Adaptive fuzzy control for multiple sliding modes in multi-infeed AC/DC transmission systems

Abstract

In traditional multi-infeed AC/DC transmission systems, decentralized and coordinated controllers are usually used to achieve AC/DC transmission control without considering the state and output of the system. Therefore, it cannot reasonably regulate the state of output based on the demand of multi-target control, which leads to poor control effect and weak adaptability. Therefore, a multi- sliding mode adaptive fuzzy controller is designed for the multi-infeed AC/DC transmission system. When the controller is designed, the state equation and the output equation of the multi-infeed AC/DC transmission system are considered. Based on the three different design parameters and the multi-sliding mode surface of the thickness of the saturated layer, the adaptive controller based on multi-sliding mode is designed. This controller is used to set up the dynamic characteristics of some observable measurements in the multi-infeed AC/DC transmission system. Based on the setting results, the results of the comprehensive decision of the system are obtained by the adaptive fuzzy controller. According to the results of a comprehensive decision, the disturbance degree of feedback point is judged. Through the fuzzy algorithm based onthe second component function, the weighting matrices of the output feedback gain matrix are modified, so that the optimal control feedback gain is variable gain, to ensure that the control effect of the system meet the multi-objective control of engineering, and realizing the multi-sliding mode adaptive fuzzy controller ofthe multi-infeed AC/DC transmission system. The experimental results show that the designed multi-sliding mode adaptive fuzzy controller has good control effect on multi-infeed AC/DC transmission system, and has strong adaptability, and it can improve the dynamic performance of the system.

Keywords

Multi-infeed AC/DC transmission system multi-sliding mode adaptive fuzzy controller state weight matrix

1 Introduction

With the wide application of DC transmission technology, there may be multiple DC transmission systems in one power grid, namely the multi-infeed AC/DC transmission system [1]. The implementation of Three Gorges and the west east power transmission project in China, more and more large-scale AC/DC interconnected systems, such as China Southern Power Grid and East China Power Grid’s multi-infeed DC system, appear in China’s power grid. Because the DC transmission distance is not limited by the stability of the synchronous operation, there are several advantages of large transmission capacity, small power loss, rapid adjustment, flexible non-synchronous communication ability, which has been widely used in large power grid interconnection, long-distance transmission capacity and cross channel transmission etc. By the end of 2008, more than 90 of DC transmission projects have been built in worldwide [2]. Because of the uneven distribution of energy resources and productivity in China, the direct current transmission plays a great role in the energy strategy of our country. According to the plan, by 2015, there will be more than 7 or more DC lines in the China Southern Power Grid. There will be more than 8 or more DC transmission lines in the East China power grid. At that time, there will be many super large scale multi-infeed AC/DC transmission system [3]. This complex large-scale AC/DC interconnected system provides strong power services while bringing new problems for power grid security and stability. It is both an opportunity and a challenge.

In the previous design of multi-infeed AC/DC transmission system, decentralized and coordinated controller is generally adopted, ignoring the parameter setting of sliding surface [4], lacking the ability to identify system characteristics, resulting in poor control effect and weak adaptability to different disturbances. Therefore, multi-sliding mode adaptive fuzzy controller of multi-infeed AC/DC system is designed. By constructing three sliding surfaces with different design parameters and saturation thickness, we can identify the observable dynamic characteristics of multi-infeed AC/DC transmission system [5], and get better system control effect and disturbance adaptability.

2 Material and methods

2.1 State equation and output equation of multi-infeed AC/DC transmission system

Equipped with N generators and L converter stations in multi-machine power systems, the differential equation and state equation of generator and network’s mathematical model are made linearization and deviation (The network equation needs to be transformed by coordinate) [6]. After finishing, simplifying and merging, the equation of state of the whole system can be finished, as shown informula (1). $\bar{X} = AX + BU$ (1)

Wherein, X is a state variable of whole system; U is the control vector; A and B are the coefficient matrices of the state vector and the control vector, respectively. According to the design method for adaptive fuzzy controller of subsystem state [7], the adaptive fuzzy excitation controller and DC controller of generator are designed, and according to the control effect or the difficulty degree of measure. In this way, the output equation of a multi-infeed AC/DC hybrid multi-machine system can be obtained as shown in formula (2). $Y = CX$ (2)

Where, Y is the partial output of the selectable control structure, and C is the coefficient matrix of the output vector.

In order to improve the quality control of multi-infeed AC/DC system, considering the state equation and output equation of the system, the design process of the multi-sliding mode adaptive module controller is described in detail in this paper.

2.2 Design of multi-sliding mode adaptive fuzzy controller

In the design of the whole multi-sliding mode adaptive fuzzy controller, three sliding mode surfaces are defined, and the formula (3) is expressed as: $s_{i} = x_{i} - x_{id}$ (3)

Where, i = 1, 2, 3, x_id represents the expected value of the state variable x_i, and x_1d = y_d.

Step 1. The formula (4) for the first sliding surface derivation (4) is expressed as: $s_{1} = s_{2} + x_{2 d} - \dot{yd}$ (4)

For the setting of formula (14), the design of x_2d is expressed by formula (5): $x_{2 d} = \dot{yd} - c_{1} \frac{s_{1}}{h_{1}}$ (5)

Where: c₁ is the design parameter, c₁ > 0; h₁ is the thickness of the saturation layer of the first sliding surface, h₁ > 0.

Step 2. The result of the second sliding mode surface derivation is described by the formula (6).

${\dot{s}}_{2} = s_{3} + x_{3 d} + f_{1} (x_{1}, x_{2}) - {\dot{x}}_{2 d}$ (6)

Where, $\begin{matrix} f_{1} (x_{1}, x_{2}) = - \frac{b}{m} x_{2} \\ - \frac{Ff (x_{2})}{m} + \frac{F (x_{1}, x_{2})}{m} \end{matrix}$ , because the function f₁ is unknown, the fuzzy logic system can be used to approximate it, that is: $f_{1} (x_{1}, x_{2}) = w_{1}^{T} φ_{1} (x_{1}, x_{2}) + ɛ_{1}$ (7)

For the setting of formula (7), the design of x_3d is expressed by formula (8): $x_{3 d} = {\dot{x}}_{2 d} - {\hat{w}}_{1}^{T} φ_{1} (x_{1}, x_{2}) - c_{2} \frac{s_{2}}{h_{2}}$ (8)

Where, c₂ represents the design parameter, c₂ > 0; h₂ is the thickness of the saturation layer of the second sliding surface, h₂ > 0; ${\hat{w}}_{1}$ represents the estimated value of the parameter vector w₁ in the fuzzy logic system, and its online regulation law is taken as a formula (9). ${\dot{\hat{w}}}_{1} = r_{1} Proj {\hat{w}}_{1} (s_{2} ϕ_{1})$ (9)

In the formula, r₁ is a positive real number, r₁ > 0.

Step 3. Set $g (x_{1}) = \frac{A_{1}}{m} (\begin{matrix} \frac{β_{e}}{v_{01} + A_{1} x_{1}} k_{q 1} \sqrt{| p_{1} |} \\ + \frac{β_{e}}{2 (v_{02} - A_{2} x_{1})} k_{q 2} \sqrt{| p_{2} |} \end{matrix}) S_{c}$ , $f_{2} (x_{1}, x_{2}) = - \frac{A_{1}}{m} (\begin{matrix} \frac{β_{e} A_{1}}{v_{01} + A_{1} x_{1}} \\ + \frac{β_{e} A_{2}}{2 (v_{02} + A_{2} x_{1})} \end{matrix}) x_{2}$ , it is known the function $\bar{g} (x_{1})$ , so that $\bar{g} (x_{1}) > g (x_{1})$ . Set $β (x_{1}) = \bar{g} (x_{1}) / g (x_{1})$ , $V_{30} = \int_{0}^{S_{3}} σ β (x_{1}) d σ$ , the algebraic transformation σ = η_{S
_n} is taken, there are $V_{30} = S_{3}^{2} \int_{0}^{1} η β (x_{1}) d η$ , the derivation of V₃₀ can be obtained formula (10):

$\begin{matrix} {\dot{V}}_{30} & = & S_{3} β (x_{1}) {\dot{S}}_{3} + \int_{0}^{S_{3}} σ \dot{β} (x_{1}) d σ \\ = & S_{3} (\begin{matrix} \bar{g} (x_{1}) u + β (x_{1}) f_{2} (x_{1}, x_{2}) \\ + S_{3} \int_{0}^{1} η \dot{β} (x_{1}) d η - β (x_{1}) {\dot{x}}_{3 d} \end{matrix}) \end{matrix}$ (10)

The multi-sliding mode adaptive controller is designed based on the multi-sliding mode surface constructed above.

The adaptive fuzzy controller based on multi-sliding mode is designed by adaptive fuzzy algorithm, and its structure is shown in Fig. 1. Through fuzzy recognition, reasoning and judgment, the numerical value of each state variable in the weighting matrix Q can be automatically adjusted, to output the most suitable weighting matrix Q, so that the feedback gain K of the optimal control quantity U^* is changed to the variable gain K, making that the control effect can meet the engineering needs of multi-target [8]. This is the “Secondary coordination”.

Fig.1

Structure of adaptive fuzzy controller based on multi sliding mode.

In the multi-infeed AC/DC transmission system, some observable arrangements are as follows: generator’s observable power angle Δδ, rotation rate Δω, electromagnetic power ΔP_e and generator’s terminal voltage ΔU_t [9]. The observable measurement of multi-infeed AC/DC transmission system includes direct current ΔI_d, adjacent AC transmission line ΔP_ac, generator speed Δω close to rectifier side, or the AC bus voltage ΔU_ac near the inverter side [10]. When these selected observables in the input multi-sliding mode adaptive fuzzy controller to make fuzzy recognition, the absolute values should be taken such as: |Δδ|, |Δω|, |ΔP_e|, |ΔU_t|, |ΔI_d| |ΔP_ac|, |ΔU_ac| [11]. Each language variable is “(B)”, “(M)”, “(S)” and the corresponding membership function μ in the generator’s power angle Δδ is as shown in Fig. 2. The membership function of other input variables is similar to that of other variables.

Fig.2

Membership function of generator power angle deviation |Δδ|.

Based on the dynamic characteristics of the observable measurements in each system, the synthetic decision results of the system are obtained by the adaptive fuzzy controller. For the generator, the selected observable measurements correspond to a fuzzy factor set U_G. For the DC system, the observable measurements of the rectifying side and the inverter side are corresponding to the fuzzy factors set U_rL and U_iL [12], namely, the formula (11): $\begin{matrix} U_{G} & = & {Δ δ, Δ ω, Δ P_{Ge}, Δ U_{t}} \\ U_{rL} & = & {Δ I_{d}, Δ P_{ac}, Δ ω} \end{matrix}$ (11) $U_{iL} = {Δ β, Δ U_{ac}}$

The influence degree of each feedback point on the local disturbance based on the fuzzy factor set is judged, such as large disturbance (B), middle disturbance (M), small disturbance (S), very small disturbance (VS) [13], the corresponding fuzzy decision set V is described by formula (12):

$\begin{matrix} V & = & {v_{1}, v_{2}, v_{3}, v_{4}} \\ = & {B, M, S, VS} \end{matrix}$ (12)

Because the weight of various factors in different types of engineering technicians is different, the judgement may also be different. It can be seen that the comprehensive judgment depends on the weight of various factors to allocate [14]. Weight assignment A can be regarded as a fuzzy subset U. Because people’s decisions are not absolutely positive or absolute negative, Comprehensive decision can be regarded as a fuzzy subset V, which is described by formula 13. $B = {b_{1}, b_{2}, b_{3}, b_{4}} \in V$ (13)

The fuzzy transformation T_f of U to V can be derived with the help of the fuzzy relation R [15], and is corresponding to a weight allocation A. Through the fuzzy relation R, a comprehensive decision result B = A ∘ R corresponding to it can be obtained.

According to the comprehensive decision result B, after judging the disturbance degree of the feedback point, the state weight matrix Q of the outputfeedback gain matrix K_d is modified through the fuzzy algorithm based on the second component function [16]. The initial value of the state weight matrix of a decentralized and coordinated controller Q₀ is described by formula (14). $Q_{0} = diag (\begin{matrix} q_{δ 1}, q_{ω 1}, q_{U_{t} 1}, q_{E_{fd} 1} \\ , \dots, q_{α_{rL}}, q_{β_{fL}}, q_{I_{d}} \end{matrix})$ (14)

In the formula, q_δ1, q_ω1, q_{U_t1}, q_{E_fd1}, ⋯, q_{α
_rL}, q_{β
_fL}, q_{I
_d} is the specific initial value in the state weight matrix.

The constraint interval K_Ui ∈ P_i = (K_Uimin, K_Uimax) of the generator’s voltage feedback magnification is given, and the fuzzy algorithm for modifying the initial value Q₀ of the state weight matrix is as follows:

Firstly, for the excitation control system of the generator, there are formulas (15–18): $q_{δ 1}^{'} = k_{δ 1} • q_{δ 1}$ (15) $q_{ω 1}^{'} = k_{ω 1} • q_{ω 1}$ (16) $q_{U_{t} 1}^{'} = k_{U_{t} 1} • q_{U_{t} 1}$ (17) $q_{E_{fd} 1}^{'} = k_{E_{fd} 1} • q_{E_{fd} 1}$ (18)

Secondly, for the multi-infeed AC/DC control system, there are formulas (19–21): $q_{α_{rL}}^{'} = k_{α_{rL}} • q_{α_{rL}}$ (19) $q_{β_{rL}}^{'} = k_{β_{rL}} • q_{β_{rL}}$ (20) $q_{I_{d}} = k_{I_{d}} • q_{I_{d}}$ (21)

The correction coefficients are k_δ, k_δ, k_Ut, k_Efd, k_αrL, k_βiL, and k_Id in the formula (21). Taking the correction calculation of the power angle’s correction factor k_δ as an example, the concrete algorithm is introduced.

It is assumed the adaptive fuzzy controller in a feedback point, based on their observable fuzzy recognition, the fuzzy decision B is obtained, B = (VB, B, M, S). On this basis, the correction coefficient of a state in the initial value Q₀ of the state weight matrix is calculated according to the fuzzy algorithm based on the second component function [17]. The fuzzy rules of the fuzzy algorithm based on the second component function are as follows.

Firstly: the value of the second component function value for each rule is solved [18], (taking |Δδ| as the example), there are formulas (22–25): $R_{1} : if | Δ δ | = VB, then u_{1} = k_{e 1} Δ δ$ (22) $R_{2} : if | Δ δ | = B, then u_{2} = k_{e 2} Δ δ$ (23) $R_{3} : if | Δ δ | = M, then u_{3} = k_{e 3} Δ δ$ (24) $R_{4} : if | Δ δ | = S, then u_{4} = k_{e 4} Δ δ$ (25)

Secondly: to find the weight of each rule, there are: λ₁ = μ_VB (|Δδ|), λ₂ = μ_B (|Δδ|); λ₃ = μ_M (|Δδ|); λ₄ = μ_S (|Δδ|)

Thirdly: the “Center of gravity method” is used to find the numerical ΔU [19] after the fuzzy calculation, and the formula (26) is described. $Δ U = (\sum_{i = 1}^{4} λ_{i} \cdot u_{i}) / \sum_{i = 1}^{4} λ_{i}$ (26)

Fourthly: by multiplying ΔU by a fixed constant k, a correction factor k_δ corresponding to the power angle 3 can be obtained [20 –22], and there is a formula (27): $k_{δ} = k \cdot Δ U$ (27)

According to the theory described above, the adaptive fuzzy controller based on multi-sliding mode is finally designed to realize the multi-sliding mode adaptive module control of multi-infeed AC/DC transmission system.

3 Experimental analysis

3.1 Experiment 1

The three-machine multi-infeed double AC/ DC transmission system is used to verify the effectiveness of the multi-sliding mode adaptive fuzzy controller designed for the above multi-infeed AC/DC hybrid transmission system. The system structure studied is shown in Fig. 3, and the simulation software is NETOMAC.

Fig.3

Structure of multi infeed AC and DC transmission system.

The state equation of the whole system is the 18 orders equation, and the three generators and their exciters are represented by the three orders model and the first order model respectively. AC/DC transmission lines and AC/ DC controllers adopt the three orders model. The control modes of two AC/DC transmission systems are rectifier-side fixed current and inverter-side arc extinguishing angle control respectively. Thus, the equation of state consists of 12 generator state quantities and six state quantitiesof AC/DC transmission system, that is: $X = [\begin{matrix} Δ δ_{1} Δ ω_{1} Δ E_{q 1}^{'} Δ E_{fd 1} Δ δ_{2} Δ ω_{2} Δ E_{q 2}^{'} \\ Δ E_{fd 2} Δ δ_{3} Δ ω_{3} Δ E_{q 3}^{'} Δ E_{fd 3} Δ α_{1} Δ β_{1} \\ Δ α_{2} Δ β_{2} Δ I_{d 1} Δ I_{d 2} \end{matrix}]$ (28)

The initial weight matrix is selected:

$\begin{matrix} Q_{0} \\ = diag (\begin{matrix} 100, 100, 2000, 10, 100, 100, 2000, 10, \\ 100, 100, 2000, 10, 5, 5, 5, 5, 5, 5 \end{matrix}) \end{matrix}$ (29)

It provides that the excitation controllers of each generator only feed back δ, ω, U_t and E_fd of the machine. The rectifying side of AC/DC system feedback is close to ω of the adjacent generator. The power of the parallel AC line is P_ac, its own AC/DC current is I_d, the inverter side feedback is β and the AC bus voltage of the inverter endis U_ac. According to the selected weight matrix, the multi-sliding mode adaptive fuzzy controller is designed, the corresponding generator excitation control of the three-machine and two-intersection AC/DC systems and the feedback gain of the AC/ DC control system can be obtained, and shown in Tables 1 and 2 respectively.

Table 1

Feedback gain of generator excitation control for three machine two DC system

Generator excitation control	K _δ	K _ω	K _U1	K _Efd
Unit number
G₁	–2.0	4.02	–42.16	–4.16
G₂	–5.05	–2.4	–43.21	–6.42
G₃	–4.68	–5.1	–45.18	–5.68

Table 2

Feedback gain of DC control in AC and DC systems

	Rectifying end			Invert end
	K _I	K _ω	K _P	K _β	K _U
HVDC1	–5.2	–2.1	–20.24	–0.13	–1.8
HVDC2	–4.1	–1.4	–12.32	–0.13	–1.6

The disturbance of the system is: a three-phase grounding fault occurs at the midpoint of the line BC, and the fault is excised after 0.15 s. Figure 4 gives the dynamic response of the system under three different control modes: conventional non-decentralized and coordinated controller, decentralized and coordinated controller and multi-sliding mode adaptive fuzzy controller.

Fig.4

Dynamic response of a multi infeed DC system with and without decentralized coordinated control and this method controlmode.

From Fig. 4, we can see that the decentralized and coordinated control or multi-sliding mode adaptive fuzzy controller can really improve the dynamic performance and the stability of the AC/DC system. The dynamic performance of the system is obviously better than that under the non-decentralized and coordinated control. Moreover, in the three control methods, the multi-sliding mode adaptive fuzzy controller has the best effect on improving the stability of the system. This is because the Q value of the state weight matrix of the adaptive fuzzy control is fixed, and the selected Q value is not optimal. By using adaptive fuzzy recognition technology, the controller can carry out “two coordination”. Each feedback point can automatically adjust the value of Q according to the degree of disturbance, so that the adaptability of fuzzy control is better, and the dynamic performance of the system is further improved.

In order to check the adaptability of the designed multi-sliding mode adaptive fuzzy controller, the location of disturbance is changed, and the disturbing point is changed to the middle point of one of the double circuit FG. The dynamic simulation results still show that the dynamic performance of the AC/DC system is better than that of the conventional non-decentralized and coordinated control. The dynamic responses of three different control modes are compared respectively. The experimental results show that the multi-sliding mode adaptive fuzzy control has the best control effect on the stability of the system, and it also has good adaptability to different disturbances.

3.2 Experiment 2

A multi-infeed AC/DC system with three machines is used to verify the effectiveness of the multi-sliding mode adaptive fuzzy controller designed in this paper. The experimental system is shown in Fig. 5. The eigenvalues of the corresponding system oscillations are shown in Table 3. The system consists of two DC tie lines. According to the control principle of DC system, each DC line is equipped with a DC supplementary controller on the rectifier side and the inverter side. Therefore, the system control signals have four of ΔU_sr1, ΔU_si1, ΔU_sr2, and ΔU_si2. Table 4 shows the controllability degree of the four control signals of the two DC link lines relative to each mode 1. The controllable degree of control signal ΔU_si2 is the highest, while other three control signals of the modes are relatively small; for the oscillation mode 2, the control signal ΔU_sr1 has the greatest impact on it.

Fig.5

Experimental system structure.

Table 3

The eigenvalues of the corresponding system oscillation

Modal number	Characteristic root	Characteristic value
		Damping	Frequency
1	0.04127±j0.97052	0.0425	0.9714
2	0.0873±j0.75491	0.11478	0.75993

Table 4

The controllability of the control signals relative to each mode

Modal number	Modal eigenvalue	Control quantity
		ΔU _sr1	ΔU _si1	ΔU _sr2	ΔU _si2
1	0.0413±j0.9705	0.081	0.474	0.148	1.617
2	0.0873±j0.7549	1.293	0135	0.978	0.208

In practical multi-infeed AC/DC system, generator’s power angle and speed are not easy to measure. If we only compare the observability of each AC quantity and the active power deviation of AC lines parallel to two DC lines, we can see that the two modes of oscillation are all higher than those of other AC system parameters, and the location of measurement is also closest to the location of DC supplementary controller. Therefore, the active power deviations (ΔP_d1 and ΔP_d2) of the parallel AC line are chosen as the feedback quantity of the two DC additional controller. The feedback gain k₁ and k₂ are obtained by the feedback control of the partial output of the selectable control structure. Assuming that there are disturbances in multi-infeed AC/DC transmission system, node 9 suddenly loses 50% of the load. After 0.5 s recovery, the control effect under different control strategies is shown in Figs. 6–8.

Fig.6

State response of power strip modulation control only on the first DC rectifying side.

Fig.7

State response to a strip modulation controller with only second DC inverter sides.

Fig.8

Double DC additional control damping system concussion.

If only a DC additional power modulation controller is installed on the rectifier side of first DC lines, the input of the controller is the active deviation of the parallel AC line L_11-10, and the control quantity is ΔU_sr1. Then the damping effect on the system oscillation is shown in Fig. 6.

From Fig. 6(a) is not difficult to see that the installation of controller can make damping mode 1, which is consistent with the previous analysis of controllability results, indicating that the feedback amount is a parallel active power deviation from the DC, and the oscillation characteristics can be introduced into the controller, which is in good agreement with the observable degree of the quantity. For the oscillation mode 2, the controllability of the control amount ΔU_sr1 has less controllable, shown in Table 4, and the observability of ΔP_d1 is not high, as shown in Fig. 6(b). Under the action of the controller, the oscillation between generator 2 and generator 3 is only slightly improved, and the damping effect is worse than that of the action of oscillation mode 1.

If only a DC additional power modulation controller is installed on the rectifier side of second DC lines, the input of the controller is the active deviation of the parallel AC line L_19-9, and the control quantity is ΔU_si2. Then the damping effect on the system oscillation is shown in Fig. 7.

The comparison between Figs. 7 and 6(b) shows that it is also an oscillation mode between generator 2 and generator 3, and the control effect of ΔU_si2 is obviously better than that of ΔU_sr1, which indicates that the adaptive fuzzy controller designed in this paper is effective. The effect of single control on the oscillation mode 2 is the same as the analysis of the front side, because the control signal ΔU_si2 has a low controllability to the mode.

At the same time, two multi-sliding mode adaptive fuzzy controllers are used. The results obtained are described in Fig. 8.

Analysis of Fig. 8 can see that although each controller collects only local feedback, the multi-sliding mode adaptive fuzzy controller designed in this paper has a very significant control over two modes. It can be seen from the figure that its control ability is not comparable to any single controller in the front. The experimental results show that the multi-sliding mode adaptive fuzzy controller designed in this paper has good control effect on the multi-infeed AC/DC system.

4 Conclusions

The multi-sliding mode adaptive fuzzy controller of multi-infeed AC/DC transmission system designed in this paper enhances the effectiveness of the previous controller for system control, and it can adapt to different disturbances and has better control performance. In this paper, an excellent controller design scheme is provided for the continuous enrichment and development of the multi-infeed AC/DC transmission system.

Footnotes

Acknowledgments

Shandong Provincial Natural Science Foundation, China (No. ZR2016EEP10) and Scientific Research Foundation of Shandong University of Science and Technology (No. 2015RCJJ073).

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