The rapid integration of renewable energy sources, characterized by inherent volatility and uncertainty, poses significant challenges to power system frequency stability. Secondary frequency regulation (SFR), executed through Automatic Generation Control (AGC) signals, is crucial for maintaining grid frequency within nominal limits. Traditional thermal power units, while being the primary workhorse for SFR, exhibit inherent limitations such as slow response, overshoot, and delay due to the large mechanical inertia of boiler-turbine systems. This paper, from my research perspective, focuses on modeling a synergistic framework where a battery energy storage system (BESS) assists a thermal power unit in providing fast and accurate SFR, thereby compensating for the thermal unit’s deficiencies and enhancing overall grid stability.
The core of this synergistic control strategy lies in accurate dynamic models of both the thermal power unit and the battery energy storage system. An effective model must capture the essential dynamics to facilitate optimization of power dispatch between the two assets. My modeling approach begins with a detailed representation of the thermal unit, followed by the development and parameter identification of an advanced battery model.
Modeling of the Thermal Power Unit for Secondary Frequency Regulation
The thermal power unit model for SFR must encompass the coordinated dynamics of the boiler, turbine, and control systems. It is constructed based on mechanism modeling principles for a 660 MW unit. The model integrates four key subsystems: the Coordination Control System (CCS), the boiler and its master controller, the turbine master controller and governor, and the turbine-generator model. Crucially, this model accounts for the influence of boiler main steam pressure on turbine output, a dynamic often oversimplified in grid-side studies.
The overall structure of the thermal unit model is governed by the following block diagram relationship. The AGC signal, combined with the speed deviation feedback, forms the unit load demand. This demand is processed by the CCS, which generates commands for both the boiler firing rate and the turbine valve opening to achieve coordinated power output while maintaining steam pressure.
1.1 Boiler and Its Master Controller Model
For a once-through boiler, the dynamics of the heating, evaporation, and superheating sections are lumped into equivalent energy storage elements. The model incorporates combustion delay, heat transfer in the water walls, and the storage effects in the evaporator and superheater. The boiler master controller typically employs a PID regulator to maintain main steam pressure (PT) at its setpoint (PT0) by adjusting the boiler firing command (BD). The key dynamic equations governing the boiler model can be summarized by the following transfer functions and parameters.
The combustion and heat release process is modeled with a time delay and lag:
$$ \text{Fuel Rate}(s) = \frac{e^{-T_D s}}{T_F s + 1} \cdot \text{BD}(s) $$
The evaporation and superheating dynamics are represented as:
$$ \text{Steam Flow}, m_s(s) = \left( \frac{1}{T_w s + 1} + \frac{C_Z}{C_{SH} s + K} \right) \cdot \text{Fuel Rate}(s) $$
$$ \text{Main Steam Pressure}, P_T(s) = \frac{1}{C_{SH} s + K} \cdot \text{Fuel Rate}(s) + \frac{1}{C_{SH} s + K} \cdot (m_s(s) – CV(s)) $$
where \( CV \) is the turbine valve opening command.
| Symbol | Parameter Description | Value |
|---|---|---|
| \(T_F\) | Combustion Time Constant (s) | 30 |
| \(T_w\) | Water Wall Time Constant (s) | 7 |
| \(T_D\) | Pulverizing Delay (s) | 40 |
| \(C_Z\) | Evaporation Storage Coefficient | 180 |
| \(K\) | Superheat Coefficient | 3.5 |
| \(C_{SH}\) | Superheater Storage Coefficient | 33 |
| \(K_{P-b}\) | Boiler Controller Proportional Gain | 1 |
| \(K_{I-b}\) | Boiler Controller Integral Gain | 0.002 |
1.2 Turbine Master Controller and Governor Model
The turbine master controller receives the error between the unit load demand and the actual generator electrical power. After PID regulation, it outputs the turbine valve opening demand (TD). This demand is sent to the Digital Electro-Hydraulic (DEH) control system, which actuates the governor valves. The model includes the speed governor droop characteristic, actuator time constants, and measurement filters. The governing equations involve:
Speed deviation contribution: \( \Delta P_{\omega} = \frac{K_F}{T_{\omega} s + 1} \cdot \Delta \omega(s) \)
Load reference: \( P_{ref} = AGC + \Delta P_{\omega} \)
Valve command: \( TD(s) = \left( K_{P-t} + \frac{K_{I-t}}{s} + K_{D-t}s \right) \cdot (P_{ref}(s) – P_e(s)) \)
Valve actuation: \( CV(s) = \frac{1}{T_{sm} s + 1} \cdot TD(s) \)
| Symbol | Parameter Description | Value |
|---|---|---|
| \(K_F\) | Inverse of Speed Droop | 20 |
| \(T_{\omega}\) | Speed Measurement Time Constant (s) | 0.02 |
| \(T_{sm}\) | Actuator Time Constant (s) | 1 |
| \(K_{P-t}\) | Turbine Controller Proportional Gain | 2 |
| \(K_{I-t}\) | Turbine Controller Integral Gain | 0.2 |
1.3 Turbine Model
The turbine model represents a multi-stage (High Pressure, Intermediate Pressure, Low Pressure) steam turbine with reheater. An improved model is adopted that includes overshoot coefficients for the HP and IP sections to more accurately capture the dynamic over-shooting behavior during rapid valve movements. The mechanical power output \(P_M\) is a function of the valve opening \(CV\) and the main steam pressure \(P_T\).
$$ P_M(s) = \left[ \frac{\lambda_g \alpha_1}{T_{p1} s + 1} + \frac{\lambda_z \alpha_2}{(T_{p1} s + 1)(T_{p2} s + 1)} + \frac{\alpha_3}{(T_{p1} s + 1)(T_{p2} s + 1)(T_{p3} s + 1)} \right] \cdot (CV(s) \cdot P_T(s)) $$
where \(\lambda_g\) and \(\lambda_z\) are the HP and IP overshoot coefficients, respectively.
| Symbol | Parameter Description | Value |
|---|---|---|
| \(T_{p1}\) | Steam Chest Time Constant (s) | 0.7 |
| \(T_{p2}\) | Reheater Time Constant (s) | 18 |
| \(T_{p3}\) | Cross-over Pipe Time Constant (s) | 0.5 |
| \(\alpha_1, \alpha_2, \alpha_3\) | HP, IP, LP Power Fractions (%) | 30, 30, 40 |
| \(\lambda_g\) | HP Overshoot Coefficient | 1.1 |
| \(\lambda_z\) | IP Overshoot Coefficient | 1.2 |
1.4 Model Simulation and Validation
The integrated thermal unit model was simulated in MATLAB/Simulink. A per-unit system based on 50 Hz and 660 MW was used. An AGC step signal, rate-limited at 3% Pn/min to respect the unit’s ramp rate constraint, was applied. The simulation results confirm the model’s validity but clearly illustrate the classic limitations of thermal units: a significant delay in the initial response, pronounced overshoot, and a slow settling time to reach the new setpoint. This performance gap motivates the integration of a fast-responding battery energy storage system.
Modeling of the Battery Energy Storage System
To accurately represent the dynamic behavior of the battery energy storage system during high-rate charge/discharge cycles typical of frequency regulation, a second-order RC equivalent circuit model is employed. This model offers a favorable balance between complexity and accuracy, capturing both the ohmic loss and the short-term/long-term polarization dynamics better than a simple first-order model.
2.1 Model Structure and Parameter Identification
The second-order RC model consists of an open-circuit voltage source \(U_{oc}\), an ohmic internal resistor \(R_0\), and two RC parallel networks (\(R_1\)–\(C_1\), \(R_2\)–\(C_2\)) representing polarization effects. The terminal voltage \(U\) under a load current \(I\) is given by:
$$ U(s) = U_{oc}(s) – \left( R_0 + \frac{R_1}{R_1 C_1 s + 1} + \frac{R_2}{R_2 C_2 s + 1} \right) I(s) $$
The time-domain equation can be discretized for parameter identification. The relationship between the voltage difference and current is linear in parameters, allowing for system identification techniques.
To prevent the “data saturation” phenomenon common in recursive least squares algorithms, where old data continuously diminishes estimation agility, the Limited Memory Least Squares (LMLS) method is used. LMLS maintains a fixed-length data window, discarding the oldest data point when a new one is added, thus ensuring the estimator remains responsive to recent system behaviors. Using Hybrid Pulse Power Characterization (HPPC) test data for a lithium-ion battery, the parameters \(R_0, R_1, C_1, R_2, C_2\) are identified at different State-of-Charge (SOC) levels. The recursive LMLS algorithm proceeds as follows for a memory length \(L\):
Define the parameter vector \(C = [k_1, k_2, k_3, k_4, k_5]^T\) from the discretized model. For each new data point at time \(k+L\), with measurement \(y(k+L) = U_{oc}(k+L) – U(k+L)\) and regression vector \(H_k\), the standard RLS update is first performed:
$$ \begin{aligned}
K_a &= P H_k / (1 + H_k^T P H_k) \\
C_a &= C + K_a (y(k+L) – H_k^T C) \\
P_a &= (I – K_a H_k^T) P
\end{aligned} $$
Then, to remove the influence of the oldest data point in the window (at time \(k\)), a second update is performed using its regression vector \(H_L\):
$$ \begin{aligned}
K_b &= P_a H_L (1 – H_L^T P_a H_L) \\
C_b &= C_a – K_b (y(k) – H_L^T C_a) \\
P_b &= (I + K_b H_L^T) P_a
\end{aligned} $$
The final parameter estimate for the window is \(C = C_b\) and \(P = P_b\). The circuit parameters are then derived from the identified \(k_i\) coefficients. The identified parameters across the SOC range are summarized below, showing relative stability in the mid-SOC range (20%-80%).
| SOC Range (%) | \(R_0\) (Ω) | \(R_1\) (Ω) | \(C_1\) (F) | \(R_2\) (Ω) | \(C_2\) (F) |
|---|---|---|---|---|---|
| 10-20 | 0.0253 | 0.0078 | 69.58 | 0.2658 | 2764 |
| 20-30 | 0.0284 | 0.0077 | 72.84 | 0.0780 | 2630 |
| 30-40 | 0.0276 | 0.0076 | 63.28 | 0.0288 | 1725 |
| 40-50 | 0.0260 | 0.0083 | 47.40 | 0.0327 | 1494 |
| 50-60 | 0.0252 | 0.0087 | 41.72 | 0.0301 | 1437 |
| 60-70 | 0.0247 | 0.0091 | 35.73 | 0.0325 | 1497 |
| 70-80 | 0.0245 | 0.0096 | 36.38 | 0.0255 | 1281 |
2.2 Model Verification
The accuracy of the identified battery model is verified by comparing its simulated terminal voltage under a dynamic current pulse profile with the experimental data. Using the parameters identified at 50% SOC, the model’s output is calculated. The average relative error \(\delta\) is used as the evaluation metric:
$$ \delta = \frac{1}{k_2 – k_1 + 1} \sum_{i=k_1}^{k_2} \frac{|U^*(i) – U(i)|}{U^*(i)} $$
where \(U^*(i)\) and \(U(i)\) are the experimental and simulated voltages, respectively. The calculated error within the 20%-80% SOC range is remarkably low at 0.0524%, confirming the high precision of the second-order RC model identified via the LMLS method. Furthermore, a simulation of the battery energy storage system responding alone to a per-unit AGC step command shows its output perfectly tracking the signal with virtually no delay or overshoot, highlighting its superior dynamic performance for SFR compared to the thermal unit.
Coupled Model for BESS-Assisted Secondary Frequency Regulation and Simulation
The integrated system model couples the thermal power unit and the battery energy storage system at the plant level. The control architecture is designed such that the total AGC command from the grid dispatch center is received by the plant’s master controller. This controller then allocates portions of the power regulation command to the thermal unit’s CCS and the battery energy storage system‘s power management system (PMS) according to a predefined strategy. Their combined output forms the plant’s total response to the AGC signal. This structure allows the battery energy storage system to augment the thermal unit’s response without modifying the unit’s internal control logic.
The simulation of the coupled model involves a 660 MW thermal unit and a 40 MW battery energy storage system. Two case studies are conducted to validate the synergistic benefits.
Case 1: Step Response. A step-change AGC signal is shared proportionally between the two resources. The results demonstrate a significant improvement over the thermal-only response. The overshoot is reduced from 34.98% to 19.62%, the undershoot from 35.5% to 24.73%, and the settling time to reach the target is shortened from 1155 seconds to 970 seconds. The instantaneous response of the battery energy storage system effectively compensates for the initial delay of the thermal unit.
Case 2: Composite Signal Response. A more realistic, composite AGC signal is formed by superimposing a high-frequency, low-amplitude component (Wave I: 0.03 pu magnitude, 0.1 pu frequency) on a low-frequency, high-amplitude component (Wave II: 0.2 pu magnitude, 0.004 pu frequency). An optimal allocation strategy is applied: the fast battery energy storage system is tasked with tracking the high-frequency Wave I, while the thermal unit handles the slower, larger Wave II. The simulation confirms that the battery energy storage system accurately follows the high-frequency fluctuations that the thermal unit, due to its ramp rate limits, cannot. This synergy prevents unnecessary wear and tear on the thermal unit from frequent small adjustments while ensuring precise overall tracking of the AGC signal.
The mathematical representation of the power allocation can be generalized as:
$$ P_{AGC}(t) = P_{th}^*(t) + P_{bess}^*(t) $$
where \(P_{th}^*(t)\) and \(P_{bess}^*(t)\) are the optimal power setpoints for the thermal unit and the battery energy storage system, respectively, determined by a real-time optimization that considers their dynamic constraints, such as the thermal unit’s ramp rate \(RR_{th}\) and the battery’s state-of-charge \(SOC(t)\) and power rating \(P_{bess}^{max}\).
$$ \text{subject to: } \left| \frac{dP_{th}^*(t)}{dt} \right| \leq RR_{th}, \quad SOC_{min} \leq SOC(t) \leq SOC_{max}, \quad |P_{bess}^*(t)| \leq P_{bess}^{max} $$
Conclusion
This modeling study successfully establishes a detailed framework for analyzing battery energy storage system-assisted secondary frequency regulation by thermal power plants. The key contributions are the development of a mechanism-based thermal unit model that incorporates critical boiler-turbine pressure dynamics and the creation of a high-fidelity, parameter-identified second-order RC model for the battery energy storage system using the Limited Memory Least Squares method. The coupling of these models demonstrates quantitatively how the integration of a battery energy storage system can mitigate the inherent limitations of thermal units—specifically, reducing overshoot/undershoot, shortening settling time, and enabling precise tracking of high-frequency load components. This synergy enhances grid frequency stability and can reduce mechanical stress on thermal plants. Future work will focus on developing real-time optimal power dispatch algorithms that explicitly consider the degradation and lifetime costs of the battery energy storage system during frequent regulation duty cycles, ensuring the economic viability of such hybrid frequency regulation assets.