# Power Management Strategy for an Autonomous DC Microgrid
## 2 Autonomous DC Microgrid Configuration and Component Description
### Supercapacitor Modelling  
The SC has a high-power density characteristic, which allows it to release a great amount of power instantly [30], but its energy capacity is relatively low. In general, there is no power constraint and SC power \(p_{SC}\) can be positive and negative, which represent discharging power to the DC bus and absorbing power from the DC bus, respectively. Although the SC is made up of many capacitor  
Figure 1: Topology of an autonomous DC microgrid system.  
cells connected in series and parallel, it can be seen as a single capacitor. Therefore, the SC energy \(E_{SC}\), depends on the SC voltage \(v_{SC}\) and the SC capacitance \(C_{SC}\) as given in Equation (2):  
\[E_{SC}=\frac{1}{2}\frac{Q_{SC}^{2}}{C_{SC}}=\frac{C_{SC}v_{SC}^{2}}{2} \tag{2}\]  
where \(Q_{SC}\) is the electric charge stored in the SC.  
Hence, the state of charge of the SC, noted \(SOC_{SC}\), can be calculated as in Equation (3):  
\[SOC_{SC}=\frac{\frac{C_{SC}v_{SC}^{2}}{2}}{\frac{C_{SC}v_{SC}\frac{RATED}{2}} {2}}\times 100\%=\left(\frac{v_{SC}}{v_{SC\_RATED}}\right)^{2}\times 100\% \tag{3}\]  
where \(v_{SC\_RATED}\) is the SC's rated voltage. In addition, it should be noted that as, in general, the studied SC has a self-discharging nature due to its resistive cell balancing, \(SOC_{SC}\) may decrease from 100% to 15% in about 40 h, as shown in Figure 2[27].  
Citing this self-discharging nature and given that the SC has to cover DG start-up, there are four definitions on the constraints of the SC's state of charge: \(SOC_{SC\_MAX\_MAX}\), \(SOC_{SC\_MAX\_MIN}\), \(SOC_{SC\_MIN\_MAX}\), and \(SOC_{SC\_MIN\_MIN}\), which are explained one by one as follows.  
1. \(SOC_{SC\_MAX\_MAX}\) and \(SOC_{SC\_MAX\_MIN}\) define a range in which the SC is considered fully recharged. This is for the purpose of avoiding frequent SC recharging caused by the self-discharging property. At each recharge, the SC is recharged until \(SOC_{SC}\) arrives at \(SOC_{SC\_MAX\_MAX}\), and then the SC is not recharged again unless \(SOC_{SC}\) decreases below \(SOC_{SC\_MAX\_MIN}\).
2. Symmetrically, \(SOC_{SC\_MIN\_MAX}\) and \(SOC_{SC\_MIN\_MIN}\) define a range in which the SC is considered to be empty. While \(SOC_{SC\_MIN\_MIN}\) is the lower limit of \(SOC_{SC}\), \(SOC_{SC\_MIN\_MAX}\) indicates that the SC still has enough energy to support the whole system alone during DG start-up. As such, the SC must be recharged if \(SOC_{SC}\) falls to \(SOC_{SC\_MIN\_MAX}\) due to self-discharging, and in such cases, the recharging power can be so weak that self-discharging is merely compensated.  
Thus, the SC should always have enough energy for DG start-up when there is not enough stored energy in electrochemical batteries. Therefore, as \(SOC_{SC}\) below \(SOC_{SC\_MIN\_MAX}\) becomes prohibited, when \(SOC_{SC}\) reaches this value, the SC is recharged like a load; the recharging power is equal to the self-discharging power, so that the SC's stored energy is maintained until the DG needs starting.  
By keeping \(SOC_{SC}\) between the four critical values, SC should always have enough energy to maintain the microgrid power balance during the start-up of DG.  
Figure 2: Evolution of \(SOC_{SC}\) during supercapacitor (SC) self-discharging, as in Reference [27].