The State of Charge (SoC) is an important parameter of a battery energy storage system (BESS), and its balance problem is also an issue worth studying in a multi-BESS network. Recently, some researchers have proposed a power allocation method, claiming that as long as the power sharing state and SoC balance state can be obtained in real-time, it can not only maintain supply and demand balance, but also ensure that any BESS will not exit early due to insufficient energy storage. Considering this, we are attempting to design a distributed dynamic average tracking algorithm based on multiagent systems (MASs) with prescribed transient and steady state performance to estimate the power sharing and SoC balance states of a BESSs network with dynamic load in a distributed manner. Two versions of the solution are designed here, where one is event triggered and the other is self triggered. These two schemes ensure the performance of the estimators under intermittent communication to varying degrees, respectively. In each constructed estimation scheme, prescribed performance control (PPC) method is implemented to ensure the expected steady-state and dynamic performance, which is encapsulated in a performance function. Thus, it achieves almost zero error estimation of the fast time-varying power sharing and the SoC balance states. In addition, consensus and average tracking performance are decoupled, which provides convenience for parameters tuning. Finally, to verify the results of the theoretical analysis, some cases are studied and relevant simulations are performed on a 4 bus system.