Citation: Gudimindla Hemachandra, Manjunatha Sharma K. Design and performance analysis of quantitative feedback theory based automated robust controller : An application to uncertain autonomous wind power system[J]. AIMS Energy, 2018, 6(4): 576-592. doi: 10.3934/energy.2018.4.576
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Nomenclature
R Rotor resistance (
p Pole pair
Abbreviations
2-DOF Two degree of freedom
ALS Automatic loop-shaping
AQFT Automated quantitative feedback theory
AWPS Autonomous wind power system
CAD Computer aided design
CSD Control system design
GA Genetic Algorithm
HFG High frequency gain
MAQFT Modified automated Quantitative feedback theory
MMQFT Multi-model Quantitative feedback theory
PFL Partial feedback linearization
PMSG Permanent magnet synchronous generator
QFT Quantitative feedback theory
Quantitative feedback theory (QFT) [1,2,3,4], is a robust control system design (CSD) approach which employs system output as an feedback variable to achieve the desired dynamic performance in presence of plant uncertainties and disturbances. In general, QFT is well applicable for handling uncertainty in frequency domain. Ever since its inception, QFT is applied to solve various real time CSD problems [2,3,5,6]. Although QFT was intially applied to different to various single input - single output (linear time variant, time invariant and nonlinear) systems, its extension to multiple input - multiple output (linear and nonlinear) systems is presented in [5,7,8].
It is well known that for adequate implementation of QFT, system gain-phase loop-shaping is imperative and can be performed either manually or automatically. On this line many computer aided design (CAD) tools are developed to perform manual loop shaping like the pioneer Air Force Institute of Technology CAD tool [2,9,10], QFT control toolbox by European space agency author's group [11,12], QFT MATLAB toolbox [13,7] and Qsyn [14]. Despite its simplicity in design, the method primarily depends on the trial and error approach which indeed results in the system performance highly dependent on the designer. Further, the complexity increases profoundly for unstable and non-minimum phase uncertain systems in order to fulfil all the necessary performance specifications resulting in the need for automatic loop shaping (ALS) methods.
Concerning the attempts in developing the ALS methods, Gera and Horowitz proposed design of QFT robust controller based on iterative procedure to derive the shape of a nominal loop transfer function (
In addition, the authors, Garcia-sanz and Guillen [20], Garcia-sanz and Oses [21], and Garcia-sanz and Molins [22], have proposed the evolutionary and Genetic algorithm (GA) based ALS. Unlike the aforesaid methods, a phase independent controller is developed using the least square type algorithm [23]. Similarly, the application of particle swarm optimization, hybrid optimization (interval consistency and hull consistency), teaching learning-based optimization algorithm, flower pollination algorithm and convex concave optimization methods are presented in [24,8,25,26,27] respectively. In spite of these many efforts, a simple design methodology to devise a controller with an overall satisfactory performance is still left unattended. With this motivation, an attempt to formulate a controller structure exhibiting the characteristics of descending modular trace within the close vicinity of the universal bound is addressed.
In this paper, a modified fitness function is formulated by considering the suitable cost function terms in order to accurately capture the desired descending modular trace close-packed to the universal bound. The minimization of the cost function is accomplished by the application of GA. The suitability and superiority of the proposed QFT based controller is demonstrated by applying it to an uncertain autonomous wind power system (AWPS). Numerical simulations are performed on MATLAB platform and results revealing the improved performance of the developed controller in comparison to the state-of-the-art methods are included. Finally, an exhaustive comparative analysis proving the proposed controller's improved performance with that of other well- established methods is carried out and the corresponding results are presented.
A permanent magnet synchronous generator (PMSG) -based AWPS shown in Figure 1 is considered as the test system to demonstrate the applicability and suitability of the proposed QFT based robust controller under stochastic wind speeds [28,29,30,31]. Assuming that the other system components excluding those within the local control loop as highlighted in Figure 1 works as intended, with suitable assumptions, the chopper equivalent circuit is obtained [28]. Neglecting the dynamics of power electronic converters, dynamics pertaining to the aerodynamic model of wind turbine and PMSG are accounted. The aerodynamic torque is given as,
τwt=12ρπR3TCτ(λ)V2τ. | (1) |
The dynamic model of PMSG with chopper equivalent variable resistance
diddt=−(R+Rs)(Ld+Ls)id+p(Lq−Ls)(Ld+Ls)iqωhdiqdt=−(R+Rs)(Lq+Ls)iq−p(Ld+Ls)(Lq+Ls)idωh+pϕmdωhdt=−1Jhτwt−τemJhτem=p[(Ld−Lq)idiq−ϕmiq] | (2) |
In real environment, the conventional control system fails to meet the necessary design specifications owing to its incapability to handle the uncertainties and disturbances that are inevitable in any practical operating conditions. In order to attribute to these requirements, a highly robust CSD is essential. A two degree of freedom (2DOF) control structure shown in Figure 2 is used to design the robust controller and pre-filter in QFT framework [23]. Where
|T1(jω))|=|UD1|=|YD0|=|YN|=|L(jω))1+L(jω))|=|Y(jω)F(jω))R(jω)|≤γ1(ω) | (3) |
|T2(jω))|=|YD2|=|11+L(jω)|≤γ2(ω) | (4) |
|T3(jω))|=|YD1|=|P(jω)1+L(jω))|≤γ3(ω)) | (5) |
|T4(jω)|=|YDX|=|P2(jω)1+L(jω)|≤γ4(ω) | (6) |
|T5(jω)|=|UDx|=|G(jω)P2(jω)1+L(jω)|≤γ5(ω) | (7) |
|T6(jω)|=|UN|=|UD2|=|G(jω)1+L(jω)|=|U(jω)R(jω)F(jω)|≤γ6(ω) | (8) |
γ7L(ω)≤|T7(jω)|=|YR|=|F(jω)L(jω)1+L(jω)|≤γ7U(ω) | (9) |
The necessary procedure to be followed in the process of designing a QFT controller is comprehensively described below :
Step 1: Select the uncertain plant and define it's uncertainty range
Step 2: Establish the performance specifications
Step 3: Generate the template for an uncertain plant
Step 4: Compute the QFT bounds by selecting the nominal plant such that the template points fulfils the performance specifications and are stable at every frequency for
Step 5: Perform the loop shaping
Step 6: Design the Pre-filters
Reduction of high frequency noises at the sensor output and plant disturbances is desired to utilise the feedback benefits and is achieved by reducing high frequency gain (HFG) expressed as follows
HFG=lims→∞srG(s) | (10) |
where
With this background the proposed design procedure of the automated QFT controller is presented in this Section. First, the problem statement formulation is discussed. Second, the modified fitness function evaluation is carried out.
The critical design specifications such as reducing HFG there by maximizing the feedback benefits along with the minimization of cost of feedback are translated into a mathematical formulation given as,
G(x,jω)=KGnz∏i=1(jω+zi)np∏k=1(jω+pk) | (11) |
where
In general, fitness function defined as combining an expression of constraints and objectives ideally defines quality of controller and its estimated behaviour. Hence it is very vital to formulate an effective fitness function and its coefficients which translates all the requisite CSD specifications into a mathematical expression. As a first step the controller excess gain-band width area on
A(ω1,ω2)=ω2∫ω1ln|G(jω)|dω | (12) |
The foregoing assumptions permit the computation of the definite integral of (12) in terms of
A(ω1,ω2)=[ln(K2G)(ω2−ω1)+nz∑i=1(ω2ln[ω22+z2iω22+p2i]−ω1ln[ω21+z2iω21+p2i]+2zi tan−1(ω2zi)−2pi tan−1(ω2pi)+2zi tan−1(ω1zi)−2pi tan−1(ω1pi)) | (13) |
Finally, the controller gain-bandwidth area measure is used to form an augmented cost function. As an outcome, the formulated fitness function is given as,
J(x)=a0 k2G+a1nz∑i=1(pi−zi)4+1pizi+a2 A(ω1,ω2)+a3 V(x)+a4 nz∑i=1tan−1(pi−zipizi) | (14) |
In the above cost function, the first two terms corresponds to high frequency gain and lead ratios respectively while the third and fourth terms refers to constraint pertaining to the area and the penalty respectively. The third term in (14) facilitates the tight control of gain at any frequency of interest. Thus, helps in diminishing the over-design at low frequencies. Similarly, additional terms can be included in order to cater other frequency ranges as well.
In general the nominal loop transmission should satisfy the bounds of intersection and are described as a function q with lower and upper parts,
Lmqu(∠L0(x,jωi),ωi)≤LmL0(x,jωi)i∈ILmL0(x,jωi)≤Lmql(∠L0(x,jωi),ωi) | (15) |
where
θx,i=∠L0(x,ωi)Vu(x,ωi)=max{logqu(θx,i,ωi)−logL0(x,ωi),0}Vl(x,ωi)=max{logL0(x,ωi)−logql(θx,i,ωi),0} | (16) |
Vω(x,ωi)={min {Vu(x,ωi),Vl(x,ωi)},if qu(θx,i,ωi)≥ql(θx,i,ωi)max{Vu(x,ωi),Vl(x,ωi)},otherwise | (17) |
Thus, the penalty function V(x) to penalise the unfeasible solutions that do not fulfil the performance specifications and assist in comprehending the degree of disparity of each bound at frequency
V(x)=∑i∈IV2ω(x,ωi) | (18) |
It is worth noting that despite the addition of the prominent and wide spread terms into the fitness function, the inclusion of the proposed fifth term greatly aids in obtaining the desired descending modular plot of the controller. The inherent characteristics of the proposed term results in impelling the locus of the closed loop transmission
The proposed QFT based robust controller and pre-filter design steps as applied to an PMSG-based AWPS is discussed in this Section. The transfer function model of PMSG based AWPS is represented as an uncertain system given by,
P(s)=k(1+2Tξs+s2T2) | (19) |
where, the uncertainty of plant parameter is
Templates are the pictorial representation of the uncertain plant's magnitude and phase response at fixed frequency. The sketch of the templates for the frequency vector
The edge point templates are used to obtain QFT bounds. Stability margins and performance specifications are transformed to frequency domain in order to represent the QFT bounds in Nichols chart. The computation of QFT bounds is accomplished by considering the quadratic inequalities and the closed loop robust stability margins are given as,
|L(jω)1+L(jω)|≤γ1∀P∈{p} | (20) |
In this case,
The upper and lower reference tracking bounds for the considered AWPS are given by,
HU(s)=16.67s+400s2+36s+400HL(s)=12000s3+80s2+1900s+12000 | (21) |
Further, all the stability and tracking bounds are grouped to calculate the worst case possibilities and are pictorially depicted in Figure 4.
The existing and proposed ALS methodology based control structures formulated using GA are respectively given by,
G1(s)=15.701(s+42.29)(s+38.43)(s+10.42)(s+354)(s+134)(s+0.01765) | (22) |
G2(s)=5.7074(s+53.06)(s+23.58)(s+13.63)(s+125.1)(s+109.6)(s+0.07097) | (23) |
It is evident from Figure 5 and Figure 6 that the open loop phase of
Performance index | [22] | Proposed method |
FG | 15.701 | 5.7074 |
J(x) |
The design of pre-filter is performed using the same procedure as that of loop-shaping and obtained transfer function is given as,
F(s)=17.614×10−6s3+0.00156s2+0.087s+1 | (24) |
The corresponding block diagram representation of developed system with proposed subsystems is shown in Figure 7.
The maximum power tracking capability of the proposed controller under step change in wind speed and stochastic variation is tested and a comparative evaluation is performed against the following methods: feedback linearization [29], QFT [28], multi-model QFT [28], GA based automated QFT [22] through MATLAB simulations. The PMSG and wind turbine parameters employed for the simulation are shown in Table 2. The input wind velocity template depicts step variations at
Wind turbine parameters | PMSG parameters |
P=3, R=3.3 | |
| |
owing to the associated mechanical structure. Among the other available methods, a smooth torque variation with an improved performance is witnessed with the application of the developed controller. Further the key observations proving the proposed controllers competency is enlisted in Table 3.
Parameter | Wind profile | PFL | QFT | MMQFT | AQFT | MAQFT |
Control input ( | 7 m/s | 27 | 25.9 | 25.82 | 25.72 | 25.52 |
7 | 33.65 + 33.63 % | 33.65 | 33.55 | 33.45 | 33.24 | |
9 | 41.55 + 17.83 % | 41.55 | 41.41 | 41.3 | 41 | |
Electromagnetic torque (N-m) | 7 m/s | 14.7 | 14.59 | 14.598 | 14.65 | 14.755 |
7 | 24 - 75 % | 24.1 + 32.78 % | 24.16 + 28.31 % | 24.23 + 13 % | 24.42 + 4.42 % | |
9 | 36 -65.67 % | 36 + 34.03 % | 36.1 + 23.96 % | 36.3 + 11.57 % | 36.5 + 4.11 % | |
Power extracted (W) | 7 m/s | 2000.54 | 2000.65 | 2001.02 | 2002.3 | 2004.75 |
7 | 4252 | 4252 | 4254 | 4256.6 | 4261 | |
9 | 7763 | 7763 | 7767.6 | 7772.2 | 7780 | |
+ Overshoot; - Decay; ± Oscillatory |
Owing to the inherent variability in the wind power generation, a stochastic wind profile is consider as the second test case. The corresponding results are shown in Figure 9. It is evident from the result figures that the proposed controller outperforms the other available CSD methods from various control system perspectives.
This paper presented a modified fitness function based automated robust controller using GA in QFT framework to extract the maximum power from PMSG based AWPS. The prominent features of the proposed controller are as follows.
1. It absolutely exhibits the highly desired decreasing modular plot and descending phase response.
2. Addition of a simple arc tangent function helps in shifting the loop-shaping curve closer to the universal bound intern significantly reducing the gain at high frequencies.
3. The usage of GA leads to the effortless acquisition of the controller parameters.
The applicability and feasibility of the developed controller is verified through extensive simulations and the results attesting its improved performance against well established methods are presented.
All authors declare no conflicts of interest in this paper.
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1. | Hemachandra Gudimindla, Manjunatha Sharma Krishnamurthy, S Sandhya, 2020, Performance analysis of automated QFT robust controller for long-term grid tied PV simulations, 978-1-7281-7213-2, 412, 10.1109/ICSTCEE49637.2020.9277021 | |
2. | Hemachandra Gudimindla, Manjunatha Sharma K, Dynamic performance evaluation of automated QFT robust controller for grid-tied fuel cell under uncertainty conditions, 2020, 42, 22131388, 100800, 10.1016/j.seta.2020.100800 | |
3. | Akash Singh, Arnab Ghosh, Comparison of Quantitative Feedback Theory Dependent Controller with Conventional PID and Sliding Mode Controllers on DC-DC Boost Converter for Microgrid Applications, 2022, 7, 2199-4706, 10.1007/s40866-022-00133-2 | |
4. | Hemachandra Gudimindla, Manjunatha Sharma K, S Sandhya, 2021, Performance Analysis of Adaptive Speed Reference Tracking QFT Robust Controller for Three Phase Grid connected Wind Turbine under Stochastic Wind Speed Conditions, 978-1-6654-2237-6, 1, 10.1109/ETI4.051663.2021.9619223 |
Wind turbine parameters | PMSG parameters |
P=3, R=3.3 | |
| |
Parameter | Wind profile | PFL | QFT | MMQFT | AQFT | MAQFT |
Control input ( | 7 m/s | 27 | 25.9 | 25.82 | 25.72 | 25.52 |
7 | 33.65 + 33.63 % | 33.65 | 33.55 | 33.45 | 33.24 | |
9 | 41.55 + 17.83 % | 41.55 | 41.41 | 41.3 | 41 | |
Electromagnetic torque (N-m) | 7 m/s | 14.7 | 14.59 | 14.598 | 14.65 | 14.755 |
7 | 24 - 75 % | 24.1 + 32.78 % | 24.16 + 28.31 % | 24.23 + 13 % | 24.42 + 4.42 % | |
9 | 36 -65.67 % | 36 + 34.03 % | 36.1 + 23.96 % | 36.3 + 11.57 % | 36.5 + 4.11 % | |
Power extracted (W) | 7 m/s | 2000.54 | 2000.65 | 2001.02 | 2002.3 | 2004.75 |
7 | 4252 | 4252 | 4254 | 4256.6 | 4261 | |
9 | 7763 | 7763 | 7767.6 | 7772.2 | 7780 | |
+ Overshoot; - Decay; ± Oscillatory |
Performance index | [22] | Proposed method |
FG | 15.701 | 5.7074 |
J(x) |
Wind turbine parameters | PMSG parameters |
P=3, R=3.3 | |
| |
Parameter | Wind profile | PFL | QFT | MMQFT | AQFT | MAQFT |
Control input ( | 7 m/s | 27 | 25.9 | 25.82 | 25.72 | 25.52 |
7 | 33.65 + 33.63 % | 33.65 | 33.55 | 33.45 | 33.24 | |
9 | 41.55 + 17.83 % | 41.55 | 41.41 | 41.3 | 41 | |
Electromagnetic torque (N-m) | 7 m/s | 14.7 | 14.59 | 14.598 | 14.65 | 14.755 |
7 | 24 - 75 % | 24.1 + 32.78 % | 24.16 + 28.31 % | 24.23 + 13 % | 24.42 + 4.42 % | |
9 | 36 -65.67 % | 36 + 34.03 % | 36.1 + 23.96 % | 36.3 + 11.57 % | 36.5 + 4.11 % | |
Power extracted (W) | 7 m/s | 2000.54 | 2000.65 | 2001.02 | 2002.3 | 2004.75 |
7 | 4252 | 4252 | 4254 | 4256.6 | 4261 | |
9 | 7763 | 7763 | 7767.6 | 7772.2 | 7780 | |
+ Overshoot; - Decay; ± Oscillatory |