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.

1. 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, hybrid alternating current/ voltage source converter based high voltage direct current (AC/VSC-HVDC) system is envisaged as an important part of renewable energy transmission, e.g., wind energy integration such as doubly fed induction generator (DFIG), which can enhance the system stability and reliability. Needless to say, design of a proper control system is critical to an effective operation of hybrid AC/VSC-HVDC system.

In general, proportional-integral (PI) or proportional-integral-differential (PID) controllers are conventionally adopted by HVDC systems thanks to its high reliability and easy implementation. However, their gains are determined through local linearization and generally tuned through trial-and-error. When power system is subject to large disturbances and deviates far from the operation point, an effective control can hardly be achieved by PI/PID controllers [14,15]. With the development of nonlinear control theory, nonlinear control was first successfully applied in converter control of affine nonlinear model of HVDC system in literatures [16,17]. However, feedback linearization method requires accurate system parameters and measurable states, which lacks of robustness against various changes in parameters and models [18]. Thus, an inverse system method was proposed to establish a third-order non-affine and nonlinear DC model based on feedback linearization, upon which a combined commutation controller has been designed to satisfy the pole configuration requirements while the stability of closed-loop system is proved [19]. Besides, an inverse system method of multivariable nonlinear control was adopted to design a feedback linearization and quadratic optimal combination controller, which can improve the stability of converter station and prevent commutation failure [20]. However, the aforementioned controllers still lack of high flexibility and reliability under different operation conditions and disturbances.

On the other hand, various artificial intelligence methods, such as fuzzy control, robust control, neural network (NN), and meta-heuristic algorithms have been widely used in HVDC transmission control systems. A sequential decentralized control technology was presented to design a fuzzy logic controller [21], which owns benefits of simple structure and can effectively improve damping characteristics of HVDC system. Besides, literature [22] designed a high-gain and low-frequency controller for inverter side connected to weak AC system based on H∞ control theory, which shows high robustness to uncertain AC system parameters and operation conditions. Besides, PI controller and NN controller based linearized HVDC systems are reported [23], which demonstrates that NN controller is superior to PI controller in terms of damping and robustness. Furthermore, a new interactive teaching-learning optimizer (ITLO) for VSC-HVDC systems with integration of offshore wind farm was designed, which can dramatically enhance controlling performance at different operating scenarios via optimal tuning of parameters of eight interactive PI loops [24].

Generally speaking, in a hybrid AC/VSC-HVDC system, VSC-HVDC system can effectively adjust power factor and enhance system stability, while the control of AC system such as excitor control, power system stabilizer (PSS), and reactive voltage control can also affect VSC-HVDC system. Hence, coordinated control between them becomes a very important task, e.g., a dual-mode control strategy for AC/VSC-HVDC hybrid transmission system control with wind farm integration was proposed [25], in which an improved direct-current vector control approach is employed for grid-side VSC control. Moreover, literature [26] proposed a novel coordinated control technique to achieve emulation of synchronous generators inertia. Meanwhile, combination of direct feedback linearization and linear optimal theory was applied to design a nonlinear optimal control rule for coordination of HVDC and generator excitor [27]. Furthermore, based on small interference linearization model, decentralized control and genetic algorithm (GA) are combined to achieve an optimal coordination of multiple controller parameters in hybrid HVDC systems [28].

However, structure of the above control schemes for hybrid AC/VSC-HVDC system are usually complex, which hinders its applications in practice. In order to realize an optimal 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.

2. 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, R1 and R2 represent equivalent resistances of coupling transformer and phase reactor, respectively; L1 and L2 represent equivalent inductances of coupling transformer and phase reactor, respectively; Us∠θs(i = 1,⋯,4) and Uc∠(θs+δ)(i = 1,2) represent generator voltages and voltages of point of common coupling (PCC); Ps and Qs denote active and reactive power of AC system; Pc and Qc are active and reactive power of VSC-HVDC system; is means the current flowing from AC system to VSC; Rdc and Ldc denote resistance and inductance of DC line, respectively.

Fig 1

Hybrid AC/VSC-HVDC system integrated with DFIG structure.

2.1 Synchronous generator model

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

2.2 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 vwind stands for the wind speed; Cp(λ,β) represents the power coefficient, in which λ can be expressed as where ωm represents wind turbine’s rotational speed. Considering the characteristics of wind turbines, a generic equation used to describe Cp(λ,β), as follows with where coefficients c1 to c6 are selected as: c1 = 0.5176, c2 = 116, c3 = 0.4, c4 = 5, c5 = 21, and c5 = 0.0068.

The DFIG dynamics can be expressed by where Tr represents the time constant of rotor; R1 and R2 mean the equivalent resistance on the stator side and rotor side respectively; Lss, Lrr and Lm 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 Te 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 Hm, as follows where Ht and Hg denote two inertia constants of wind turbine and DFIG, respectively.

The electromechanical dynamics is then computed by where ωm denotes lumped-mass system’s rotational speed, which meets ωm = ωr; D = 0.05 p.u. means lumped system’s damping; and Tm stands for the mechanical torque, i.e., Tm = Pm/ωm.

2.3 VSC-HVDC system model

As demonstrated in Fig 1, VSC-HVDC system model can be described as follows [33,34]: where id1 and id2 represent DC currents of VSC on both sides; Ud1 and Ud2 denote DC voltages of VSC on both sides; Cd1 and Cd2 mean DC capacitances of VSC on both sides.

3. CBAS algorithm

3.1 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 = 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.

Searching: Stochastic searching direction of beetles is defined by

where rnd(.) means a stochastic function and Ddim 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 xr and xl 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.

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.

Detecting: An iterative model is presented which takes both odour detection and searching behaviour into consideration, as follows:

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:

3.2 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 approximate the current best solution, which can be described by where subscript i means the ith beetle; and represent dynamic weights of global exploration and local exploitation, respectively; C stands for a constant coefficient; r1 is a stochastic value from [0, 1]; and denotes current best solution until the (k-1)th iteration.

Fig 2

Optimization principle of CBAS algorithm.

3.2.2 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 kmax means maximum iteration number; ωmax and ωmin denote the maximum and minimum weights, respectively.

Note that global exploration weight will gradually decrease as iteration number grows based on Eq (19), while local exploitation weight 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 dmax and δmax denote the maximum antennae length and maximum step size, respectively.

3.3 Optimization process

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

Fig 3

Optimization process of CBAS algorithm.

4. CBAS algorithm design for hybrid AC/VSC-HVDC system integrated with DFIG

4.1 Control design of synchronous generator

Fig 4 shows the conventional lead-lag (LL)+ automatic voltage regulator (AVR) controller of synchronous generator structure, the dynamic model of AVR and excitor can be expressed as follows: where Ef denotes excitor voltage; Vref represents voltage reference; V means synchronous generator terminal voltage; Upss represents the voltage of PSS; Ke = 200 denotes excitor gain; TR = 0.01 stands for the time constant of excitor.

Fig 4

The conventional LL+AVR controller of synchronous generator structure.

The transfer function of PSS can be described as follows [34]: where KPSS denotes PSS gains; Tw = 10 represents the time constant of wash-out process; T1 and T2 denote two first-order time constants of lead-lag phase; T3 and T4 denote two second-order time constants of lead-lag phase; Δω stands for rotor speed difference.

4.2 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 vqr2 and vdr2 are regulated to acquire the final controller outputs vqr and vdr 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] where s denotes the DFIG slip and σ represents the leakage coefficient.

Fig 5

The overall control structure of DFIG.

4.3 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.

Fig 6

Control structure of VSC-HVDC system.

4.4 Optimal coordinated controller gains optimization

Here, CBAS algorithm is utilized to optimize PSS gains of synchronous generator KPSS, T1, T2, T3 and T4, as well as PI controller gains of VSC-HVDC system and RSC of DFIG, namely, KP1, KI1, KP2, KI2, KP3, KI3, KP4, KI4, KPd_1, KId_1, KPd_2, KId_2, KP_RSC_1, KI_RSC_1, KP_RSC_2, KI_RSC_2, KP_RSC_3, KI_RSC_3, KP_RSC_4 and KI_RSC_4. 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 where W represents corresponding weight coefficient under each operation condition, which are set as 0.3, 0.5 and 0.2, respectively; δ and denote rotor angle difference and its reference of generator #i and #j; ω and represent rotor speed difference and its reference of generator #i and #j; Simulation time T = 120 s.

4.5 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.

Fig 7

Overall control flow for hybrid AC/VSC-HVDC system integrated with DFIG.

5. 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.

Table 1

System parameters.

Synchronous generator No. (in p.u.)	xdi	xdi′	Td0i′	Hi	Di
1,2,4	1.8	0.3	8.0	13	0
Wind turbine	ρ	R	vwind	λopt	Hm
	1.225kg/m3	58.59m2	12m/s	6.325	4.4s
DFIG (in p.u.)	Rs	Lm	Rr	Lss	Lrr
3	0.005	4.0	1.1Rs	1.01Lm	1.005Lss
	Ls′	Tr	R1	R2
	L_ss−Lm2/L_rr	Lrr/Rr	Rs+R2	(Lm/Lrr)2Rr
Transformer (in p.u.)	Resistance			Impedance
	0.0025			0.0167
Line No. (in p.u.)	Resistance			Impedance
1, 2	0.0025			0.025
3, 4	0.001			0.01
5–10	0.005			0.2245
Load No. (in p.u.)	Active power			Reactive power
1	9.67			-1
2	18.67			-2.5
VSC (in p.u.)	Resistance			Impedance
	0.005			0.2245
AC system base voltage	uACbase			132kV
DC cable base voltage	uDCbase			150kV
System base power	Sbase			100MVA

Table 2

Parameters of four algorithms.

Algorithm	Parameters
PSO	Accelerated constant	Speed range	Population	kmax
	1.49445	[-0.5, 0.5]	50	20
GA	Crossover rate	Mutation rate	Population	kmax
	0.8	0.1	50	20
BAS	Step size	Sensing diameter	Population	kmax
	0.9	0.9	50	20
CBAS	Step size	Sensing diameter	Population	kmax
	0.9	0.9	50	20

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.

Fig 8

Comparison of the best convergence performance of four algorithms obtained in 10 runs.

Fig 9

Statistical results of fitness function acquired in 10 runs (in p.u.).

Table 3

Optimized controller gains.

AlgorithmController		Manual tuning	PSO	GA	BAS	CBAS
PSS of synchronous generator	KPSS	8.969	10.157	5.621	5.954	8.672
	T1	0.01	0.216	1.755	1.491	0.226
	T2	0.237	0.126	0.449	0.302	0.148
	T3	1.568	2.162	1.265	2.556	0.234
	T4	0.704	0.157	2.238	0.759	0.114
RSC of DFIG	KP_RSC_1	0.234	0.510	0.866	0.643	0.422
	KI_RSC_1	8.996	6.254	3.987	5.226	4.598
	KP_RSC_2	0.436	0.911	0.538	0.294	0.198
	KI_RSC_2	7.357	5.694	8.327	5.102	3.269
	KP_RSC_3	0.369	0.128	0.567	0.912	0.248
	KI_RSC_3	7.356	6.982	4.328	2.397	5.264
	KP_RSC_4	0.346	0.289	0.861	0.674	0.423
	KI_RSC_4	3.259	8.256	4.336	6.196	4.233
Rectifier of VSC-HVDC system	KI1	7.355	5.068	9.561	8.636	9.048
	KP1	325.153	382.077	306.369	379.032	361.342
	KI3	-6.255	-5.066	-7.121	-7.484	-7.221
	KP3	-300.54	-382.076	-390.453	-357.35	-317.498
	KPd_1	2.548	1.262	2.648	1.949	1.828
	KId_1	2.687	1.841	1.194	4.264	3.383
Inverter of VSC-HVDC system	KI2	57.156	100.154	140.278	85.161	88.636
	KP2	3.169	1.165	3.437	4.392	2.041
	KI4	2.144	1.681	4.851	0.513	0.873
	KP4	300.195	230.263	250.215	180.248	150.441
	KPd_2	0.087	0.047	0.095	0.054	0.093
	KId_2	0.371	0.285	0.414	0.295	0.474

5.1 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.

Fig 10

System responses acquired under the three-phase short-circuit fault.

5.2 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), while Fig 11 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.

Fig 11

System responses obtained under load disconnection.

5.3 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.

Fig 12

System responses obtained under DFIG loss.

5.4 Comparative analysis

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) [40-42] of each algorithm in three scenarios are given by Table 6, in which and x* denotes the reference of variable x, respectively [43,44]. In particular, IAE of CBAS algorithm is merely 32.28%, 55.41%, 48.81%, and 56.94% of that of manual tuning [45-47], PSO, GA, and BAS algorithm, respectively (bold colour indicates the best results in Tables 4–6).

Table 4

Statistical results of convergence performance acquired by four algorithms in 10 runs.

Algorithm	Execution time (hours)			Convergence time (hours)			Iteration number
	Max.	Min.	Mean	Max.	Min.	Mean	Max.	Min.	Mean
PSO	3.541	1.165	2.549	3.186	0.873	2.109	18	15	16.547
GA	3.236	1.512	2.274	2.075	1.209	1.835	17	16	16.135
BAS	3.164	1.015	2.141	2.848	0.711	1.625	18	14	15.183
CBAS	1.154	0.714	0.894	0.808	0.393	0.556	14	11	12.436

Table 5

Statistical results of fitness function acquired by four algorithms in 10 runs (in p.u.).

Algorithm	Maximum	Minimum	Mean
PSO	7263.154	4432.132	5133.164
GA	6941.324	4231.512	5741.236
BAS	6871.642	3814.264	4943.345
CBAS	2654.365	1861.165	2364.146

Table 6

IAE indices of five algorithms acquired in three cases (in p.u.).

Cases	IAE indices	Manual tuning	PSO	GA	BAS	CBAS
Three-phase short-circuit fault	IAEδ	1.27	0.74	0.84	0.72	0.41
Load disconnection	IAEδ	4.54×10−2	2.84×10−2	2.62×10−2	2.14×10−2	5.82×10−4
DFIG loss	IAEδ	3.28	1.49	1.54	1.35	0.97

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:

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;

CBAS algorithm is utilized to optimally tune the interacted PSS gains of synchronous generator, PI gains of VSC-HVDC system, and PI gains of DFIG, such that the overall power oscillations can be minimized under various operation conditions;

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 algorithm is merely 46.05%, 41.18%, and 47.82% of that of PSO, GA, and BAS algorithm, respectively.

Besides, three future studies are given as follows:

Hardware experiment should be undertaken to validate the feasibility;

Advanced methods are encouraged to be constructed to overcome system uncertainties;

More load models, e.g., electric vehicle load need to be considered.

Abbreviations

Nomenclature.

δi	generator rotor angle	xr, xl	location in right-hand and left-hand searching area
ωi	generator rotor speed	ω1k, ω2k	dynamic weights of global exploration and local exploitation
R1	equivalent resistance of coupling transformer	xbestk−1	current best solution until the (k-1)th iteration
R2	equivalent resistance of phase reactor	dmax, δmax	maximum antennae length and maximum step size
Rdc, Ldc	resistance and inductance of DC line	Wi	weight coefficient under a certain operation condition
Idi, Iqi	d-axis and q-axis generator current	δij	rotor angle difference of generator i and j
isi	current flowing from AC grid side to VSC	δij*	reference value of δij
Yij	equivalent admittance between the ith and jth nodes	kmax	maximum iteration number
Gii	equivalent self-conductance of the ith machine	Abbreviations
Ef	excitor voltage	DFIG	doubly-fed induction generator
Upss	voltage of PSS	PID	proportional-integral-differential
Ke	excitor gains	CBAS	cooperative beetle antenna search
TR	time constant of excitor	ITLO	interactive teaching-learning optimizer
Tw	time constant of wash-out	PSS	power system stabilizer
id1, id2	DC currents	LL	lead-lag
Cd1, Cd2	DC capacitances	AVR	automatic voltage regulator