Optimal coordinated control of hybrid AC/VSC-HVDC system integrated with DFIG via cooperative beetle antennae search algorithm

Nowadays, with the significant integration of various renewable energy, hybrid alternating current/ voltage source converter based high voltage direct current (AC/VSC-HVDC) system integrated with doubly-fed induction generator (DFIG) has achieved rapidly development in smart grid. A proper control system design for hybrid AC/VSC-HVDC system plays a very crucial role for a reliable and effective power transmission. Hence, this paper designs a novel cooperative beetle antenna search (CBAS) algorithm for optimal coordinated control of hybrid AC/VSC-HVDC system integrated with DFIG. Compared with original beetle antennae search (BAS) algorithm, CBAS algorithm can significantly improve searching efficiency via an efficient cooperation with a group of multiple beetles instead of a single beetle. Particularly, CBAS algorithm can effectively escape from local optimums thanks to its dynamic balance mechanism, which can maintain appropriate trade-off between global exploration and local exploitation. Moreover, three case studies are undertaken to validate the effectiveness and superiorities and effectiveness of CBAS algorithm compared against that of other traditional meta-heuristic algorithms. Especially, the average results of fitness function acquired by CBAS algorithm is merely 46.05%, 41.18%, and 47.82% of that of PSO, GA, and BAS algorithm, respectively.


Introduction
With the rapid development and wide application of renewable energy [1], new materials [2], and advanced power electronics [3], requirements for higher power supply quality, reliability, and operation efficiency are ever-increasing in the past decade. High voltage direct current (HVDC) transmission technology owns elegant merits of flexible operation [4], fast power regulation [5], high reliability [6], and improved system transient stability [7]. It has achieved wide application in long-distance power transmission [8], asynchronous interconnection [9], and submarine power transmission [10]. Particularly, voltage source converter based high voltage direct current (VSC-HVDC) system shows higher superiorities in many aspects compared with that of current source converter based HVDC (CSC-HVDC) system [11,12], such as higher flexibility. VSC-HVDC transmission technology is an ideal transmission method for large-scale grid-connected wind farms. Such grid-connected methods can better improve the transmission capacity of wind power and system operation stability [13]. In recent years, control design of hybrid AC/VSC-HVDC system integrated with DFIG, this paper designs a novel bio-inspired meta-heuristic algorithm called cooperative beetle antennae search (CBAS) algorithm, to achieve optimal gains tuning of PSS gains of synchronous generator, PI gains of VSC-HVDC system, and PI gains of DFIG under various operation scenarios. Compared to the original beetle antennae search (BAS) [29,30] algorithm which mimicking searching mechanism of long-horn beetles, a cooperative group of multiple beetles instead of a single beetle is introduced by CBAS algorithm to realize a dynamic balance between local exploitation and global exploration, upon which an optimal control gains tuning are simultaneously achieved for hybrid AC/VSC-HVDC system integrated with DFIG. The rest of this paper is organized as: Section 2 presents the modelling of hybrid AC/ VSC-HVDC system integrated with DFIG; Then, basic principle of CBAS algorithm is introduced in Section 3; Section 4 elaborates detailed design of CBAS algorithm based optimal coordinated control for hybrid AC/VSC-HVDC system integrated with DFIG; Section 5 undertakes three case studies to validate its effectiveness. At last, conclusions are presented in Section 6.

Hybrid AC/VSC-HVDC system integrated with DFIG modelling
Hybrid AC/VSC-HVDC system integrated with DFIG is illustrated in Fig 1 based on typical 4 machines 11 bus (4M11B) systems, which includes three synchronous generators (#1, #2, #4) and one DFIG (#3). Meanwhile, VSC-HVDC is connected between bus 7 (rectifier side) and bus 9 (inverter side) and operates in parallel with two AC lines to transmit power. Here, R 1 and R 2 represent equivalent resistances of coupling transformer and phase reactor, respectively; L 1 and L 2 represent equivalent inductances of coupling transformer and phase reactor, respectively; U si ffθ si (i = 1,� � �,4) and U ci ff(θ si +δ i )(i = 1,2) represent generator voltages and voltages of point of common coupling (PCC); P si and Q si denote active and reactive power of AC system; P ci and Q ci are active and reactive power of VSC-HVDC system; i si means the current flowing from AC system to VSC; R dc and L dc denote resistance and inductance of DC line, respectively.

Synchronous generator model
The nth machine in a multimachine power system with n machines represents the reference machine, which is expressed by [31] where the meaning of all variables/parameters contained in Eq (1) and Eq (2) can be referred to literature [31].

DFIG model
The mechanical power that the wind turbine can capture is given as follows [32] where ρ means the air density; R denotes wind turbine's radius; and v wind stands for the wind speed; C p (λ,β) represents the power coefficient, in which λ can be expressed as where ω m represents wind turbine's rotational speed. Considering the characteristics of wind turbines, a generic equation used to describe C p (λ,β), as follows where coefficients c 1 to c 6 are selected as: c 1 = 0.5176, c 2 = 116, c 3 = 0.4, c 4 = 5, c 5 = 21, and c 5 = 0.0068. The DFIG dynamics can be expressed by where T r represents the time constant of rotor; R 1 and R 2 mean the equivalent resistance on the stator side and rotor side respectively; L 0 s ; L ss , L rr and L m denote the equivalent inductance on the stator side, stator inductance, rotor inductance and mutual inductance respectively. Other parameters can be referred to literature [32].
The electromagnetic torque T e generated by the DFIG is described as The shaft system can be simply described as a single lumped-mass system with a lumped inertia constant H m , as follows where H t and H g denote two inertia constants of wind turbine and DFIG, respectively. The electromechanical dynamics is then computed by where ω m denotes lumped-mass system's rotational speed, which meets ω m = ω r ; D = 0.05 p.u. means lumped system's damping; and T m stands for the mechanical torque, i.e., T m = P m /ω m .

VSC-HVDC system model
As demonstrated in Fig 1, VSC-HVDC system model can be described as follows [33,34]: where i d1 and i d2 represent DC currents of VSC on both sides; U d1 and U d2 denote DC voltages of VSC on both sides; C d1 and C d2 mean DC capacitances of VSC on both sides.

BAS algorithm
BAS algorithm is a novel biology-based meta-heuristic algorithm, which is mainly based on special food detecting and searching behaviour of long-horn beetles characterized by extremely long antennae in nature [29]. Such long antennae are a very common symbol in most beetle species, and it is composed of various types of olfactory receptor cells. The main function of large antennae is to expand detection range, within this range, beetles can better capture the odour of prey and detect sex pheromones that may be suitable for mating [29]. Basically, beetle uses two antennae to randomly detect nearby areas, and the detection direction depends on which side has a higher odour. In BAS algorithm, at the kth time, the location of each beetle is considered as a vector x k (k = 1,2,. . .). Meanwhile, the fitness function is represented by f(x), which means odour concentration locates at x, while its maximum value directly relies on where odour begins to diffuse, called source point. Inspired by stochastic searching mechanism of beetles, two stages are mainly contained, namely, searching and detecting. a. Searching: Stochastic searching direction of beetles is defined by where rnd(.) means a stochastic function and D dim stands for location dimensions, respectively. Besides, for more accurately replicating actual searching behaviour of beetle's antennae, right-hand and left-hand searching behaviours are adopted, as follows: where x r and x l denote location in the right-hand and left-hand searching area, respectively; and d is sensing length of antennae, which initial value should be large enough to avoid premature convergence at the initial phase, and decreases over time.
b. Detecting: An iterative model is presented which takes both odour detection and searching behaviour into consideration, as follows: where δ denotes step size that indicates convergence rate, while initialization of δ and searching area should be equal; and sign(.) means sign function, respectively. Particularly, the updating rule of parameters which directly influences searching behaviour, e.g., antennae length d and step size δ, can be expressed as follows:

CBAS algorithm
3.2.1 Cooperative group. BAS algorithm only adopts a single beetle to seek a potentially better solution, which is easy to fall into local optimums. In order to overcome such drawbacks, CBAS algorithm employs a cooperative group with multiple beetles to find potential better solutions, as demonstrated in Fig 2. Hence, CBAS algorithm not only contains a detecting stage (i.e., global search) like BAS algorithm, but also a local searching behavior to

PLOS ONE
Optimal Coordinated Control of Hybrid AC/VSC-HVDC System Integrated with DFIG via CBAS algorithm approximate the current best solution, which can be described by where subscript i means the ith beetle; o k 1 and o k 2 represent dynamic weights of global exploration and local exploitation, respectively; C stands for a constant coefficient; r 1 is a stochastic value from [0, 1]; and x kÀ 1 best denotes current best solution until the (k-1)th iteration.

Dynamic balance between local exploitation and global exploration.
Like other meta-heuristic algorithms, it is significant to achieve a stable and desirable optimization of a dynamic balance between local exploitation and global exploration. For example, if CBAS algorithm attaches more attention to local exploitation, it will easily result in a low-quality local optimum; otherwise, it will result in a low optimization efficiency to seek a better solution. In order to realize a dynamic balance between local exploitation and global exploration, weights in Eq (18) are designed to be time-varying as iteration increases, yields where k max means maximum iteration number; ω max and ω min denote the maximum and minimum weights, respectively. Note that global exploration weight o k 1 will gradually decrease as iteration number grows based on Eq (19), while local exploitation weight o k 2 will gradually increase since their sum is equal to be 1 in Eq (20). According to such improvement, global exploration ability of CBAS algorithm can be significantly improved in initial optimization stage, which can effectively boost searching efficiency and probability of high-quality solutions. As iteration number increases, CBAS algorithm tends to concentrate on local exploitation, which can further improve solution quality.
Furthermore, parameters of BAS algorithm, including antennae length d and step size δ, are prone to considerably decrease with an exponential type in Eqs (16) and (17), upon which a broad global exploration cannot be achieved smoothly. To remedy such problem, an exponential reduction is displaced by a linear reduction in CBAS algorithm, as follows: where d max and δ max denote the maximum antennae length and maximum step size, respectively.

Optimization process
In general, the optimization process of CBAS algorithm is given in Fig 3.  :

Control design of synchronous generator
where E f denotes excitor voltage; V ref represents voltage reference; V t means synchronous generator terminal voltage; U pss represents the voltage of PSS; K e = 200 denotes excitor gain; T R = 0.01 stands for the time constant of excitor. The transfer function of PSS can be described as follows [34]: where K PSS denotes PSS gains; T w = 10 represents the time constant of wash-out process; T 1 and T 2 denote two first-order time constants of lead-lag phase; T 3 and T 4 denote two secondorder time constants of lead-lag phase; Δω stands for rotor speed difference.

PLOS ONE
Optimal Coordinated Control of Hybrid AC/VSC-HVDC System Integrated with DFIG via CBAS algorithm

Control design of DFIG
The power of DFIG can be described as follows Control design of rotor side converter (RSC) of DFIG is the major task, in which outer control loops are utilized for regulation of DFIG active and reactive power independently. In particular, two currents related to the compensation terms v qr2 and v dr2 are regulated to acquire the final controller outputs v qr and v dr in inner control loops. Based on this operation framework, four interactive PI loops are used to obtain the optimal control performance, as shown in Fig 5, which corresponding symbols can be expressed as [32] s where s denotes the DFIG slip and σ represents the leakage coefficient.

Control design of VSC-HVDC system
Here, rectifier side adopts outer-loop control of constant AC voltage and constant DC voltage regulation, while inverter side adopts outer-loop control of constant active power and constant reactive power regulation [35], which is illustrated in Fig 6.

PLOS ONE
Optimal Coordinated Control of Hybrid AC/VSC-HVDC System Integrated with DFIG via CBAS algorithm

Optimal coordinated controller gains optimization
Here, CBAS algorithm is utilized to optimize PSS gains of synchronous generator K PSS , T 1 , T 2 , T 3 and T 4 , as well as PI controller gains of VSC-HVDC system and RSC of DFIG, namely, K P1 , In order to realize an optimal control performance, the above gains are adjusted under the following three typical operating conditions, e.g., (a) three-phase short-circuit fault, (b) load disconnection, and (c) DFIG loss. Moreover, the objective function of the designed system is expressed as

PLOS ONE
Optimal Coordinated Control of Hybrid AC/VSC-HVDC System Integrated with DFIG via CBAS algorithm subject to where W k represents corresponding weight coefficient under each operation condition, which are set as 0.3, 0.5 and 0.2, respectively; δ ij and d � ij denote rotor angle difference and its reference of generator #i and #j; ω ij and o � ij represent rotor speed difference and its reference of generator #i and #j; Simulation time T = 120 s.

Overall control flow for hybrid AC/VSC-HVDC system integrated with DFIG
To this end, overall control flow of CBAS algorithm for hybrid AC/VSC-HVDC system integrated with DFIG is shown in Fig 7.

Case studies
In this section, control performance of CBAS algorithm in hybrid AC/VSC-HVDC system integrated with DFIG is compared to that of manual tuning, particle swarm optimization (PSO) algorithm [36], GA [37], and BAS algorithm [29] under the above cases. Note that all approaches are executed in 10 independent runs to acquire statistical results and convergence graphs [38,39], while the best solutions are used as the optimal gains. In addition, AC system frequency is set as 50 Hz and parameters of hybrid AC/VSC-HVDC system integrated with DFIG are tabulated in Table 1 while algorithm parameters are given in Table 2. Besides, ode23 was selected as the solver, and the sampling rate was set to 0.001 s.
Moreover, convergence of four algorithms is shown in Fig 8, which indicates that CBAS algorithm owns the fastest convergence under all three evaluation indices. Fig 9 illustrates boxplot of different methods, i.e., distribution of simulation results, which shows that CBAS algorithm can distribute within the smallest range with minimal lower and upper bounds among all algorithms. It verifies that CBAS algorithm owns the highest convergence stability and searching ability. As a result, CBAS algorithm can effectively avoid local optimum trapping. Furthermore, convergence rate of CBAS algorithm can be considerably improved by its multiple beetles based cooperative searching mechanism. At last, the optimized control gains are tabulated in Table 3.

PLOS ONE
Optimal Coordinated Control of Hybrid AC/VSC-HVDC System Integrated with DFIG via CBAS algorithm

Three-phase short-circuit fault
To validate control performance of CBAS algorithm under varying operation conditions, a three-phase short-circuit fault occurs on the middle of transmission line 7 when t = 1s, and removed at 1.1s, as illustrated in Fig 1. Besides, as shown in Fig 10, simulation results of corresponding system responses can explicitly validate that CBAS algorithm can suppress the power oscillation most effectively and efficiently. In contrast, manual tuning reveals the largest overshoot of active power and the slowest convergence rate compared to that of other algorithms.

Load disconnection
This test aims to investigate the effectiveness and reliability of various controllers under load disconnection. Hence, load 1 is disconnected when t = 1s (highlighted in Fig 1)

BAS
Step size Sensing diameter Population k max 0.9 0.9 50 20

CBAS
Step size Sensing diameter Population k max 0.9 0.9 50 20 https://doi.org/10.1371/journal.pone.0242316.t002 depicts corresponding system responses. Here, BAS algorithm can hardly maintain an effective control performance because single beetle searching strategy easily falls into a local optimum, along with slow convergence speed. In contrast, CBAS algorithm can compensate active/reactive power imbalance with the highest tracking speed and the lowest tracking error compared to that of other algorithms. Besides, tracking results of rotor angle difference δ also demonstrate that CBAS algorithm can stably and rapidly restore the disturbed system compared against other approaches based on its cooperative searching mechanism to maintain proper balance between local exploitation and global exploration.

DFIG loss
In this case, severe power oscillation is caused by DFIG loss when t = 1s, which is usually caused by the internal failure of the generator (such as damage to the mechanical parts of the generator leading to the start of the protection device), while corresponding system responses are presented in Fig 12, which illustrates that all algorithms are subjected to such active power oscillations. However, CBAS algorithm can effectively suppress such malignant oscillations as it can adjust rotor angle difference with the slightest overshoot and highest convergence speed.  Table 4 reveals that CBAS algorithm owns the fastest convergence rate while Table 5 shows that CBAS algorithm owns the minimum fitness function in 10 runs. Lastly, the integral of absolute error (IAE) [40][41][42] of each algorithm in three scenarios are given by Table 6, in which IAE x ¼ R T 0 jx À x � jdt and x � denotes the reference of variable x, respectively [43,44]. In particular, IAE δ of CBAS algorithm is merely 32.28%, 55.41%, 48.81%, and 56.94% of that of manual tuning [45][46][47], PSO, GA, and BAS algorithm, respectively (bold colour indicates the best results in Tables 4-6).

Conclusions
This paper designs a novel CBAS algorithm for an optimal coordinated control of Hybrid AC/ VSC-HVDC system integrated with DFIG, which owns the following three contributions/novelties: 1. Compared to original BAS algorithm, CBAS algorithm can remarkably improve optimization efficiency via a cooperative group of multiple beetles instead of a single beetle. Besides, it can also acquire a high-quality optimum through a dynamic and proper balance between local exploitation and global exploration; In addition, the convergence time of CBAS algorithm is only 55.27% of BAS algorithm; 3. Case studies validate that CBAS algorithm can achieve active power demand variation tracking with the highest tracking speed and lowest tracking error. Moreover, it can also effectively and efficiently restore the disturbed system. Statistical results of fitness function further verify that CBAS algorithm can find the best quality optimum with the highest convergence stability and reliability compared to that of manual tuning and other meta-heuristic algorithms. Particularly, the average results of fitness function acquired by CBAS  Besides, three future studies are given as follows: 1. Hardware experiment should be undertaken to validate the feasibility; 2. Advanced methods are encouraged to be constructed to overcome system uncertainties; 3. More load models, e.g., electric vehicle load need to be considered.