Joint optimization of BS-VS association and power control in secure HSR communication systems

In this paper, the secure transmission for high-speed railway (HSR) communication system is studied. The considered HSR wireless communication system consists of a macro base station, B roadside base stations, and K vehicle stations (VSs) on the top of the train, and the eavesdropping user is a mobile unmanned aerial vehicle (UAV). We consider maximizing the sum of the minimum security rate of each time slot as the objective function, and the constraint conditions contain the quality of service (QoS), switch requirements and the total power constraint. The original optimization problem is mixed-integer and non-convex, it is intractable to solve directly. The block coordinate descent (BCD) method is applied, and the original problem can be decoupled into two sub-problems. The one is the joint BS-VS association problem, and the other is the power control problem. The first sub-problem of optimizing BS-VS association can be solved by the successive convex approximation (SCA) algorithm, and the second one of optimizing transmit power can be solved by the lagrangian dual method. Simulation results show that the proposed algorithms have good convergence.

cooperative distributed secure spatial modulation was established, which could improve the security performance of HSR communication system. A joint optimal secrecy capacity algorithm was proposed to maximize the system security capacity, and the eavesdropping users were also stationary [25].
In this paper, the secure transmission for HSR communication system is investigated. The contributions are mainly summarized as follows: (1) The considered HSR wireless communication system includes a macro base station (MBS), B roadside base stations (RBS) and K vehicle stations (VSs) on the top of the train, and the eavesdropping user is a mobile UAV. The objective function is to maximize the sum of the minimum security rate in each time slot. The constraint conditions are the QoS, switch requirements, and total power constraint.
(2) The original optimization problem is mixed-integer and non-convex. The block coordinate descent (BCD) method is applied and the original problem can be decoupled into two sub-problems. The one is the joint BS-VS association problem, and the other is the power control one.
(3) The first subproblem of optimizing BS-VS association can be solved by the successive convex approximation (SCA) algorithm, and the second subproblem of optimizing transmit power can be solved by the Lagrangian dual method [26]. Simulation results verify the effectiveness of the proposed algorithm.
The rest of this paper is organized as follows. In Section 2, the secure HSR communication system model is introduced and the joint BS-VS association and power control problem is formulated. Then, we proposed an efficient iterative algorithm to maximize the minimum secrecy rate by using some convex optimization techniques in Section 3. Finally, numerical results show the effectiveness of the proposed solution in Section 4, and Section 5 concludes this paper.

System model
The architecture of a HSR wireless communication network is illustrated in Fig. 1. A heterogeneous cellular network consisting of a MBS and B RBS is considered to communicate with K single antenna VSs installed on the train roof, where all RBSs are deployed at regular intervals in the coverage area of MBS and all BSs are configured with single antenna. The set of BSs is defined as B = {0, 1, . . . , B} where 0 represents the MBS and {1, 2, . . . , B} are RBSs, and the set of VSs is defined as K = {1, 2, . . . , K}. Moreover, there is a malicious eavesdropping UAV equipped with single antenna in the cell, and its trajectory can be known [27].
It is assumed that the running time of the high-speed train in the cell of MBS is T, which is equally split into N time slots for the sake of convenience in analyzing the system model. The set of time slots is defined as N = {1, 2, . . . , N}. In each time slot, the BSs send the requested data to the VSs and each VS is assumed to only associate with a RBS or the MBS. Let x ik [ n] ∈ {0, 1} be a binary variable for VS k ∈ K at time slot n ∈ N to indicate VS-to-BS i ∈ B association, i.e., x ik [ n] = 1 if VS k is associated with BS i and x ik [ n] = 0 otherwise. Since each VS is only associated with one BS at each time slot, we have (1) In order to avoid the high frequent switch and reduce associated overhead in the process of running, the VS is inclined to associate with the same BS in a period of consecutive time slots. Therefore, we give the following constraint where c and d are the constants to determine a tradeoff between low-switch and highperformance. d is the maximum number of connections and c is the number of time slots. The larger the number of connections in the fewer slots, the higher the performance of the system.
If VS k is associated with BS i at time slot n, the achievable data rate of the downlink transmission for VS k is given by interference from BSs to all other VSs except k, and σ 2 k is the noise power. It is supposed that the eavesdropping UAV is located at the horizontal coordinate w u [ n] with the fixed altitude H at time slot n, which are imperfectly known at the BSs. The location of BS i ∈ B is denoted as l i . We assume that UAV is also the network user and so it can be considered as the internal eavesdropping user. Since the propagation conditions between BSs and the eavesdropping UAV can be approximated as free space, we adopt the free-space path loss model to characterize the channel. Thus, the channel power gain between BS k and the UAV is dominated by the line-of-sight link and is written as where = ξ i ξ u ( ζ 4πd 0 ) 2 denotes the channel power gain of BS-to-UAV link at the reference distance d 0 = 1 m, with ζ being the wavelength of the transmit signals, ξ i and ξ u being the transmitting gain of the antenna of BS i and receiving gain of the antenna of UAV, respectively. Therefore, the data rate for the eavesdropping UAV to eavesdrop the signal from the VS k can be given as Since the VS k ∈ K does not know that the received signal is eavesdropped or not by the eavesdropping UAV, the secure communication between the associated BS and VS k need to be guaranteed at any time slot. The secrecy rate for VS k at time slot n is given by where [ a] + = max{a, 0}.

Problem formulation
The aim of this paper is to maximize the sum of N secrecy rates any of which is the minimum secrecy rate among all VSs at one time slot. We seek to optimize the BS-VS association x {x ik [ n] , ∀i ∈ B, ∀k ∈ K, ∀n ∈ N } and power control p p ik [ n] , ∀i ∈ B, ∀k ∈ K, ∀n ∈ N under of guaranteeing QoS, less switch requirements and total power. The optimization problem can be formulated as where γ min k is the minimum QoS requirement for VS k ∈ K and p max i is the peak transmit power of BS i ∈ B.
The joint BS-VS association and power control problem in P1 have some challenges to solve, since two key variables are coupled in P1 and P1 is in a mixed-integer and non-convex form. To solve P1, we apply the BCD method and decouple P1 into two subproblems. We solve the first subproblem of optimizing BS-VS association by applying the SCA technique, and then leverage the lagrangian dual method to solve the second subproblem of optimizing transmit power.

BS-VS association subproblem
When the power control vector p is fixed, the optimization problem P1 can be rewritten as To begin with, P2 is reformulated as P2.1 by introducing the auxiliary variables χ[ n] , n ∈ N and slacking the constraint (7d).

P2.1 : max
In order to ensure the associated solutions are binary, we introduce the auxiliary variables y {y ik [ n] , ∀i ∈ B, ∀k ∈ K, ∀n ∈ N } and give the following constraints.
The constraint (10) is transformed into a penalty item to be added to the objective function [28]. Then, we get the following problem.

P2.2 : max
where ω n is penalty factor for ∀n ∈ N . To solve problem P2.2, we propose the double-layer iterative algorithm where y with the fixed x and χ[ n] is optimized in outer layer and the problem P2.2 with y is solved in inner layer. In outer layer, the optimal solution y * is obtained by solving the unconstrained optimization problem, as follows: By deriving the first-order derivative of problem P2.3 with respect to y ik [ n], the following equation can be obtained as (2021) 2021:64 Page 7 of 20

Fig. 2 The deployment locations of all BSs and VSs
Then, the optimal solution of y is obtained as for ∀i ∈ B, ∀k ∈ K, ∀n ∈ N . In inner layer, problem P2.1 with the fixed y is non-convex due to constraint (9b).
Since f 1 and g 1 are concave functions, f 1 (x) − g 1 (x) is a subtraction form of two concave functions. We adopt the SCA technique to re-express R k [ n] −R ku [ n] in the t + 1-th iteration as Constraint (7b) is rewritten as Since the left item of constraint (7e) is convex, we introduce the function We adopt the SCA technique to re-express the above inequality by using the first order Taylor expansion of f 2 around a feasible point {x ik [ n] } at the (t+1)-th iteration. Constraint (7e) is re-expressed as (21).
Problem P2.2 is reformulated as which is a convex optimization problem.

BS transmit power subproblem
When the BS-VS association vector x is given, P1 can be split to N subproblems, which is equivalently formulated as where the symbol [ n] indicating time slot n is omitted for the stake of convenience.
To convexify the objective function, we introduce two kinds of auxiliary variables ρ and τ k , k ∈ K and P3 is reformulated as Since (24b) is non-convex, we introduce the inequality where the inequality holds if η =η, and get that Thus, constraint (24b) is rewritten as The second item in above inequality is coupled with respect to the optimization variables. Its first order Taylor expansion around a feasible point (ρ t , τ t k , p t b k k ) at the (t + 1)-th iteration is given as According to (29), inequality (24b) can be derived as a further tight constraint (30) at the (t + 1)-th iteration. Constraint (24c) is non-convex and expressed as To convexify (24c), the first order Taylor expansion of 2 τ k − 1 around a feasible point τ k at the (t + 1)-th iteration is given as Thus, (24c) is converted into Letting k = 2 τ t k 1 − ln 2τ t k − 1 and k = 2 τ t k ln 2, we have The right item of inequality (34) is coupled with respect to the optimization variables τ k and p b l l . We derive its first order Taylor expansion around a feasible point (τ k , p b k k ) at the (t + 1)-th iteration, which is similar to the process of (29). The lower bound of the right item of inequality (34) is given as (35). According to (35), inequality (24c) can be derived as a further tight constraint (36) at the (t + 1)-th iteration.

Theorem 1 For any given (λ, μ, κ, ν), the optimal solution of power control for all VSs is given by (43).
Proof When (λ, μ, κ, ν) is given, the Eq. 42 is obtained by deriving the first-order derivative of L with respect to p b k k for ∀k ∈ K. Thus, the optimal solution of power allocation for all VSs can be get by (43). (λ, μ, κ, ν), the optimal auxiliary variables ρ * and τ * k for any VS k ∈ K can be obtained by

Theorem 2 For any given
Proof: It is assumed that (λ, μ, κ, ν) have been given. By deriving the first-order derivative of L with respect to ρ and τ k respectively, we can obtain the following equations and Thus, the optimal solution ρ * and τ * k , k ∈ K can be obtained by (44).

Algorithm 1
The BS-VS association algorithm for solving P2.
Loop for t = 1, 2, · · · , 3: Constraint (17) is updated based on the feasible point x t−1 . 4: Obtain the optimal BS-VS association solution x t by solving the convex optimization problem P2.4. 5: If the convergence holds, end loop and let x t 1 = x t . 6: t = t + 1 End loop 7: t 1 = t 1 + 1 8: If the convergence holds, end loop and output the optimal solution x * = x t .

End loop
According to the aforementioned analysis, the proposed iterative algorithms are described in Algorithm 1 where the BS-VS association x with given p is optimized and in Algorithm 2 where the power control p with given x is solved. The joint optimization 1: Initialization: Initialize the power control vector p 0 {p 0 ik , ∀i ∈ B, ∀k ∈ K}. Loop for t 1 = 1, 2, · · · , 2: Use the feasible point p t 1 −1 to update the θ t 1 and ϑ t 1 from (26). 3: Set the initial feasible points (p * b k k , ρ * , τ * k ). Loop fort = 1, 2, · · · , 4: Let ρ t = ρ * , τ t k = τ * k , p t b k k = p * b k k . Constraints (30) and (36) are updated based on the feasible points (ρ t , τ t k , p t b k k ) and (θ t , ϑ t ).
End loop 12: t 1 = t 1 + 1 13: If the convergence holds, end loop and output the optimal solution p * = p t 1 .

Algorithm 3
The joint optimization algorithm for solving P1.
1: Initialization: Initialize t = 0, and p 0 . Repeat 2: Obtain the BS-VS association solution for x t by Algorithm 1 with given p t−1 . 3: Obtain the power control solution for p t by Algorithm 2 with given x t and p t−1 .
Until convergence.
procedure is presented in Algorithm 3. In order to verify the convergence of Algorithm 3, , where x t+1 is the globally optimal solution for P2.1 with fixed p t . Since P2.1 is equivalent to P2, the inequality φ 2 (x t , p t ) ≤ φ 2 (x t+1 , p t ) holds. From step 6 to step 9 in Algorithm 2, the solutionp t+1 to P3.2 with a feasible point (p t ,ρ t ,τ t ) is obtained. Constraints (30) and (36) are relaxed by updating (p t ,ρ t ,τ t ) and (θ t ,θ t ). Whenp t+1 =p t , we have φ 3.2 (x t+1 ,p t+1 ,p t ) ≤ φ 3.2 (x t+1 ,p t+1 ,p t+1 ) = φ 3.1 (x t+1 , p t+1 ), where p t+1 is the globally optimal solution for P3.1 with fixed x t+1 . Since P3.1 is equivalent to P3, the In Algorithm 3, the obtained solutions in step 2 and step 3 have the following relationship: It is obviously obtained that φ 1 is monotonically non-decreasing with respect to the iteration number and is upper bounded by a finite value. Therefore, Algorithm 3 is guaranteed to converge.

Parameters setting
In this subsection, we present numerical results to validate our analysis and demonstrate the effectiveness of the proposed algorithms. We consider a railway  Figure 2 gives the corresponding coordinate positions of all BSs and VSs. Figure 3 shows the influence of the different flight speeds of UAV on the BS-VS association performance with the given power in Algorithm 1. It can be seen from Fig. 3 that for the given flight speed of UAV, when the number of iterations increases from 1 to 3, the system performance gradually increases. When the number of iterations is greater than 3, the system performance tends to be stable, which shows that Algorithm 1 has the good convergence and stability. It can also be seen from Fig. 3 that the flight speeds of UAV has an important impact on the system performance. The lower the flight speed of UAV, the higher the system performance. This is because the high flight speed of UAV increases the chance of eavesdropping VS. Figure 4 gives the convergence performance of Algorithm 2 when the maximum transmission power of BS is different. As can be seen from Fig. 4, at first, the system performance gradually increases with the increase of iterations. When the number of iterations is between 3 and 10, the system performance jumps. For example, when Pmax = 35dbm, the system performance after the fourth iteration is higher than that of the following iterations. When the number of iterations is more than 10, the performance of Algorithm 2 tends to be stable. It can also be seen from Fig. 4 that the higher the power, the better the system performance. Figure 5 shows the convergence performance of Algorithm 3 for the different maximum transmission power of BS. It can be seen from Fig. 5 that, with the increase of the number of the iterations, the system performance is monotonically increasing and then tends to be stable. When the maximum transmission power of BS is 35dbm, the system performance tends to be stable after two iterations. When the maximum transmission power of BS is 38dbm, 40dbm, and 42dbm, the system performance tends to be stable after four iterations. Similar to Fig. 4, the higher the maximum transmission power of BS, the better the system performance. When the maximum transmission power of BS is 42dbm, the maximum system performance of Algorithm 2 in Fig. 4 is about 79.8 bit/s/Hz, while the maximum system performance of Algorithm 3 with the joint optimization in Fig. 5 is 114 bit/s/Hz. This is because Algorithm 3 in Fig. 5 jointly considers the power optimization and the BS-VS association, while Algorithm 2 in Fig. 4 only considers the power optimization for the given BS-VS association. Figure 6 shows the connection state of VSs and the system performance with different c and d. As can be seen from Fig. 6, when c = 6 and d = 4, there is only one VS-BS association in the first seven time slots and it remains unchanged. In the 8th time slot, there are three VS-BS association. In the 9th and 10th time slots, there is only one VS-BS association. In the 11th time slot, there are three VS-BS association. It can be seen that when c = 6, d = 4, the switch frequency is lower, while when c = 4, d = 2, the switch frequency is relatively high. It can be seen from the relationship between the system performance and the number of iterations in Fig. 6 that, when the switch frequency is low, the system performance will be reduced.

Conclusion
In this paper, the secure transmission for heterogeneous HSR communication system was studied, which contained a MBS, B RBSs, and K VSs on the top of the train, and a mobile eavesdropping UAV. Our goal was to maximize the sum of the minimum security rate of each time slot, and the constraint conditions contained the QoS, switch requirements, and the total power constraint. The original optimization problem was intractable to solve directly as it was mixed-integer and non-convex. The BCD method was applied, and the original problem could be decoupled into two sub-problems. The one was the joint BS-VS association problem, and the other was the power control problem. The first subproblem of optimizing BS-VS association could be solved by the SCA algorithm, and the second sub-problem of optimizing transmit power could be solved by the Lagrangian dual method. In the future, we will consider the impact of beamforming and AN on the system security performance.