## 2 Problem Description and Motives  
In terms of a solar cell, when the sun irradiates the P-N junction semiconductor, the electric energy is generated by the movement of electron holes. The PV panel consists of several solar cells in series-parallel connection, as shown in Figure 1. The output current can be expressed as Equation (1):\[I=n_{p}I_{ph}-n_{p}I_{sat}[\exp(\frac{qV}{kTAn_{s}})-1]\,, \tag{1}\]  
where \(I\) is the output current of PV panel, \(V\) is the output voltage of PV panel, \(q\) is the electric quantity of electron, \(I_{ph}\) is the photocurrent, \(I_{sat}\) is the reverse saturation current, \(T\) is the cell temperature, \(n_{p}\) is the parallel number of PV panels, \(n_{s}\) is the serial number of PV panels, \(k\) is the Boltzmann constant, and \(A\) is the ideal factor coefficient of PN junction. When \(I_{sat}\) and \(I_{ph}\) are sensitive to the cell temperature, this is expressed as Equations (2) and (3):  
\[I_{sat}=I_{r}[\frac{T}{T_{r}}]^{3}\exp(\frac{qE_{g}}{kA}[\frac{1}{T_{r}}-\frac {1}{T}]) \tag{2}\]  
\[I_{ph}=[I_{sc}+k_{sc}(T-T_{r})]\frac{S}{1000}\,. \tag{3}\]  
\(T_{r}\) is the reference temperature at ordinary temperature (25 \({}^{\circ}\)C), \(I_{r}\) is the reverse saturation current of semiconductor at ordinary temperature, \(E_{G}\) is the energy in energy gap, \(I_{sc}\) is the short-circuit current of semiconductor at ordinary temperature, \(k_{sc}\) is the temperature coefficient of short-circuit current, and \(S\) is the illumination. The output power of PV power system can be expressed as \(P=VI\). The nonlinear characteristics of power-voltage (PV) curves in the PV power system are shown in Figure 2.  
In order to reach the maximum point of PV curve, the PV power system implements MPPT so that the voltage and current signals sent from the PV panel are extracted to the MPPT algorithm for calculation. A boost converter is connected, and the duty ratio of the switch is controlled by the voltage and current generated by PV panel for MPPT. The fractional order chaos synchronization dynamic error detector proposed in this paper puts the results of MPPT and the voltage signal from the PV panel into the chaos system for analysis and calculation. The result can control the MPPT method to continue or suspend tracking, so as to reduce the power loss.  
Figure 1: Internal equivalent diagram of PV panel.  
The output power of the PV panel varies with climatic conditions, such as solar illumination and the cell temperature of the PV panel. A diagram of structure used in this paper to reach the MPPT in a PV power system is shown in Figure 3. The voltage and current from the PV panel are connected to the DC-DC boost converter directly [12, 13, 14, 15]. The duty cycle of switching pulse width modulation (PWM) is adjusted by the MPPT and chaos synchronization dynamic error detector. Finally, the DC-DC boost converter is connected to the load, the system output power is measured.  
Many MPPT algorithms have been compared and discussed [8]. The common methods include INC, P&O, voltage feedback method, and power feedback method. Even genetic algorithm and Particle Swarm Optimization have been used. The fractional order incremental conductance method used in this paper is described in [7].