Comparison between Kalman filter and incremental conductance algorithm for optimizing photovoltaic energy
 Saad Motahhir^{1}Email authorView ORCID ID profile,
 Ayoub Aoune^{1},
 Abdelaziz El Ghzizal^{1},
 Souad Sebti^{1} and
 Aziz Derouich^{1}
https://doi.org/10.1186/s4080701700468
© The Author(s) 2017
Received: 11 May 2017
Accepted: 4 December 2017
Published: 16 December 2017
Abstract
The purpose of this paper is to present a performance comparison between two maximum power point tracking algorithms. These two algorithms are incremental conductance (INC) which is an improved version of the perturb and observe algorithm, and the second algorithm is the Kalman filter applied to a photovoltaic system. In this work, a photovoltaic panel is modeled in PSIM tool; a Boost converter controlled by the maximum power point tracker is put between the PV panel and the load. Then the two algorithms are implemented by using C language and C block provided by PSIM tool. Next, several tests under stable and variable environmental conditions are made for the two algorithms, and results show a better performance of the Kalman filter compared to the INC in terms of response time, efficiency and steadystate oscillations.
Keywords
Introduction
The solar energy is now widely used in different areas like in power stations, standalone systems, and rural electrification. One of the technologies used to generate this energy is the PV panel which converts light from the sun into electricity. In order to get the maximum power available from the PV panel, researchers are working on MPPT algorithms (ElKhozondar et al. 2016). Over the last decade, many MPPT algorithms were used in order to find the MPP. These algorithms differ in many aspects such as cost, effectiveness, response time, required sensors, correct tracking in case of temperature or solar irradiance change, and implementation complexity (ElKhozondar et al. 2016). The most commonly used algorithms are fuzzy logic control, neural networks, Perturb and observe, and Incremental Conductance (Verma et al. 2016; Reisi et al. 2013). For fuzzy logic control, it has some advantages like dealing with imprecise inputs, handling nonlinearity, and fast convergence. However, this algorithm requires hardware with high specifications, high on complexity, and the effectiveness of this algorithm depends a lot on choosing the right error computation and an appropriate rule base (Reisi et al. 2013). The neural networks can track the MPP accurately and rapidly. However, this technique presents many disadvantages like the fact that the data needed for the training process have to be specifically acquired for every PV array and location, also the PV characteristics change with time, so the neural network has to be periodically trained (Reisi et al. 2013). On the other hand, P&O and INC are the most used due to their easy implementation and their low requirements in hardware and number of sensors (Motahhir et al. 2016, 2017). However, P&O algorithm presents some disadvantages such as high oscillation around the MPP because the perturbation continuously changes in both directions to maintain the MPP (Motahhir et al. 2016). Therefore, INC algorithm was proposed to minimize the steadystate oscillation by comparing the slope of the P–V curve with zero, theoretically, when the peak of the P–V curve is found, no further perturbation of duty cycle or no oscillation in the output power (Motahhir et al. 2017). However, the algorithm speed is slow when fixed step size is used. Therefore, the variable step size in INC algorithm was proposed in (Liu et al. 2008), which means, the fixed step size is multiplied with the slope of the P–V curve. Thus, the duty cycle step size becomes smaller when the PV system operates near to the MPP (peak of the P–V curve). Meanwhile, the step size is larger and possesses fastertracking speed when the PV system operates far from the MPP.
The remainder of the paper is organized as follows, after the introduction; section two provides a general recall on a PV panel. Section three focuses on presenting the Boost converter and design and implementation of the Kalman filter based MPPT and the INC algorithm. Finally, section four presents the results and discussions.
PV panel model
Characteristics of the MSX60 PV panel
Characteristics  Value 

Maximum power (P _{max})  60 W 
Voltage at P _{max} (V _{mp})  17.1 V 
Current at P _{max} (I _{mp})  3.5 A 
Guaranteed minimum P _{max}  58 W 
Shortcircuit current (I _{sc})  3.8 A 
Opencircuit voltage (V _{oc})  21.1 V 
Temperature coefficient of opencircuit voltage  − (80 ± 10) mV/°C 
Temperature coefficient of shortcircuit current  (0.065 ± 0.015) %/°C 
Temperature coefficient of power  − (0.5 ± 0.05) %/°C 
NOCT^{2}  47 ± 2 °C 
These curves change according to the variation of temperature and solar irradiance, and therefore the position of the MPP changes, this is why we need an algorithm for finding the MPP under any circumstances.
MPPT controller
Boost converter
Incremental conductance algorithm
Kalman filter based MPPT algorithm
Kalman filter design
The Kalman filter is a set of mathematical equations that provides an efficient computational (recursive) solution of the leastsquares method. The filter is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown (Ramchandani et al. 2012). In 1960, Kalman published his famous paper describing a recursive solution to the discrete data linear filtering problem (Kalman 1960). Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation.
The second set of equation is called “the measurement update” or also the “correction state,” and it is used to correct the value predicted during “the time update” step. This set of equations is constituted from three equations presented below (Aoune et al. 2016):
These two steps “the time update” and “the measurement update” are repeated during each iteration which causes noise to reduce and the error covariance to become zero (Aoune et al. 2016).
Design the MPPT algorithm using Kalman filter
In order to find the MPP using the Kalman filter, we need to design this filter to look for the voltage at the MPP. Therefore, based on the P–V curve shown in Fig. 3, the power increases with a gradual positive slope until reaches one optimal point then after that it decreases with a negative slope. Using this analysis, the voltage at MPP can be predicted using Eq. (9) and the priori error covariance is computed by Eq. (10), where A is 1 and B is M.
The measurement update:
Results and discussions
Results comparison between the proposed method and INC algorithm at temperature 298.15 K
Irradiance (W/m^{2})  Kalman filter based MPPT  INC algorithm  

Response time (ms)  Efficiency (%)  Oscillations (W)  Response time (ms)  Efficiency (%)  Oscillations (W)  
1000  5  99.38  0.8  30  96.64  3 
500  4  99.25  0.4  24  96.72  1.6 
800  5  99.23  0.7  29  96.62  2.7 
Results comparison between the proposed method and INC algorithm at irradiance 500 W/m^{2}
Temperature (K)  Kalman filter based MPPT  INC algorithm  

Response time (ms)  Efficiency (%)  Oscillations (W)  Response time (ms)  Efficiency (%)  Oscillations (W)  
298.15  4  99.40  0.4  30  96.98  2 
283.15  4  99.49  0.4  28  96.92  1.6 
290.15  4  99.42  0.45  28  96.96  1.8 
Tests under standard test conditions
INC algorithm
As shown in the figures above, by using the INC algorithm, the system takes 20 ms to stabilize around the MPP with an efficiency of 96.64%. Also, the power ripple is more than 3 W, which leads to high oscillations around the MPP.
Kalman filter based MPPT
Tests under variable solar irradiance
INC algorithm
Kalman filter based MPPT
Tests under variable temperature and fixed solar irradiance (500 W/m^{2})
INC algorithm
Kalman filter based MPPT
Recapitulation
After presenting the results obtained by the INC algorithm and Kalman filter based MPPT, these results are summarized in Tables 2 and 3.

The response time is the time needed by the method to achieve the maximum power.

The efficiency is a factor that defines how close the results obtained to the ideal maximum power.

Oscillations are how much the power oscillates when the MPP is reached.
Based on the results obtained in this work and the criteria cited before, we can see that the proposed method presents better results than the INC algorithm. From the tables above, we notice that the response time of the proposed method is 6 times lower than the INC algorithm response time. Kalman filter based MPPT presents better efficiency around 99.38%, and the oscillations are 5 times lower. These results show that Kalman filter based MPPT presents better performances than INC algorithm. Therefore, the latter can be considered as one of the powerful MPPT algorithms.
Conclusion
In this work, we compared the Kalman filter based MPPT with the INC algorithm in order to prove Kalman efficiency under stable and variable environmental conditions. The results obtained showed a better performance for the proposed method when compared to the INC algorithm. The efficiency obtained by Kalman is better than the obtained by the INC. The oscillations generated by the INC are high compared to Kalman. We can conclude that Kalman filter is better when used it as an MPPT than the INC algorithm. The next stage of our research will be the implementation of the proposed method in a real hardware device especially in a microcontroller or in a DSP. In addition, in our future research, we intend to design the MPPT algorithm using the extended Kalman filter (EKF) which can handle the nonlinearity of the PV characteristics.
List of symbols
 a :

Diode’s ideality factor
 I _{d} :

Diode current (A)
 I :

Panel output current (A)
 I _{s} :

Panel reverse saturation current (A)
 I _{ph} :

Panel photocurrent (A)
 I _{o} :

Output current of the boost converter (A)
 K :

Constant of Boltzmann (J K^{−1})
 N _{s} :

Number of cells connected in series
 q :

Electron charge (C)
 R _{s} :

Series resistance (Ω)
 R _{sh} :

Shunt resistance (Ω)
 R :

Load (Ω)
 T :

Junction temperature (K)
 V :

Panel output voltage (V)
 V _{o} :

Output voltage of the boost converter (V)
Greek letter
 α :

Duty cycle
Abbreviations
 INC:

Incremental conductance
 MPP:

Maximum power point
 MPPT:

Maximum power point tracking
 P&O:

Perturb and observe
 PV:

Photovoltaic
 PWM:

Pulse width modulation
 STC:

Standard test conditions
Declarations
Authors’ contributions
SM modeled the PV system and implemented the incremental conductance algorithm, AA and SM implemented the Kalman Filter based MPPT. SM, AA and AE wrote the paper. SS and AD contributed to reviewing the paper. All authors of this research paper have directly participated in the planning, execution, or analysis of this study. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Funding
The authors declare that they have no funding for the research.
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Authors’ Affiliations
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