthermore,KT is the expected value of power grid reliability at time T.The physical meaning is that the total number of faulty lines caused by the typhoon does not exceed KT.

(2)Other constraints are shown in Eqs.(10)-(18).

2.3 Power grid planner model

BSU placement and capacity should be considered by grid planners; moreover,transmission line reinforcement measures are needed to defend against typhoons.Common reinforcement measures include changing overhead lines to underground cables,upgrading poles and towers with more materials,and upgrading substation equipment[29].Considering these issues,line reinforcement measures are taken as the decision-making variables of the power grid planner.

2.3.1 Objective function

The power grid planner optimizes the line reinforcement to resist the invasion of the typhoon and minimize the load loss,thereby constructing the objective function of the power grid planner:

2.3.2 Constraints

The constraints of the power grid planner model mainly include line reinforcement investment,branch power flow,power balance,conventional unit output,BSU output,node voltage,and branch transmission capacity constraints.

(1)Investment constraints for line reinforcement

The investment constraints for design line reinforcement are

where wl indicates whether power line l is reinforced.wl =1 indicates that line l is reinforced,whereas wl = 0 indicates that line l is not reinforced.B is the line reinforcement investment budget value; the total investment amount of reinforced lines does not exceed B.

(2)Other constraints are shown in Eqs.(10)-(18).

2.4 Constrained linearization

The branch power flow constraints shown in Eqs.(10)and (11)are nonlinear constraints.The linear-programming approximation of the AC power flow method [29] is introduced to achieve constraint linearization.The linearized constraints are

where cl is the approximate value of cos θl,d is the interval between tangent points,d = 2θ/(h + 1),h is the number of tangents (please refer to a previous report [29] for the specific meanings of h and d),and Vn and c are nodes.The ideal voltage value of n,ΔVn,is the line deviation between the actual and ideal voltage values.

3 Solution

3.1 Nash equilibrium existence of multiagent DAD dynamic game

Because the strategies the attacker (typhoon)can take are limited,and,in any situation,defenders have an optimal defensive strategy,the defender’s maximum loss can be calculated in each round.Therefore,the DAD dynamic game proposed has only an optimal solution when both the attacker and defender are rational [20].

3.2 Case analysis

3.2.1 Real-example analysis

A real example of a regional power grid in the Guangdong coastal area is selected,as shown in Fig.6,and a multipletime-step typhoon transit model is designed according to the temporal and spatial dynamic characteristics of a typhoon[19].The objective function of the destructiveness of typhoon transit is to maximize the grid load loss.Based on the typhoon model presented in Section 2.1,the optimal solution is provided for three time steps (t1,t2,and t3),as shown in Fig.7.

Fig.6 Typhoon Hato caused power grid failure in Guangdong,China

Fig.7 Line faults and load shedding after typhoon transited

In each step,a typhoon causes transmission lines to be out of service,and the number of outage lines is limited by the expected value of the power grid reliability KT.The faulty line in the previous time step is still out of service in the subsequent time step.The line fault outage situation shown in Fig.7 is obtained under the premise of the expectation set {Kt1 = 2,Kt2 = 2,Kt3 = 2}.This means that the numbers of line faults at time steps t1,t2,and t3 are not more than 2,2,and 2,respectively.The optimal solution obtained by solving the model corresponds to the maximum load loss under the budget set.

In this example,when a typhoon runs to the range shown at time step t1,lines 22-45 and 43-44 will fall out of service because of faults.When the typhoon continues to run to the position shown at time step t2,lines 22-24 and 22-52 will fall out of service owing to faults.When the typhoon finally reaches the t3 position,it destroys lines 4-10 and 14-20.The total grid load in this area is 3144 MW.After the typhoon passes,the total grid load loss reaches 1380 MW.The load loss node is shown in the shaded part of Fig.7.

continue

Methods BSU planner optimal BSU placement Power grid planner optimal measures Typhoon attacking strategy＆ load loss A-D static gaming 57 53-57 53-57,58-59,22-24,22-52,4-10,14-20 Load loss 571 MW Multiagent DAD dynamic gaming 57 22-24 45-53,53-57,22-52,24-25,4-10,14-20 Load loss 546 MW

Fig.7 shows that,after lines are out of service because of faults,the original power system is divided into multiple grid islands,large-scale load loss exists,and grid islands run independently.Nodes without a power supply stay are denoted in the gray part in Fig.7.

3.2.2 Results based on multiagent DAD game

Based on the example in Section 4.2.1,the planning of the multiagent DAD dynamic game results in two scenarios.

Scenario 1: Power grid reliability expectation set K1 ={Kt1 = 2,Kt2 = 2,Kt3 = 2},BSU investment budget value S1= 1,line reinforcement investment budget B1 = 1.

Scenario 2: Power grid reliability expectation set K2 ={Kt1 = 2,Kt2 = 2,Kt3 = 2},BSU investment budget value S2= 1,line reinforcement investment budget B2 = 2.

The game process for Scenario 1 is shown in Table 1.In the first round,the BSU planner solves the optimal distribution point (node 34),and then the typhoon launches an attack based on the grid time division afterthe distribution point and obtains the worst-case fault for lines 45-53,58-59,22-24,22-52,4-10,and 14-20.Subsequently,the power grid planner obtains the optimal reinforced lines 45-53 based on the BSU distribution and line fault conditions.Finally,the typhoon attacks the updated grid topology again based on the BSU distribution and line reinforcements updated in this round.The BSUs connected to node 34 and after lines 45-53 are reinforced,and the maximum load loss is 614 MW.The grid topology at the end of the first round of the game is shown in Fig.8.

Table 1 Gaming procedure under Scenario 1

Rounds BSU placement Typhoon attacking strategy ＆load loss Lines that need reinforcements Typhoon attacking strategy ＆load loss 1 34 45-53,58-59,22-24,22-52,4-10,14-20 Load loss:804 MW 45-53 53-57,58-59,22-24,22-52,4-10,14-20 Load loss:614 MW 2 57 45-53,53-57,22-24,22-52,4-10,14-20 Load loss:571 MW 22-24 45-53,53-57,22-52,24-25,4-10,14-20 Load loss:546 MW

Table 2 Gaming procedure under Scenario 2

Rounds BSU placement Typhoon attacking strategy ＆load loss Lines that need reinforcements Typhoon attacking strategy ＆load loss 1 34 45-53,58-59,22-24,22-52,4-10,14-20 Load loss 804 MW 45-53,22-24 53-57,58-59,24-24,22-52,4-10,14-20 Load loss 589 MW 2 57 45-53,53-57,22-24,22-52,4-10,14-20 Load loss 571 MW 45-53,22-24 53-54,53-56,22-52,24-25,4-10,14-20 Load loss 546 MW 3 54 53-57,58-59,22-24,22-52,4-10,14-20 Load loss 614 MW 58-59,22-24 41-45,41-42,22-52,24-25,4-10,14-20 Load loss 506 MW

Fig.8 First-round DAD dynamic game result under Scenario 1

Fig.9 shows the grid status after the second round.Fig.10 shows the load loss situation of all 15 rounds of the game.The defense strategy in the second round minimizes the possible load loss caused by the typhoon.Starting from the third round,the game enters a convergence stage,which means that,no matter how the defense strategy varies,the maximum load loss would not exceed that in the second round of the game.

Fig.9 Second-round DAD dynamic game result under scenario 1

Fig.10 DAD dynamic game load shedding under Scenario 1

The game processes of rounds 1-3 of Scenario 2 are shown in Table 3; the load losses in each round are shown in Fig.11.In the third round of the game,the optimal defense strategy is obtained: the BSU is connected to node 54,and lines 58 and 59 and 22-24 are reinforced.After this strategy is adopted,the maximum load loss caused by a typhoon is 506 MW.Starting from the fourth round of the game,the game enters the convergence stage.

Table 3 Optimal defense measures in three kinds of game

Methods BSU planner optimal BSU placement Power grid planner optimal measures Typhoon attacking strategy＆ load loss Static gaming 34 22-45 45-53,58-59,22-24,22-52,4-10,14-20 Load loss 804 MW

Fig.11 Dynamic game load shedding under Scenario 2

However,the proposed DAD-based grid defense strategy differs from the latest research.In one study [30],a multistage multizone DS line state set was adopted to reflect the spatial and temporal characteristics of severe contingencies,and the DAD model was used to minimize the weighted load shedding.In another study [31],a multiple-attack scenario DAD model was proposed by extending the conventional trilevel DAD model.Uncertainties from offensive resources occur in the first level,whereas the attacker and system operator are placed in the middle and low levels.Both methods are quite advanced,but the necessary data for the real example in Section 4.2.1 cannot be obtained.A comparative analysis was performed by comparing the static game,defenderattacker (DA)game,and proposed DAD game to illustrate its effectiveness and correctness.

3.3 Comparative analysis

To verify the effectiveness and accuracy of the proposed method,the power grid reliability expectation set K1 = {Kt1= 2,Kt2 = 2,Kt3 = 2},BSU investment budget value S1 =1,and line reinforcement investment budget B1 = 1 were used to compare the static game,DA dynamic game,and multiagent DAD dynamic game.The results are shown in Table 4.In the static game,the BSU planner connects the BSU to node 34,and the grid planner reinforces lines 22-45.After the defense strategy is formulated,the maximum load loss caused by a typhoon attacking the power grid is 804 MW.When the DA dynamic game is adopted,the BSU is connected to node 57,the grid planner reinforces lines 53-57,and the maximum load loss is 571 MW.The multiagent DAD dynamic game connects the BSU to node 57 and reinforces lines 22-24,with a maximum load loss of 546 MW.The gaming procedure comparisons are shown in Fig.12.

Fig.12 Load shedding comparison of three kinds of game

In the same given set,the multiagent DAD dynamic game can obtain the optimal solution through two rounds of the game,whereas the static game requires three rounds,and the DA dynamic game requires one round.However,the static game results are poor,and the DA dynamic game enters the convergence process prematurely.Defenders do not note the interaction of grid topology information,and decision making is not based on real-time grid topology information.The results of the calculation example show that the lost load obtained by the multiagent dynamic DAD dynamic game is the lowest among the three methods,and the optimality is better than the attacker-defender dynamic game method.

4 Conclusion

A power source-power grid coordinated typhoon defense strategy based on multiagent dynamic game theory was proposed.A power grid struck by a typhoon in the coastal area of Guangdong was used as an example.The results show the following.(1)By treating typhoons as rational virtual gamers,in the process of dynamic games,typhoons update their own attack strategy immediately according to the defense.As a result,an optimal defense strategy can be obtained.It has an improved ability to resist the worst-case scenario.(2)The multiagent DAD dynamic game obtains better results than the static game and DA dynamic game,and the calculation efficiency is better than that of the DA dynamic game.Besides,the proposed method in this study can also be used on natural disaster resistance enhancements,e.g.,destructive winds in western China,which may cause serve damage to power grid.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No.U1766204).

Declaration of Competing Interest

We declare that we have no conflict of interest.