# Coordinated Control of Virtual Power Plants to Improve Power System Short-Term Dynamics
## 3 System Modeling  
The power system model considered in this paper includes stochastic processes, which take into account wind and solar generation as well as load variations. Since the proposed coordinated control requires the transmission of remote signals, time delays are also con  
Figure 4: Frequency control of WGs.  
sidered to model the communication system. The resulting overall model of the system is the following set of Stochastic Delay Differential-Algebraic Equations (SDDAEs) [18; 32]:  
\[\dot{x} =\mathbf{f}(\mathbf{x},\mathbf{y},\mathbf{y}_{d},\mathbf{\eta},\mathbf{u})\,,\] \[\mathbf{0} =\mathbf{g}(\mathbf{x},\mathbf{y},\mathbf{y}_{d},\mathbf{\eta},\mathbf{u})\,, \tag{3}\] \[\dot{\mathbf{\eta}} =\mathbf{a}(\mathbf{x},\mathbf{y},\mathbf{\eta})+\mathbf{b}(\mathbf{x},\mathbf{y},\mathbf{\eta}) \,\mathbf{\zeta}\,,\]  
where \(\mathbf{f}\) and \(\mathbf{g}\) are the differential and algebraic equations, respectively; \(\mathbf{x}\) are the state variables, e.g., rotor angles/speeds of synchronous machines, the dynamic states of loads, etc.; \(\mathbf{y}\) are the algebraic variables, e.g., bus frequency and bus voltage magnitudes/phases, and the active power output of generators; \(\mathbf{y}_{d}=\mathbf{y}(t-\tau)\) are the delayed algebraic variables; \(\mathbf{\eta}\) are the stochastic perturbations; \(\mathbf{u}\) are the input variables, e.g., faults and line outages; \(\mathbf{a}\) and \(\mathbf{b}\) are the drift and the diffusion terms of the Stochastic Differential Equation (SDE) respectively and \(\mathbf{\zeta}\) represents the white noise, i.e., the formal time derivative of the Wiener processes. The stochastic models of voltage-dependent load, wind, and solar irradiance are based on [32; 33; 34], respectively. The interested reader can find a detailed description of these stochastic models in these references. The delay included in (3) models communication network issues, e.g., latency, packet dropout and jitter involved in the transmission of the measurements and control signals in the proposed coordinated control of the VPP. Following the requirements of the VPP communication system described in [35; 36], in this paper, the communication delay in (3) is modeled as a Wide-Area Communication (WAC) delay, as follows [37; 38]:  
\[\tau(t)=\tau_{f}+\tau_{p}(t)+\theta(t), \tag{4}\]  
where \(\tau\) is the total communication delay, \(\tau_{f}\) is the fixed constant delay associated with sensors (e.g., Phase-Locked Loop (PLL) or Phasor Measurement Unit (PMU)) and data processing; \(\tau_{p}\) is the transmission delay; and \(\theta\) is the random jitter.  
The transmission delay is formulated as:  
\[\tau_{p}=\tau_{po}+\frac{S}{B}, \tag{5}\]  
where \(\tau_{po}\) is the propagation delay determined by the transmission medium (e.g., optical fiber, WIFI, or WLAN), \(S\) is the packet size of each data, and \(B\) is the transmission channel bandwidth. Finally, the random jitter \(\theta\) in (4) includes the effect of noise, network topology, routing protocol, and background traffic.  
In the following we assume that the frequency signal \(\Delta\omega_{\text{POC}}\) and power injection \(p_{\text{inj}}\) transmitted from meters to the control center are delayed, as follows:  
\[\Delta\omega_{\text{POC},d} =\Delta\omega_{\text{POC}}(t-\tau(t))\,, \tag{6}\] \[p_{\text{inj},d} =p_{\text{inj}}(t-\tau(t))\,,\]  
where \(\Delta\omega_{\text{POC},d}\) and \(p_{\text{inj},d}\) are the delayed measurements, hence \(\mathbf{y}_{d}=[\Delta\omega_{\text{POC},d},p_{\text{inj},d}]\). Please note that all communication systems delays are included in \(\Delta\omega_{\text{POC},d}\) and \(p_{\text{inj},d}\), thus there is no need to model the delay in the signal \(y_{p}\) that is sent to DERs and ESSs.