AVR fuzzy PID control system based on MCU

Abstract

This paper analyzes the AVR excitation control system and gives the structure of the MCU control system, using the fuzzy PID control system with programming analysis. The results of MATLAB simulation show that the fuzzy PID control system is superior to the conventional PID control system in dynamic performance and system stability. Therefore, the fuzzy PID excitation control system has good performance.

Keywords

Fuzzy control generator excitation system MCU

1. Introduction

With the technological progress, the increase of diesel generator frequency makes it more and more miniaturization, which the requirements of excitation regulator (AVR) are also getting higher and higher. Synchronous generator excitation control system is an important part of the generator, its main task is basing on the generator operating conditions and providing an adjustable DC current to the synchronous generator excitation device to meet the need of various generator running Mode. It is necessary for the excitation device of excellent performance and high reliability to ensure the safety of synchronous generator’s power generation and to improve the stability of the power system. In the normal operation of the generator or accident conditions, the excitation control system plays a very important role. The excitation control system of the excellent performance can not only ensure the safe operation of the generator and provide qualified power, but also effectively improve the generator and the economic and technical indicators of its connected power system.

In recent years, the research and application of fuzzy control, artificial neural network and genetic algorithm have brought new development to the control field, which has made the development of intelligent excitation control, with the emerging of fuzzy excitation control, excitation control of artificial neural network and excitation control based on expert system and genetic algorithms and their combination. Its basic characteristics are that it is a kind of intelligent concept model and it combines the control theory and human experience with intuitive reasoning rather than relying on the exact mathematical model of the object system, and it has the abilities and advantages of non-linear, parallel computing, adaptive, self-learning, self-organization, etc.

At present, most of the excitation regulator of portable diesel engine adopt the circuit regulation of separating electronic components, but the advance of the integrated electronic devices and digital level needs to be updated from the following aspects. For one thing, it demands for changing the original circuit selection for IGBT, while using PWM control system; for another, it is necessary to change the original PID control for fuzzy control system based on MCU, making the product miniaturization and intelligent.

2. The control principle of AVR system

The excitation control system is mainly composed of two parts: one is the excitation power unit, which provides the DC excitation current to the excitation winding of the synchronous generator; the other part is the excitation regulator, which constitutes the control part of the excitation device and automatically adjusts the output excitation current of power unit according to the changes of the synchronous generator voltage and operating conditions, to meet the operation requirements of the synchronous generator. The overall block diagram of the excitation control system is shown in Fig. 1.

Figure 1.

The overall block diagram of the excitation control system.

Figure 2.

The structure chart of SCM control system.

3. The AVR principle of SCM control

The system adopts the SCM and the fuzzy PID control system, and uses semi-controllable devices of thyristor to implement, of course, with further development of full control device of IGBT. That is using the separative excitation system and alternator coaxial with the host as the AC excitation power, supplying to excitation through the rectification of silicon rectifier or thyristor. As a result of using static rectification component instead of rotating DC exciter, it increases the reliability and convenience of maintenance and management. The structure chart is shown in Fig. 2.

The system adopts the AT89C2051 SCM. (1) The generator’s AC voltage of 220 V is changed into 7.5 V by transformer in power part, which is rectified to the DC voltage of $+$ 5 V by the rectifier bridge later for the use of SCM and system; (2) The AT89C2051 chip in main control part is low-voltage and high-performance SCM of 8-bit, with a programmable, erasable and read-only memory (E2PROM) of 2 KB Flash; (3) Keyboard part uses three keys to achieve the operation, a button to achieve switching of the setting voltage and the actual voltage, the other two keys to increase and decrease the setting voltage value. LED displays the current voltage after power on, the voltage can be set when the switch button is pressed, and press it again to return to the current voltage display; (4) The input of 1-bit is changed into parallel output of 8-bit by the 74HC164 and display in LED in display part; (5) The current and voltage detected by the voltage transformer are connected to the SCM through the A/D conversion, with the fuzzy control of the excitation system; (6) SCM forms the control pulse by the D/A conversion, and control the excitation current and so on with the trigger circuit by the amplification of thyristor.

Consider the following aspects for the excitation control system. (1) Excitation limit, that is, the protection restrictions when the generator is subject to relatively large interference; (2) The sensitivity and reliability of relay protection are supposed to improve; (3) Rapid demagnetization, the relay protection device action is taken when the internal winding of generator have a short circuit, cutting off the main switch of generator and excitation power. However, the winding inductance of generator excitation is relatively large, storing a lot of electromagnetic energy, which will induce overvoltage when cutting off the excitation power. In order to reduce the damage of short-circuit current on the internal winding of generator, the excitation device is required to have a function of rapid demagnetization.

4. The composition of Fuzzy PID control system

The fuzzy control controller is composed of three parts. The first part is the fuzzification, the precise input quantity supervised is transformed into a fuzzy set. The second part is the fuzzy reasoning, and the fuzzy set is output according to the fuzzy control rules and the input fuzzy set. The third part is anti-fuzzification, a representative exact value is selected as the control quantity according to the output fuzzy set.

PID controller is based on the exact mathematical model of the controlled object, but in the actual excitation system, there are often such cases.

(1)
It is difficult to obtain the exact mathematical model of the excitation system;
(2)
The excitation model is time-varying in many cases;
(3)
In practice, the excitation model is simplified for mathematical convenience, and the accuracy of the control system is sacrificed. Compared with the conventional controller such as PID controller, the classical fuzzy controller has the characteristics of the controlled object without establishing the mathematical model and considering its non-linearity and time-varying nature, but it also needs to be further improved, such as the poor stability of the fuzzy controller control accuracy, etc. For this purpose, numerous of excitation controller of fuzzy PID are designed to modify the shortcomings of precision controller and achieve good control results.

Figure 3.
The structure of the excitation controller of fuzzy PID.

Figure 4.
The membership function of the input and output variables.

The fuzzy PID controller is composed of fuzzy controller and PID controller. The input variables of the fuzzy controller are the deviation value of voltage $e$ and the deviation change rate of voltage $e_{c}$ . The output variables are the three parameters KP, KI and KD of the PID controller. Firstly, the input variables are caculated by fuzzification, and then the output variables KP, KI and KD are obtained by the logic reasoning and decision according to fuzzy-control rules. The structure of the excitation controller of fuzzy PID is shown in Fig. 3.
4.1 Determine the universe of input and output variables

The paper adopts the fuzzy controller of double input and three output. The universe of the controller’s input variables (deviation value of voltage $e$ and deviation change rate of voltage $e_{c}$ ) and output variables (three parameters KP, KI, KD of PID controller) are defined as follows.

$\displaystyle e\in\{-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6\}$ $\displaystyle e_{c}\in\{-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6\}$ $\displaystyle K_{P}\in\{-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6\}$ (1) $\displaystyle K_{I}\in\{-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6\}$ $\displaystyle K_{D}\in\{-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6\}$

The respective domain of voltage error e and its change rate ec and three parameters KP, KI, KD of PID controller is divided into seven levels: NB (negative big), NM (negative medium), NS (negative small), ZO (zero), PS (positive small), PM (positive medium), PB (positive big). The membership function is shown in Fig. 4.

4.2 Determine the quantification factor

Assuming that the actual universe of $e$ and $e_{c}$ are written as $[-e_{p},e_{p}]$ and $[-e_{cp},e_{cp}]$ , corresponding to the fuzzy state [ $-$ 6, 6], so the input of quantization factors is as follows:

$\displaystyle G_{e}=\frac{e_{p}}{6},G_{ec}=\frac{e_{p}}{6}$ (2)

The $e_{p}$ and $e_{cp}$ are selected according to the actual situation.

The precise quantities of deviation $e(k)$ and the change rate of deviation $e_{c}(t)$ are quantified to the input of the fuzzy controller according to the following formula:

$\displaystyle e=\frac{e(k)}{G_{e}},e=\frac{e(k)}{G_{ec}}$ (3)

The fuzzy controller of quantization is input through the definition of the membership function shown in Fig. 4, transforming the fuzzy speech into fuzzy language set.

4.3 Determine the fuzzy-control rules

For variant $e$ and $e_{c}$ , tuning principles of fuzzy-control correction of $K_{P}$ , $K_{I}$ , $K_{D}$ are as follows.

(1)
When $e$ is big, the bigger $K_{P}$ and smaller $K_{D}$ are taken to make the tracking of system be better. $K_{I}=$ 0 is obtained to avoid a larger overshoot of system response at the same time.
(2)
When $e_{c}$ is medium, $K_{P}$ should be smaller to make the system have a smaller overshoot. In this case, the value of $K_{D}$ has a greater impact on the system, which should take smaller value, while the value of $K_{I}$ should be appropriate.
(3)
When $e$ is small, $K_{P}$ and $K_{I}$ should be bigger to make the system has good stability; The smaller $K_{D}$ is obtained when $e_{c}$ is bigger, otherwise $e_{c}$ smaller.

According to the above tuning principles, the control rules of output variables $K_{P}$ , $K_{I}$ , $K_{D}$ can be obtained as shown in Tables 1–3 below.

Table 1
Table of KP control rules

[height=0.8cm,width=2.4cm] $e$ $e_{{c}}$ NB NM NS ZO PS PM PB

NB PB PB PM PM PS ZO ZO

NM PB PB PM PS PS ZO NS

NS PM PM PM PS ZO NS NS

ZO PM PM PS ZO NS NM NM

PS PS PS ZO NS NS NM NM

PM PS ZO NS NM NM NM NB

PB ZO ZO NM NM NM NB NB

Table 2
Table of KI control rules

[height=0.8cm,width=2.4cm] $e$ $e_{{c}}$ NB NM NS ZO PS PM PB

NB NB NB NM NM NS ZO ZO

NM NB NB NM NS NS ZO ZO

NS NB NM NS NS ZO PS PS

ZO NM NM NS ZO PS PM PM

PS NM NS ZO PS PS PM PB

PM ZO ZO PS PS PM PB PB

PB ZO ZO PS PM PM PB PB

Table 3
Table of KD control rules

[height=0.8cm,width=2.4cm] $e$ $e_{{c}}$ NB NM NS ZO PS PM PB

NB PS NS NB NB NB NM PS

NM PS NS NB NM NM NS ZO

NS ZO NS NM NM NS NS ZO

ZO ZO NS NS NS NS ZO ZO

PS ZO ZO ZO ZO ZO ZO ZO

PM PB NS PS PS PS PS PB

PB PB PM PM PM PS PS PB

(4)
Select the fuzzy reasoning method

The fuzzy reasoning synthesis rules adopted in this paper follow the maximal one – minimal synthesis rule and adopt the Mamdani type fuzzy reasoning algorithm.
(5)
Determine defuzzification method of the output quantity

The control quantity given by fuzzy-control algorithm (precise quantity) cannot directly control the object. The actual output should be treated with defuzzification, which is converted to the basic universe accepted by the control object while the center of gravity method is adopted by the treatment of defuzzification.

5. Programming and Implementation

[height=0.8cm,width=2.4cm] $e$ $e_{{c}}$	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	PS	ZO	NS
NS	PM	PM	PM	PS	ZO	NS	NS
ZO	PM	PM	PS	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

[height=0.8cm,width=2.4cm] $e$ $e_{{c}}$	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	NS	ZO	ZO
NM	NB	NB	NM	NS	NS	ZO	ZO
NS	NB	NM	NS	NS	ZO	PS	PS
ZO	NM	NM	NS	ZO	PS	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PB
PM	ZO	ZO	PS	PS	PM	PB	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

[height=0.8cm,width=2.4cm] $e$ $e_{{c}}$	NB	NM	NS	ZO	PS	PM	PB
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NS	ZO
NS	ZO	NS	NM	NM	NS	NS	ZO
ZO	ZO	NS	NS	NS	NS	ZO	ZO
PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
PM	PB	NS	PS	PS	PS	PS	PB
PB	PB	PM	PM	PM	PS	PS	PB

5.1 Anti-interference measures

(1)
A/D line: in the A/D conversion circuit, the use of high frequency filter capacitor. In the wiring, power supply and ground wire of each chip to bypass capacitor, and should be as close as possible to the A/D chip, generally used in the 0.001 F–0.1 F, the wiring should shorten the length of the wire line transmission, digital signal and analog signal lines should be kept away from the. The analog sampling signal of the microcomputer excitation regulator is easy to be interfered by the field, and it is likely to be mixed with high amplitude spikes. For this interference, we filter and limit the signal before entering the A/D board.
(2)
Isolation and shielding: isolation and shielding is an effective measure to restrain interference coupling. In this system, the analog part and the digital part are completely isolated by light disaster, which can effectively restrain the influence of external interference on the digital system. In addition, the sampling transformer is placed on a single board, and the sampling signal is transmitted to the CUP circuit board through the shielding wire to shield electromagnetic interference.
(3)
Decoupling capacitor: integrated circuit components coupled with the coupling capacitor is also very critical, the digital circuit in the typical value of the decoupling capacitor is 0.1 F, the wiring, the closer the integrated circuit components and the power of the better. When debugging, the external interrupt 1 pin NIT on the work of adding a capacity of 0.1 F, so that the normal operation of the program; in the zero crossing comparator LM339 input to add the capacitance of 0.1 F in order to remove interference, the effect is very good.
(4)
“watchdog” (WDT): CPU is a high speed device, operation, easy interference controller, when the electromagnetic interference signal into the CPU, will be the wrong execution, caused by malfunction or erroneous results, one of the most vulnerable is the program pointer PC, common problem is the program run to fly. As a result, the watchdog has become one of the indispensable members of the microcomputer system. It is usually used to prevent the running of the CUP program by combining the software with the special circuit. In this study, the use of watchdog timing function with appropriate procedures to facilitate the formation of WDT, in the PC anomaly can be timely and effective force “reset”.

5.2 Programming and debugging

This system uses power electronic excitation rectifier circuit half controlled thyristor trigger circuit, integrated trigger circuit, full bridge output control of the 12 pulse, the pulse voltage regulation by MCU D/A converter output voltage control alpha angle, voltage regulation, regulation and ultimately achieve the excitation current.

The parameters of fuzzy excitation control system is determined through programming, experiments and trial and error.

$\displaystyle K_{P}\cdot\min=26.3,K_{P}\cdot\text{mid}=30.1,K_{P}\cdot\max=45.15$ $\displaystyle K_{I}\cdot\min=0.008,K_{I}\cdot\text{mid}=0.01,K_{I}\cdot\max=0.07$ $\displaystyle K_{D}\cdot\min=0.00875,K_{D}\cdot\text{mid}=0.0125,K_{D}\cdot% \max=0.00874$

Among them, the PID regulator uses incremental of PID, the formula is shown in Eqs (4)–(7).

$\displaystyle u(k)=K_{P}e(k)+K_{I}T\sum_{j-1}^{k}e(j)+\frac{K_{D}}{T}[e(k)-e(k% -1)]$ (4) $\displaystyle\Delta u_{k}=u(k)-u(k-1);\Delta e(k)=e(k)-e(k-1)$ (5) $\displaystyle\Delta^{2}e_{k}=\Delta e_{k}-\Delta e_{k-1}=e(k)-2e(k-1)+e(k-2)$ (6) $\displaystyle\Delta u_{k}=K_{P}\Delta e_{k}+K_{I}Te_{k}+\frac{K_{D}}{T}\Delta^% {2}e_{k}$ (7)

The above data is treated with fuzzification, while AVR is controlled through programming, with programming block diagram shown in Fig. 5.

Figure 5.

Programming master block diagram.

Figure 6.

The simulation model of conventional PID excitation control system.

Figure 7.

The simulation model of fuzzy PID control system.

Figure 8.

The simulation results of conventional PID excitation control system.

Figure 9.

The simulation results of fuzzy PID excitation control system.

In the commissioning phase, it is considered that the most powerful and effective means to improve the transient stability of power system is to improve the excitation capability of the excitation system, that is to say, to increase the voltage and the voltage rise rate.

The working principle of maximum current instantaneous limiter is detection of excitation current value, and compared with the maximum current value when it is less than the limit value, if not limiter action; greater than a limit value, the output limiter immediately move to implement the limiting voltage control Strong excitation, forcing excitation power unit decreased rapidly, the excitation current as soon as possible drop. When the return current is below the limit value, the limiter is removed.

After the completion of the program, and SCM control system for joint debugging, and constantly modify and improve the system and procedures, until the commissioning results are satisfactory.

6. Simulation verification

The fuzzy control scheme of PID is simulated by Matlab and compared with the traditional control scheme of PID. The simulation parameters are as follows. The time constant of the synchronous generator is 6, the amplification factor of the generator and power unit are 1 and 2, the time constant of the power unit is 0.3, the time magnification and time constant of the measure-compare unit are taken 1 and 0.05. The simulation model of conventional PID excitation control system is shown in Fig. 6, with adding fuzzy control shown in Fig. 7.

By comparing the conventional excitation control system and the fuzzy PID excitation control system, it can be seen that the conventional PID excitation control system has a larger overshoot. And the fuzzy PID control system is superior to the former in improving the dynamic performance and stability of the generator system.

7. Conclusion

By analyzing the control system of AVR single chip computer and its characteristics, a set of fuzzy-PID control system is designed, and the system is programmed and analyzed. The simulation analysis is carried out on MATLAB software. The simulation results show that the fuzzy-PID excitation control system is superior to the conventional PID excitation control system in improving the dynamic performance and stability of the generator. It proves that the fuzzy-PID control system is stable and reliable. The system meets the control requirements of excitation control system and achieves good control effect.

Footnotes

Acknowledgments

The authors acknowledge special of universities in Fujian (Project number: JK201006).

References

Zhang

, Fuzzy Control Applied on Alternator Excited System of Ships, Dalian Maritime University, 2006, p. 3.

Gao

et al., Matching of excitation regulator parameter and protection setting value in 1 000 MW generator, Power Construction 12(33) (2012), 80–82.

Zhang

and Huang

, Design of excitation control system for hybrid-excitation PMSG based on 80C196KC, Journal of Zhuzhou Institute of Technology 20(2) (Mar. 2006), 52–55.

Morison

Wang

and Kundur

, Power system security assessment, IEEE Power and Energy Magazine 2(5) (2004), 30–39.

Hirata

Ohnishi

and Yamamoto

, A design of nonlinear PID control systems by using local model identification, in: The 30th Annual Conference of the IEEE Industr Ial Electronics Society, Korea, (3), 2004, pp. 2464–2469.

Gerasimov

Esipovich

and Zekkel

, Adjustment of Power System Stabilizing Channels Structure of Generator’s Automatic Excitation Regulators, IEEE Power techology, Russia, 2005, pp. 1–4.

and Wang

, Robust adaptive control of synchronous enerators with SMES unit via hamiltonian function method, International ournal of Systems Science 38(3) (2007), 187–196.

Kiyong

Rao

and Burnworth

, Application of Swarm Intelligence to A Digital Excitation Control System, 2008 IEEE Swarm Intelligence Symposium, St. Louis MO USA, 2008, pp. 1–8.

Zhu

Shen

and Chen

, Effects of the Excitation System Parameters on Power System Transient Stability Studies, in: IEE 2nd International Conference on Advances in Power System Control, Operation and Management, Hong Kong, 1993, pp. 532–535.

10.

Murdoch

and Venkataraman

, Excitation system rotective lim tiers and their effect on volt/var control-design, omputer modeling, and field testing, IEEE Transaction on Energy Conversion 15(4) (2000), 440–450.

11.

Tan

and Wang

, Design of fuzzy control system for coal humidity based on MCU, Journal of Xiangtan Normal University (Natural Science Edition) 31(1) (2009), 66–70.

12.

and Zhang

, Multivariable fuzzy control technology for heat setting machine, Textile Journal 26(4) (2005), 79–82.

13.

Zhao

Z.Y.

Tomizuka

and Isaka

, Fuzzy gain scheduling of PID controller, IEEE transactions on systems, man, and cybernetics, 1993.

[height=0.8cm,width=2.4cm] $e$ $e_{{c}}$	NB	NM	NS	ZO	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	PS	ZO	NS
NS	PM	PM	PM	PS	ZO	NS	NS
ZO	PM	PM	PS	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

[height=0.8cm,width=2.4cm] $e$ $e_{{c}}$	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NB	NM	NM	NS	ZO	ZO
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ZO	NM	NM	NS	ZO	PS	PM	PM
PS	NM	NS	ZO	PS	PS	PM	PB
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[height=0.8cm,width=2.4cm] $e$ $e_{{c}}$	NB	NM	NS	ZO	PS	PM	PB
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NS	ZO
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ZO	ZO	NS	NS	NS	NS	ZO	ZO
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PM	ZO	ZO	PS	PS	PM	PB	PB
PB	ZO	ZO	PS	PM	PM	PB	PB

[height=0.8cm,width=2.4cm] $e$ $e_{{c}}$	NB	NM	NS	ZO	PS	PM	PB
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NM	PS	NS	NB	NM	NM	NS	ZO
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PM	PB	NS	PS	PS	PS	PS	PB
PB	PB	PM	PM	PM	PS	PS	PB

[height=0.8cm,width=2.4cm] $e$ $e_{{c}}$	NB	NM	NS	ZO	PS	PM	PB
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NS	PM	PM	PM	PS	ZO	NS	NS
ZO	PM	PM	PS	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

[height=0.8cm,width=2.4cm] $e$ $e_{{c}}$	NB	NM	NS	ZO	PS	PM	PB
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