Use of ANN with Levenberg-Marquardt algorithm in ...

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Use of ANN with Levenberg-Marquardt algorithm in the maximum power point tracker of photovoltaic application Saadi .A, Moussi .A Larhyss Laboratory, University Of Mohamed Kheider Biskra, Algeria Institute Of Electrotechnics, University Of Mohamed Kheider Biskra, Algeria E-mail: [email protected] Abstract: This paper consists of the optimisation of the system’s efficiency by using the artificial neural network (ANN), exactly with Levenberg-Marquardt Algorithm. The data bases of ANN regulator are concluded with MPPT technique. The drive off - line ensures the suitable values of the weights and skews for the ANN necessary to the system. To facilitate the work of ANN controller, the execution of learning and recognition programs with a great deal of examples, the choice of the suitable inputs, output and the technique of the back propagation helps us to apply it with a distinctive, simple and highly competent structure. Through the recognition operation, Levenberg-Marquardt Algorithm ensures a good robustness of the control system and a high degree of accuracy. The ANN has proved its success in the level of recognition and was esteemed by 100%. Key Words: Levenberg-Marquardt Algorithm, Back-Boost Converter, MPPT Technique, Robustness and Photovoltaic Pumping System. 1. Introduction In remote regions of Algeria there are a number of small isolated and scattered communities, like Saharian areas and rural villages without access to the grid electricity. Supplying and irrigating water for small areas has remained the preoccupation of the population. To supply electrical power for pumping system, fuel generators are widely used but they suffer high maintenance cost, irregular availability of fuel, and are mostly used to give a limited energy service during few hours per day only. So the employment of photovoltaic water pumping systems is an appropriate solution to supply motors-pump set by electrical power. Algeria is characterised by its good geographical situation, it receives important quantity of solar radiation during the years and seasons [1],[2]. Solar energy is a source with no pollution, no noise and renewable [3]. But, its major problem resides in the initial cost of the system and how to choose the different components with little maintenance [4]. To reduce these problems; two ways can be used. The first is the advancement in the technology of materials used in photovoltaic PV systems starting from the generator materials up to the loads. The second one is the good energy management of the PV systems. These latter are, by nature, non-linear power sources that need accurate on-line identification of the optimal operating point [5]. Also, the power from PV array varies depending on the environmental factors such as solar radiation, cell temperature and the purity of the climate [6]. Aiming to optimise such systems to ensure optimal functioning of the unit; new techniques are used nowadays, such as the maximum point power tracker (MPPT) [7],[8],[9]. Intelligent solutions, based on artificial intelligence (AI) technologies to solve complicated practical problems in various sectors, are becoming more and more practical [10]. The need to meet demanding control requirements in increasingly complex dynamical control of the system under significant uncertainty makes the use of Artificial Neural Networks (ANN) in control very attractive. In this work, the optimising operation of the PV pumping system is based on the optimisation of power, by optimisation of duty cycle of chopper where two programs are used in the network: learning process with Levenberg-Marquardt algorithm was executed, here, and the recognition process verified the robustness of system. 2. PV Pumping Water System Photovoltaic pumping water system constitutes the example of application of this study. It includes primarily the following cascade: a photovoltaic generator, a DC-Dc chopper, a drive system which includes a permanent magnet DC motor driving a centrifugal pump fig.1. In the PV pumping system, the group of motor-pump is considered as the load that must be supplied by a nominal power from the photovoltaic generator. The equation of [11] help to calculate the values of the solar radiation which is near to experimental. In this study, the model chosen is the model of the temperature of junction (model using the NOCT). It considers the ambient temperature and incidental solar radiation on the surface of the photovoltaic module. The I-V characteristics of photovoltaic generator are calculated with equations (1), (2) & (3). (1) I = I g − I sat exp(q (V + R s I ) (AKT))− 1

[

]

Where I is the PV generator current (A), q is the electron charge, V is the generator voltage (V), Rs is the series resistance of the PV generator ( Ω ), A is the ideality factor, K is the Boltzman constant, T is the solar cell temperature, and Ig is the PV generator current under a given solar radiation (A). The relationship is written as follows:

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I g = [I sc + K I (Tc − 25)](λ 100)

(2)

Where λ is the solar radiation (mW/cm ), Isc is the short-circuit current (A) and kI is the short-circuit temperature coefficient. And Isat is the saturation current of PV generator (A). Its relationship is given by: q E GO (3) 3 I = I [T T ] exp (1 T − 1 T ) 2

sat

or

r

KT

r

Where Tr is the reference temperature and EGo is the band-gap energy of the semiconductor. DC/DC converters of different types where used in solar energy system, they deliver the highest power to the load from PV generator with efficiencies about 97%, 93% and 91% for buck, buck-boost and boost chopper respectively [12].

Solar

Load

DC/DC Motor

PV Generator

Duty cycle Isc Voc

pump

ANN

Figure1: Schematics of a photovoltaic pumping system The duty ratio of the chopper ( α ) is given by equation (4) and the chopping ratio (Y) is given by relations (5), with t1 the on-time of chopper switch (s) and T1 is the switching period of chopper (s). t (4) α = 1 T1 Y =

Vm α = V 1− α

(5)

Voltage equation given below: (6) Um = 2R I m + 2k e ω Where R is the armature circuit resistance, Im is the motor current, ke is the Flux coefficient, ω is the motor speed. Torque equation: (7) C m = 2k e I m Where Cm is the electromagnetic developed torque. (8) C r = A c ω2 Where Cr is the hydrodynamic load torque of the centrifugal pump, Ac is the constant of the pump torque. Centrifugal pump equations are (9) & (10): (9) H = C1 ω2 − C 2 ω Q − C 3 Q 2 Where H is the total head (m), Ci is the constants depending on the pump dimensions; Q is the water flow rate (m3/h) (10) H= Hg +∆H Where H g is the geodetic head (m), ∆ H is the pressure losses in the whole canalisation.

3. Artficial Neural Network The structure of an artificial neuron in a neural network i s inspired by the concept of biological neuron. Basically, neuron is a processing element (PE) in nervous system of brain that receives and combines signals from other similar neurons through thousands of input paths called dendrites. Neural networks, or simply neural nets, are computing systems which can be trained to learn a complex relationship between two or many variables or data sets. Basically, they are parallel computing systems composed of interconnecting simple processing nodes. Neural net techniques have been successfully applied in various fields such as function approximation, control systems and signal processing. As in this work, our PV pumping system is considered

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non-linear system due to PV generator. The characteristics (I-V) of this latter are non-linear, and influenced by the changes of the climate (solar radiation and temperature). Thus, the use of ANN is proposed. Neural network structure in this study contains three layers, as shown in fig. 2. Input layer: receives the external data that include two inputs: short circuit current and open circuit voltage of testing photovoltaic module. Hidden layer: contains several hidden neurons which receive data from the input layer Output layer: responds to the system which is the duty cycle of chopper. xi=(x1, x2, x3, …, xn)t , input vector applied to the layer, The whole of the hidden neuron input‘j’ is : n

net hj =

i =1

w ij x i + θhj

With: x i = (x 1 , x 2 , x 3 , ..., x n )

(11) t

(12)

Such as i=1, 2, …, Nh

xi is the input vector applied to the layer, and wij is the weights of (i) input neuron connection, and θ j represent the bias h

of hidden layer neurons. xi=(x1, x2, x3, …, xn)t , , The neurons of hidden layer can be written as follows:

y hj = f (

n i =1

w ij x i + θ hj )

(13)

The output as:

y ko = f (

N

h

j =1

w

jk

y hj + θ ko )

(14)

With θ k : bias of neurons output layer The function and results of artificial neural network are determined by its architecture that has different kinds. The ANNs can generally be classified as feedforward and feedback (or recurrent) types. In a feedforward network, the signals from neuron to neuron flow only in the forward direction; whereas, in a recurrent network, the signals flow in forward as well as backward or lateral direction. The examples of feedforward network are o f Perceptron, Adaline and Madeline, Back propagation network, Radial basis function network (RBFN), General regression network, Modular neural network (MNN), Learning vector quantification (LVQ) network and Probabilistic neural network (PNN). The examples of recurrent neural network (RNN) are Hopfield network, Boltzmann machine, Kohonen’s selforganising feature map(SOFM), Recirculation network and Brain-state-in-a-box (BSB), Adaptive resonance theory (ART) network and Bi-directional associative memory (BAM) [13]. We can conclude unlimited neural network architectures. The more several hidden layers and neurons in each layer are added; the more complex they become. The realisation of the back propagation network is based on two main points: learning and knowledge. Levenberg-Marquardt is an algorithm that trains a neural network faster than the usual gradient descent backpropagation method. It will always compute the approximate Hessian matrix, which has dimensions n × n [14]. o

Hidden layer

y hj wijh

Input layer

wojk Output layer

yko

(xi)

Bias Bias

θ ko

Figure 2: Multilayer perceptron network.

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4. Levenberg – Marquardt Algorithm Many algorithms exist for determining the network parameters. In neural network literature, the algorithms are called learning or teaching algorithms, in system identification they belong to parameter estimation algorithms. The most well-known are back-propagation and Levenberg-Marquardt algorithms. Backpropagation is a gradient based algorithm, which has many variants. Levenberg-Marquardt is usually more efficient, but needs more capable computer memory. Here we will concentrate only on using the algorithms [15],[16],[17]. As for the algorithm quasi-Newton, this algorithm does not require the calculation of the Hessien, it only approximates it.

H = J T .J

(15)

The Jacobien is calculated by:

g = J T .e

(16) Where (J) is the Jacobian matrix, which contains first derivatives of the network errors with respect to the weights and biases, and (e) is a vector of network errors is given by:

e=

1 2

n i =1

(d

o i

− y io

)

2

(17)

o

Where d k is the desired output. The Jacobian matrix can be computed through a standard backpropagation technique that is much less complex than computing the Hessian matrix. The adjust weights as:

ϕ k +1 = ϕ k − [J T J + µ I ] J T e −1

(18)

Where: ϕ = ( w, θ ) and I: identity matrix. When the scalar µ is zero, this is just Newton’s method, using the approximate Hessian matrix. When µ is large, this becomes gradient descent with a small step size. Newton’s method is faster and more accurate near an error minimum, so the aim is to shift towards Newton’s method as quickly as possible. Thus, µ is decreased after each successful step (reduction in performance function) and is increased only when a tentative step would increase the performance function. In this way, the performance function will always be reduced at each iteration of the algorithm. If e ( k ) 1 (19) µ = µ × (µ − inc ) e ( k − 1)

If e ( k ) e ( k − 1)

1

µ

= µ

× (µ − dec )

(20)

Algorithm of Levenberg-Marquadt Stage 1: to initialise the weights and skews by small value random as well as the Parameters: µ = 0,001(default value), µ - dec= 0,1 (default value), µ - Inc = 10(default value), Stage 2: To present the vector of input and desired output. Stage 3: To calculate the output of the network by using the expression (13) & (14). Stage 4: To calculate the error of output (17) Stage 5: To calculate the error in the hidden layers Stage 6: To calculate the gradient of the error compared to the weights Stage 7: To calculate Hessien approximated by using the expression (16). Stage 8: To adjust the weights according to the expression (18). Stage 9: One test: by using the expression (19) & (20). Stage 10: If the condition the error or the iteration count is reached or µ reached µ-max, outward journey at stage 11 if not outward journey at stage 2. Stage 11: End.

5. Discussion The use of MPPT technique is the best solution for the determination of the necessary and desired output of ANN which is the duty cycle of a chopper (Y). The ANN has two inputs: the open circuit voltage and the short circuit current of PV generator. There are many factors and stimuli that play important roles in correctness of the concluded solutions of the synaptic weights for the executed program, and the choice of accelerated, precise algorithm makes the application of ANN a good solution of non-linear system. The optimal values of duty cycle are the calculated output of the ANN. The dynamic (off-line) ANN was applied with a sufficient number of learning examples (1680 examples) and the possible maximum epochs (5000 epoch). To execute the ANN method, we focus on two programs, the first: learning program (with the help of Levenberg-Marquardt algorithm), the second: recognition program (the resulted weights were used to calculate the desired output which are the chopper duty cycles, in other words, recognition program is for verification). Levenberg-Marquardt algorithm is affected by Matlab software package (version 5) with manual network toolbox.

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Fig.3 shows the fast convergence of the Levenberg-Marquardt algorithm where the learning error decreases more than 10-7, especially after 1000epoch. The absolute error between the calculated output and desired output is drawn in Fig.4. The error of duty cycle chopper was about 7.5 × 10-4 in maximum. This figure proves the good learning of the proposed ANN. The absolute recognition error of recognition program is presented in Fig.5 which is about 5 × 10-4 the weak solar radiation, (less than 100 (w/m2)). At this value of solar radiation, the motor is paused, and the PV pumping system does not start. Then the exact recognition error of duty cycle does not exceed 0.25 × 10-3. Through Fig.6 we observe that whenever the number of examples increases, the error decreases to 4.5 × 10-3 the weak solar radiation (less than 100 (w/m2)) and the exact absolute recognition error does not exceed 0.5 × 10-3, this value is the double than one given by 1680 examples case. According to the Fig.7 the relative recognition error is about 0.02% for the average solar radiation. Fig. 8 ensures the precision of ANN, and presents the compatibility of calculated and desired charts of chopper duty cycle.

Figure 3: Learning error of ANN

Figure 4: Learning absolute error between the calculated and desired duty cycle output

Figure 5: Absolute recognition error of duty cycle of chopper (1680 examples)

Figure 6: Absolute recognition error of duty cycle of chopper (366 examples)

Figure 7: Relative recognition error of duty cycle of chopper (1680 examples)

Figure 8: calculated and desired duty cycle output of recognition program

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The calculated duty cycle values were used in calculating chopper output power, and the maximum power is shown in Fig.9. The power of output chopper that supplied the group motor-pump is the maximum power, and the value of relation error power is weak, whereas, it is neglected. And it does not exceed 1.5 × 10-3, as shown in Fig.10. Thus, the efficiency which is the ratio of the power of output chopper, relating to the voltage power is 100% whatever is the value of solar radiation. It is presents in Fig.11. The Fig.12 presents the charts of water quantity pumping system; we observe that the system starts pumping the water at 172.5(w/m2).

Figure 9: Optimal output power of chopper with ANN method

Figure 11: Efficiency of system with ANN method

Figure 10: Relative error on power of chopper

Figure 12: Charts of pumping water quantity of MPPT technique and ANN method

Moreover, the ANN chart is compatible to the MPPT chart, yet in Fig.13 the relative error was about 0.5 × 10-3 % in maximum at high solar radiation values. In the weak values of solar radiation like 213.5(w/m2), the relative error of water quantity is 3.5 × 10-3 %.

Figure 13: Relative error of pumping water quantity

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6. Conclusion The off-line or dynamic artificial neural network with one hidden layer is the simplified architecture. Whatever is the value of solar radiation, Levenberg-Marquardt algorithm proved the good recognition process, good robustness and good adaptation. ANN can also keep its simplicity and easiness to be applied with a high quality, precision and effectiveness. Thus, the ANN has provided a maximum power delivered to motor-pump. In addition, the system has the perfect efficiency, and pumped a maximum quantity of water. References [1] Maafi A. A survey on photovoltaic activities in Algeria. Renewable Energy., 2000,Vol. 20: 9-17. [2] Hadj Arab A; F Chenlo; Benghanem M. Loss-Of-Load Probability Of Photovoltaic Water Pumping Systems., Solar Energy, 2004, 76: 713-723. [3] Benard M. Electricity Generation from Renewable Energy. Renewable Energy., 1998, 15:264 -269. [4] Johan H.R; and al.. Integrated Photovoltaic Power Point Tracking. IEEE trans.on Industrial Elecronics., 1997,vol 44, N°6: 769-773. [5] Moussi A; Saadi A; Asher G. M. Photovoltaic Pumping Systems Trends. OPTIM, Brasov, Romania.,2004,Vol. I: 259-266. [6] Cha,In.Su; and al... MPPT For Temperature Compensation Of Photovoltaic System With Neural Networks. 26thPVSC, IEEE., 1997: 1321-1324. [7] Hua C;Lin J. A modified tracking algorithm for maximum power tracking of solar array.Energy Conversion and Management. , 2004,45: 911-925. [8] Yu,G.J; Jung Y.S; Choi J Y; Kim G.S. A novel two-mode MPPT control algorithm based on comparative study of existing algorithms., Solar Energy ,2004,76: 455-463. [9] Nagayoshi H. I2V curve simulation by multi-module simulator using I2V magnifier circuit. Solar Energy Materials & Solar Cells., 2004, 82:159-167. [10] Jebaraj S; Iniyan S. A review of energy models. Renewable and Sustainable Energy Reviews., 2004, Model 1:131. [11] Abidin Firatoglu Z; Bulent Y. New approaches on the optimization of directlycoupled PV pumping systems. Solar Energy., 2004,77: 81-93. [12] Hua C; Shen C. Control of DC/DC Converter For Solar Energy System With Maximum Power Tracking. IECON’97 3rd New Yorc USA., 1997: 827-832. [13] Bimal Bose K; Life F. Artificial Neural Network Applications In Power Electronics. IECON' 01 The 27th Annual Conference of the IEEE Industrial Electronics Society., 2001: 1631-1638. [14] Fancourt,C.L; Principe,J.C. Optimization in companion search spaces: the case of cross-entropy and the Levenberg-Marquardt algorithm., Acoustics Speech and Signal Processing IEEE, 2000, vol.2: 1317-1320. [15] Lai-Wan C; Chi-Cheong S. Training recurrent network with block-diagonal approximated Levenberg-Marquardt algorithm., Neural Networks International Joint Conference IJCNN, Vol. 3, 1999: 1521-1526. [16] Ampazis N; Perantonis, S.J. Levenberg-Marquardt algorithm with adaptive momentum for the efficient training of feedforward networks. International Joint Conference Neural Networks IJCN., 2000,Vol.1: 126-131. [17] Mujadi E. ANN Based Peak Power Tracking for PV Supplied DC Motors. Solar Energy., 2000,Vol. 69, N°.4: 3403-3410.