## Lead Batteries/KiBa Model (KiBaM)

*Hint: **The KiBa model is still available for backwards compatibility. However, it has been deprecated and its use is no longer recommended, due to its inaccuracy and lack of comparability compared to PerMod. Furthermore, KiBaM is not applicable to lithium ion batteries. It can be enabled in the advanced settings.*

In the KiBaM, batteries are assumed to be a two-tank electrical energy storage device, the first containing readily available electrical energy, the second containing slower chemical energy which is converted into electrical energy at a limited rate.

The model implemented in Polysun is that suggested by Vrettos & Papathanassiou (Operating policy and optimal sizing of a high penetration RES-BESS system for small isolated grids, 2011) which describes energies and performances. Voltage variations are not displayed. The paper (Vrettos, Witzig, Kurmann, Koch, & Andersson, 2013) directly addresses how the model is implemented in Polysun.

According to Vrettos & Papathanassiou (Operating policy and optimal sizing of a high penetration RES-BESS system for small isolated grids, 2011), we work at a constant battery voltage. The available and the chemically bound energy at the end of a time step are given by:

\(E_{1,t + 1} = E_{1,t}\ e^{- k \cdot \mathrm{\Delta}t} + \frac{{(E}_{0,t} \cdot k \cdot c – P) \cdot (1 – e^{- k \cdot \mathrm{\Delta}t})}{k} – \frac{P \cdot c \cdot (k \cdot \mathrm{\Delta}t – 1 + e^{- k \cdot \mathrm{\Delta}t})}{k}\)

\(E_{2,t + 1} = E_{2,t}\ e^{- k \cdot \mathrm{\Delta}t} + E_{0,t}(1 – c) \cdot (1 – e^{- k \cdot \mathrm{\Delta}t}) – \frac{P \cdot (1 – c) \cdot (k \cdot \mathrm{\Delta}t – 1 + e^{- k \cdot \mathrm{\Delta}t})}{k}\)

\(E_{0,t} = E_{1,t} + E_{2,t}\)

where ∆t is the time step duration in hours, E1,t E2,t and E0,t are the available, chemically bound and total electrical energy stored in the battery. P is the charge/discharge power. c=E1,t/E2,t is the capacity ratio parameter. k is the rate constant parameter that corresponds to the rate at which chemically bound energy becomes available for output. Conventionally, in this battery model P is positive during discharging and negative during charging.

The KiBaM also models the maximum charge (Pch,max) and discharge (Pdis,max) power as a function of the stored energy:

\(P_{dis,\max} = \frac{k{\cdot E}_{1,t} \cdot e^{- k \cdot \mathrm{\Delta}t} + E_{0,t} \cdot k \cdot c \cdot \left( 1 – e^{- k \cdot \mathrm{\Delta}t} \right)}{1 – e^{- k \cdot \mathrm{\Delta}t} + c \cdot \left( k \cdot \mathrm{\Delta}t – 1 + \ e^{- k \cdot \mathrm{\Delta}t} \right)}\)

\(P_{ch,max} = \frac{- k \cdot c \cdot E_{\max} + k \cdot E_{1,t}\ {\cdot e}^{- k \cdot \mathrm{\Delta}t} + E_{0,t} \cdot k \cdot c \cdot \left( 1 – e^{- k \cdot \mathrm{\Delta}t} \right)}{1 – e^{- k \cdot \mathrm{\Delta}t} + c \cdot \left( k \cdot \mathrm{\Delta}t – 1 + \ e^{- k \cdot \mathrm{\Delta}t} \right)}\)

where Emax is the nominal battery capacity. With this notation, the State of Charge (SOC) is defined as:

\({SOC}_{t} = \frac{E_{0,t}}{E_{\max}}\)

In Polysun, batteries are connected to the AC side and are equipped with their own inverter. Inverters are depicted by means of simple efficiencies.

The self-discharge of batteries is depicted as a linear decrease in charge.

New battery models can also be added to the catalog. If the c and k parameters are not available, they may be calculated from three discharge curves, each with a constant power load. The calculation may be performed using the Battery Parameter Finder programme^{[1]}.

[1] http://www.umass.edu/windenergy/research.topics.tools.software.kibam.php