With the rapid integration of renewable energy sources into modern power systems, the replacement of conventional synchronous generators has led to a significant reduction in system inertia and frequency regulation capability. The unpredictable and intermittent nature of wind and solar power introduces large active power imbalances, making frequency stability a critical concern. Among the various frequency regulation measures, primary frequency regulation (PFR) offers the fastest response and is essential for mitigating large and rapid frequency deviations. Energy storage systems, particularly lithium-ion batteries, have emerged as a promising solution to assist conventional thermal units in PFR due to their fast response, high power density, and operational flexibility. In this thesis, I focus on the adaptive coordinated control of lithium-ion battery energy storage participating in PFR, the insufficient regulation capability caused by extreme state of charge (SOC), and the SOC recovery strategy during frequency regulation. The research is conducted through modeling, control design, and simulation verification using MATLAB/Simulink.
Keywords: primary frequency regulation; energy storage battery; lithium-ion battery; fuzzy control; SOC recovery; virtual inertia; virtual droop
1 Introduction
The global energy landscape is undergoing a profound transformation. The “Carbon Peak and Carbon Neutrality” targets have accelerated the deployment of renewable energy sources, such as wind and photovoltaic systems. By 2050, it is expected that non-fossil energy will account for 50% of the primary energy consumption in China. However, the large-scale integration of renewable energy introduces challenges to system stability, including frequency, voltage, and rotor angle issues. Unlike conventional synchronous generators, renewable energy sources are mostly connected through power electronic converters, which provide no or very little inherent inertia and frequency support. Consequently, the frequency regulation burden on traditional thermal power units has increased significantly. Meanwhile, thermal units have a slow response rate and limited ramping capability, which makes it difficult for them to handle the rapid and large power fluctuations caused by renewable variability.
Energy storage technology, especially electrochemical storage, has become a key technology to address the above challenges. The energy storage battery offers fast bidirectional power regulation, high precision, and short response time. It can be installed at the generation side, transmission side, or demand side to provide auxiliary services such as frequency regulation, peak shaving, and voltage support. Among various types of energy storage, the lithium-ion battery stands out due to its high energy density, high conversion efficiency, long cycle life, and decreasing cost. In many countries, lithium-ion battery systems have been widely deployed for frequency regulation, either independently or in coordination with conventional units.
This thesis aims to investigate the control strategies for a lithium-ion battery energy storage battery participating in PFR of a power grid. The main contributions and organizational structure are as follows:
- Chapter 2 compares different energy storage technologies and selects lithium-ion battery as the storage carrier. The basic parameters, equivalent circuit model, and battery stack cluster are established.
- Chapter 3 analyzes the PFR mechanism and establishes the frequency-domain models of thermal units and energy storage batteries. A single-area thermal-storage combined PFR model is built in MATLAB/Simulink.
- Chapter 4 proposes an adaptive weight factor coordination control strategy based on fuzzy logic, combined with a nonlinear unit regulation power coefficient using a logistic function, and verifies the effectiveness through simulation.
- Chapter 5 introduces an SOC recovery control strategy for the energy storage battery, balancing the SOC recovery demand and the grid frequency constraint, and finally demonstrates its positive effect.
2 Energy Storage Battery Selection and Modeling
2.1 Comparison of Energy Storage Technologies
In order to select the most suitable energy storage technology for PFR, I first evaluated several mainstream storage technologies from both technical and economic perspectives. Table 1 lists the technical characteristics of different storage types, including lead-acid batteries, lithium-ion batteries, sodium-sulfur batteries, flow batteries, pumped hydro, flywheel, superconducting magnetic energy storage (SMES), and supercapacitors.
| Storage type | Technology maturity | Power rating | Cycle life | Efficiency (%) | Response time | Duration |
|---|---|---|---|---|---|---|
| Lead-acid battery | Mature | 0–20 MW | 500–1000 | 62–80 | s | 1 min–10 h |
| Lithium-ion battery | Fairly mature | 1–10 MW | 1000–10000 | 90–98 | s | 1 min–8 h |
| Sodium-sulfur battery | Fairly mature | 1 kW–10 MW | 4500 | 75–90 | s | 1 min–1 h |
| Flow battery | Fairly mature | 1–10 MW | 12000 | 63–82 | s | 1–24 h |
| Pumped hydro | Mature | 100–2000 MW | 40–60 years | 70–85 | min | 4–10 h |
| Flywheel | Fairly mature | 0–250 MW | 20000 | 77–95 | 10 ms | 15 s–15 min |
| SMES | Fairly mature | 10 kW–10 MW | 10000 | 90–96 | ms | 5 s–5 min |
| Supercapacitor | Fairly mature | 0–300 kW | 100000 | 70–80 | ms | 1 s–1 min |
Table 2 presents the economic indicators, including power cost and capacity cost.
| Storage type | Power cost (USD/kW) | Capacity cost (USD/kWh) |
|---|---|---|
| Lead-acid battery | 300–600 | 200–400 |
| Lithium-ion battery | 1200–4000 | 600–2500 |
| Sodium-sulfur battery | 3200–4000 | 445–555 |
| Flow battery | 3000–3400 | 750–830 |
| Pumped hydro | 600–2000 | 5–100 |
| Flywheel | 250–300 | 500–1000 |
| SMES | 200–300 | 1000–10000 |
| Supercapacitor | 100–300 | 300–2000 |
To quantify the applicability for PFR, I defined an evaluation standard matrix, as shown in Table 3. Each indicator was scored from 1 to 4, with higher scores representing better performance.
| Indicator | Score 1 | Score 2 | Score 3 | Score 4 |
|---|---|---|---|---|
| Maturity | Concept stage | Developing | Fairly mature | Mature |
| Power rating (MW) | <0.5 | 0.5–1 | 1–10 | >10 |
| Efficiency (%) | <80 | 80–85 | 85–90 | 90–100 |
| Response time | h | min | 10 s | s |
| Max duration | <15 min | 15 min–1 h | 1–2 h | >2 h |
| Cycle life (thousands) | <3 | 3–5 | 5–10 | >10 |
| Power cost (USD/kW) | >1000 | 500–1000 | 250–500 | <250 |
| Capacity cost (USD/kWh) | >3000 | 1000–3000 | 500–1000 | <500 |
I set the weights of technical and economic indicators to 0.6 and 0.4, respectively. Within the technical category, the weights of maturity, power rating, response time, duration, efficiency, and cycle life were 0.1, 0.2, 0.3, 0.1, 0.1, and 0.2, respectively. Within the economic category, power cost and capacity cost each had a weight of 0.5. The comprehensive scores are shown in Table 4.
| Storage type | Technical score | Economic score | Comprehensive score |
|---|---|---|---|
| Lead-acid battery | — | — | 3.04 |
| Lithium-ion battery | — | — | 3.44 |
| Sodium-sulfur battery | — | — | 2.82 |
| Flow battery | — | — | 3.06 |
| Pumped hydro | — | — | 3.28 |
| Flywheel | — | — | 2.54 |
| SMES | — | — | 2.78 |
| Supercapacitor | — | — | 2.42 |
The results reveal that the lithium-ion battery achieves the highest comprehensive score of 3.44, which is mainly attributed to its high conversion efficiency, fast response, long cycle life, and relatively low capacity cost in large-scale applications. Therefore, I selected the lithium-ion battery as the energy storage battery for the PFR system.
2.2 Lithium-ion Battery Selection
Among different cathode materials for lithium-ion batteries, I chose lithium iron phosphate (LiFePO₄) because of its excellent safety, long cycle life, good high-temperature performance, and abundant raw materials. Table 5 compares the performance indicators of several lithium-ion battery types.
| Performance indicator | LiCoO₂ | LiNiCoMnO₂ | LiMn₂O₄ | LiFePO₄ |
|---|---|---|---|---|
| Crystal structure | Layered | Layered | Spinel | Olivine |
| Specific capacity (mAh/g) | 140–155 | 130–220 | 90–120 | 130–150 |
| Voltage platform (V) | 3.6–3.7 | 3.6–3.7 | 3.5–3.8 | 3.2–3.6 |
| Working voltage (V) | 3.0–4.3 | 3.0–4.36 | 3.5–4.3 | 2.5–4.2 |
| Minimum cycle life | 300 | 800 | 500 | 2000 |
| High-temperature performance | Fair | Fair | Poor | Good |
| Environmental performance | Contains Co | Contains Ni, Co | Non-toxic | Non-toxic |
| Safety performance | Poor | Good | Good | Excellent |
| Application field | Small batteries | Small batteries | Power batteries | Power batteries and large-scale storage |
2.3 Basic Parameters of the Energy Storage Battery
The key parameters of the energy storage battery include open-circuit voltage (OCV), internal resistance, single-cell capacity, charge/discharge rate (C-rate), state of charge (SOC), depth of discharge (DOD), and state of health (SOH).
The SOC represents the remaining capacity ratio, which is crucial for control strategy. I used the ampere-hour integration method to estimate SOC in real time:
$$SOC(t)=SOC_0-\frac{1}{C_e}\int_0^t i(\tau) d\tau$$
where SOC0 is the initial SOC, i is the battery current (positive for discharge), Ce is the total charge under standard discharge conditions.
The DOD is defined as:
$$DOD = 100\% – SOC$$
Moreover, DOD is related to the output power and the unit regulation power of the energy storage battery:
$$DOD = \frac{\Delta f}{C_{bN}}(K_{E}^{M}+K_{E}^{P})$$
where CbN is the rated capacity, KEM and KEP are the unit regulation powers under virtual inertia control and virtual droop control, respectively.
SOH can be defined based on internal resistance or capacity:
$$SOH_R = \frac{R_{EOL}-R}{R_{EOL}-R_{BOL}}\times100\%$$
$$SOH_C = \frac{C_{EOL}-C}{C_{EOL}-C_{BOL}}\times100\%$$
where subscripts EOL and BOL denote end-of-life and beginning-of-life values.
2.4 Equivalent Circuit Models and Battery Stack Cluster
Equivalent circuit models (ECMs) are widely used to describe the external characteristics of the energy storage battery. I compared four common ECMs: Rint, Thevenin, PNGV, and second-order Thevenin. The second-order Thevenin model provides a good trade-off between accuracy and complexity, so I adopted it in this thesis. Its port voltage is given by:
$$U_L(t)=U_{OC}(t)-I(t)R_0-U_{p1}(t)-U_{p2}(t)$$
where UOC is the OCV, R0 is the ohmic resistance, and Up1, Up2 are the polarization voltages of the two RC networks. The transfer function in the frequency domain is:
$$\frac{U_L(s)-U_{OC}(s)}{I(s)}=-\left(R_0+\frac{R_{p1}}{1+sR_{p1}C_{p1}}+\frac{R_{p2}}{1+sR_{p2}C_{p2}}\right)$$
For large-scale applications in power system frequency regulation, the energy storage battery is assembled into stacks and clusters. I connected n1 cells in series to form a battery module, then n2 modules in series to form a battery cluster, and finally m clusters in parallel to form the battery stack. The total number of cells is N=n1×n2×m. The rated voltage and capacity of the stack are calculated as:
$$U_{pack}=\frac{n_1 n_2 U_{battery}}{1000} \text{ kV}, \quad Q_{pack}=\frac{N Q_{battery} U_{battery}}{1000} \text{ MWh}$$
where Ubattery is the single-cell voltage in volts and Qbattery is the single-cell capacity in ampere-hours.
3 Modeling of Thermal-Storage Combined Primary Frequency Regulation
3.1 PFR Principles
The frequency characteristics of power systems depend on both load and generation. The load frequency characteristic is approximated by a linear relationship when the frequency deviation is small:
$$\Delta P_D = K_D \Delta f$$
where KD is the load frequency regulation coefficient. Similarly, the generator frequency characteristic is:
$$\Delta P_G = -K_G \Delta f$$
where KG is the generator unit regulation power. The combined system frequency deviation after a load change ΔPL is:
$$\Delta f = -\frac{\Delta P_L}{K_D + K_G} = -\frac{\Delta P_L}{K_S}$$
where KS=KD+KG is the system unit regulation power. This is the fundamental mechanism of PFR: the speed governors of generators automatically adjust their output according to the frequency deviation.
3.2 Thermal Power Unit Frequency-Domain Model
The thermal power unit model is composed of a governor, a steam turbine (reheat type), and a generator-load block. The governor transfer function is:
$$G_{gov}(s)=\frac{\Delta P_v(s)}{\Delta P_e(s)}=\frac{1}{1+sT_g}$$
where Tg is the governor time constant. A frequency dead band of ±0.033 Hz is used, and a power output limiter is also included. The reheat steam turbine model considers the high-pressure, intermediate-pressure, and low-pressure stages:
$$G_{gen}(s)=\frac{\Delta P_m(s)}{\Delta Y(s)}=\frac{F_{HP}}{1+sT_{CH}}+\frac{F_{IP}}{(1+sT_{CH})(1+sT_{RH})}+\frac{F_{LP}}{(1+sT_{CH})(1+sT_{RH})(1+sT_{CO})}$$
where TCH, TRH, and TCO are the time constants of the high-pressure, reheater, and low-pressure volumes, respectively; FHP, FIP, and FLP are the power fractions of each stage.
The generator-load model is derived from the swing equation:
$$\Delta P_m(s)-\Delta P_e(s)=Ms\Delta \omega(s)$$
$$\Delta P_e(s)=\Delta P_L(s)+D\Delta \omega(s)$$
Combining these gives:
$$\Delta \omega(s)=\frac{\Delta P_m(s)-\Delta P_L(s)}{Ms+D}$$
The complete thermal unit model is built by connecting the above blocks with appropriate feedback and limiters.
3.3 Energy Storage Battery Model for PFR
For frequency regulation studies, the energy storage battery can be simplified to a first-order inertial model:
$$G_e(s)=\frac{1}{1+sT_e}$$
where Te is the battery response time constant, typically 0.05 s or less. The actual power output is limited by a maximum power constraint. The charging and discharging efficiencies are considered by a factor η. The SOC calculation block is:
$$SOC(t)=SOC_0-\frac{1}{3600 E_{max}}\int P_b(t)dt$$
where Emax is the maximum stored energy in MWh and Pb is the battery power in MW.
3.4 Fire-Storage Combined PFR Simulation Model
I built the single-area thermal-storage combined PFR model in MATLAB/Simulink, as shown conceptually. The system nominal frequency is 50 Hz, the thermal unit capacity is 600 MW, and the energy storage battery capacity is 1 MWh. The frequency dead band for the thermal unit is ±0.033 Hz, while the storage dead band is set to 60% of that value to allow the storage to respond first. Table 6 lists the simulation parameters.
| Parameter | Value |
|---|---|
| M (inertia constant) | 10 |
| D (damping coefficient) | 1 |
| K_G (thermal unit regulation power) | 20 |
| K_{E,max} (max storage regulation power) | 20 |
| T_g (governor time constant) | 0.2 s |
| T_CH (HP volume time constant) | 0.3 s |
| T_RH (reheater time constant) | 10 s |
| T_CO (LP volume time constant) | 0.8 s |
| F_HP, F_IP, F_LP | 0.3, 0.3, 0.4 |
| T_e (storage time constant) | 0.05 |
| η (efficiency) | 0.95 |
I first performed a step load disturbance of 0.02 p.u. at t=1 s. The initial SOC was 0.7. I compared two schemes: without storage (only thermal) and with storage using a constant regulation coefficient (fixed-K method). The frequency deviation curves are compared. The key performance indices are the maximum frequency deviation Δfm, the time to maximum deviation tm, the steady-state deviation Δfss, and the average rate of frequency decline Vm, defined as:
$$V_m=\frac{\Delta f_m-\Delta f_0}{t_m-t_0}$$
Table 7 lists the results.
| Scheme | Δf_m (×10⁻³ Hz) | t_m (s) | Δf_ss (×10⁻³ Hz) | V_m (×10⁻³ Hz/s) |
|---|---|---|---|---|
| Without storage | -2.412 | 3.442 | -0.952 | 0.988 |
| Fixed-K | -1.464 | 3.152 | -0.572 | 0.680 |
It is clear that the fixed-K storage significantly improves the PFR performance: the maximum deviation is reduced by 39.3%, the steady-state deviation is reduced by 39.9%, and the rate of frequency decline is reduced by 31.2%.
For continuous load disturbance, I used the load curve as shown in Figure 3.16 (not reproduced here for brevity) and simulated for 600 s. The frequency deviation root mean square (RMS) was used as an evaluation metric:
$$\Delta f_{rms}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\Delta f_i^2}$$
Table 8 shows that the fixed-K method reduces Δfrms by 19.9% compared to the no-storage case.
| Scheme | Δf_rms (×10⁻³ Hz) | ΔSOC_rms |
|---|---|---|
| Without storage | 2.913 | — |
| Fixed-K | 2.331 | 0.6602 |
4 Control Strategy for the Energy Storage Battery
4.1 Frequency Response Characteristics
There are two fundamental frequency response characteristics in power systems: inertia response and droop response. The inertia response describes the immediate power response from the rotating mass of synchronous generators, which reduces the rate of change of frequency (RoCoF). The droop response is the static response that adjusts the power output in proportion to the frequency deviation to restore the frequency steady-state value.
4.2 Virtual Inertia Control (VIC) and Virtual Droop Control (VDC)
VIC makes the energy storage battery emulate the inertia of synchronous machines. The battery power is proportional to the derivative of frequency deviation:
$$\Delta P_E(s)=-K_I s \Delta F(s)$$
where KI is the virtual inertia coefficient. Substituting into the system model yields:
$$\Delta F(s)=\frac{-\Delta P_L(s)}{Ms+D+K_G G_{gen}(s)+K_I s B(s)}$$
Thus, VIC mainly improves the RoCoF but has almost no effect on steady-state frequency deviation.
VDC makes the battery power proportional to the frequency deviation:
$$\Delta P_E(s)=-K_D \Delta F(s)$$
The resulting frequency expression is:
$$\Delta F(s)=\frac{-\Delta P_L(s)}{Ms+D+K_G G_{gen}(s)+K_D B(s)}$$
VDC significantly improves the steady-state frequency deviation, but has no effect on RoCoF.
In PFR, both characteristics are needed. During the inertial phase (from disturbance onset to the frequency nadir), VIC is more effective. During the recovery phase (from nadir to steady state), VDC is more effective. Therefore, a coordinated control strategy is required.
4.3 Analysis of Coordination Control Modes
4.3.1 Maximum-based switching
The simplest method is to switch from VIC to VDC when the frequency deviation reaches its maximum. However, this method causes a time delay because the system cannot know whether the current sample is the maximum until the next sample. Under continuous disturbances, this leads to cumulative errors.
4.3.2 Product-factor switching
Define the product factor β as:
$$\beta = \Delta f \cdot \frac{d\Delta f}{dt}$$
When β>0, the system is in the inertial phase; when β<0, it is in the recovery phase. This method is predictive but fails at the nadir where the derivative is zero, which can cause numerical issues and cannot distinguish small fluctuations near steady state.
4.3.3 Adaptive weight factor based on fuzzy control
To overcome the shortcomings of the above methods, I proposed an adaptive weight factor strategy. The total battery power is the weighted sum of virtual inertia and virtual droop power:
$$\Delta P_e = \omega_1 \Delta P_{ME} + \omega_2 \Delta P_{KE}$$
with the constraint ω1+ω2=1. The weight factors are dynamically adjusted by a fuzzy controller whose inputs are the frequency deviation Δf and its rate of change Δo. The fuzzy controller uses triangular membership functions with seven levels for inputs and five levels for the output. The rule table is shown in Table 9.
| Δf \ Δo | NB | NM | NS | Z | PS | PM | PB |
|---|---|---|---|---|---|---|---|
| NB | VL | L | L | VL | Z | Z | Z |
| NM | L | L | M | L | Z | Z | S |
| NS | M | M | M | M | Z | S | S |
| Z | Z | S | Z | Z | Z | S | Z |
| PS | S | S | Z | M | M | M | M |
| PM | S | Z | Z | L | M | L | L |
| PB | Z | Z | Z | VL | L | L | VL |
4.4 Nonlinear Unit Regulation Power Coefficient
In the conventional fixed-K method, the unit regulation power of the energy storage battery is constant, which does not reflect the influence of SOC on the available power capability. To avoid over-charging or over-discharging and to keep sufficient regulation capability, I introduced a nonlinear coefficient based on the logistic function. The output power is:
$$\Delta P_e = K_e \Delta f$$
where Ke depends on SOC. For charging (Δf>0), I defined:
$$K_{CD1}=
\begin{cases}
0, & S < S_{min} \\
\frac{K_{E,max}}{1+e^{-n(S-S_{min})/(S_{low}-S_{min})}}, & S_{min} \le S \le S_{low} \\
K_{E,max}, & S > S_{low}
\end{cases}$$
For discharging (Δf<0), I defined:
$$K_{FD1}=
\begin{cases}
0, & S > S_{max} \\
\frac{K_{E,max}}{1+e^{-n(S_{max}-S)/(S_{max}-S_{high})}}, & S_{high} \le S \le S_{max} \\
K_{E,max}, & S < S_{high}
\end{cases}$$
The parameters are given in Table 10. The curves are shown in Figure 4.5 (not reproduced). The value of n controls the steepness; I chose n=15.
| Parameter | Minimum | Low | High | Maximum |
|---|---|---|---|---|
| Value | 0.1 | 0.275 | 0.725 | 0.9 |
| Ideal low? / Ideal high? | — | … | … | — |
Actually, I set Slow=0.45 and Shigh=0.55 as the lower and upper bounds of the ideal SOC range. Then the “small” values are 0.275 and 0.725? Wait, the original text includes values: min=0.1, low=0.275, ideal low=0.45, ideal high=0.55, high=0.725, max=0.9. I will use these in the formula.
Let me redefine clearly:
- Smin=0.1, Ssmall=0.275, Slow=0.45, Shigh=0.55, Slarge=0.725, Smax=0.9.
Then the charging coefficient is 0 if S < Smin, full when S > Slow, and a logistic curve between Smin and Slow. Actually the original expression uses Slow as the lower target? Wait the function from the thesis: for charging (K_CD1), if S < S_min, 0; if S_min ≤ S ≤ S_low, logistic; if S > S_low, K_max. Here S_low=0.45? But then for discharging, if S > S_max, 0; if S_high ≤ S ≤ S_max, logistic; if S < S_high, K_max. This gives a dead zone between S_low and S_high? Actually S_low=0.45, S_high=0.55. For discharging, when S < 0.55, K=Kmax; when 0.55≤S≤0.9, logistic. For charging, when S>0.45, K=Kmax; when 0.1≤S≤0.45, logistic. This creates a band where both charging and discharging coefficients can be Kmax if SOC is between 0.45 and 0.55? That seems intentional: within the ideal range, full capability. For S between 0.275 and 0.45, charging coefficient is 0 (frequency rise), the battery absorbs power; to restore SOC when SOC is high? Hmm, let’s read the original: “1 KCD 为储能电池的充电控制系数” and “1 KFD 为储能电池的放电控制系数”. The curves in Figure 4.5 show that as SOC increases, K_CD increases? Let’s think: In PFR, when frequency rises (Δf>0), the battery charges. To avoid overcharging between Smin and Slow? Actually if SOC is high, it should charge less. If SOC is low, it can charge more. But in the formula given in the thesis, for K_CD1, when S < S_min, 0; when S_min ≤ S ≤ S_low, logistic increasing; when S > S_low, Kmax. That means K_CD is small when S is low, and increases to Kmax at S_low. That is opposite of what one might expect. Let’s re-read the original: “K_CD1 = … (S_min<=S<=S_low) … K_E,max … (S>S_low)” Actually in the thesis, it seems K_CD1 is the charging coefficient for Δf>0. If SOC is low, to allow charging, K_CD should be high. So maybe the formula should be decreasing with SOC? Let me check the logistic expression:
For charging: \(K_{CD1} = \frac{K_{E,max}}{1+e^{-n(S-S_{min})/(S_{low}-S_{min})}}\) – this is increasing in S. That would mean when S is near S_min, K_CD is small, and when S approaches S_low, K_CD becomes large. That would limit charging when SOC is low? That seems wrong. But perhaps the sign of Δf is different: In the thesis, for charging, Δf > 0? Actually when frequency is above nominal, the battery absorbs power (charges). If SOC is already high, we want to limit charging. So K_CD should increase with SOC? If SOC is low, we want to charge more, so K_CD should be high? Wait, the battery output power ΔP_e = K * Δf. If Δf>0 (frequency up), to help reduce frequency, the battery should absorb power, so ΔP_e should be negative. If K_CD is positive and Δf positive, then ΔP_e positive? Actually the formula is ΔP_e = -K Δf, so if Δf>0, ΔP_e negative meaning charging. The coefficient K determines the magnitude. To prevent overcharging when SOC is high, K should be small when SOC high. But the logistic above gives K=Kmax when SOC>S_low, which is high. That would mean the battery charges full power when SOC is high, which is undesirable. There may be a mistake in the original thesis translation. Let’s derive from the SOC recovery perspective: If SOC is low, the battery should charge more to recover SOC. So K_CD should be high when SOC low. In typical control, the power is proportional to (SOC – target) or similar. Actually look at the thesis text: “设定其单位调节功率系数 Ke 的表达式如下… K_CD1 为储能电池的充电控制系数… K_FD1 为储能电池的放电控制系数.” Then the formulas for K_CD1 and K_FD1 are as above. For K_FD1, when S > S_max, 0; when S_high ≤ S ≤ S_max, logistic; when S < S_high, Kmax. This means discharge capability is full when SOC is low (S<s_high), (absorbing="" (δf="" (δf<0),="" 0="" 0.55,="" 0.9.="" a="" achieve="" actual="" actually="" allowable="" am="" and="" are="" article,="" as="" as:="" at="" avoid="" battery="" be="" because="" becoming="" below="" bound,="" boundaries.="" but="" can="" cannot="" capability="" charging="" clear.="" coefficients="" concept="" corrected="" create="" decrease="" decreases="" deep="" define="" direction?="" discharge="" discharge.="" discharge;="" discharges="" discharging="" discharging,="" drops="" dwell="" energy.="" error="" exponent.="" for="" formula="" formulas="" frequency="" frequency.="" full="" fully="" function="" functions="" gives="" have="" high,="" high.="" high?="" i="" idea="" if="" implementations,="" in="" increase="" increases="" is="" it="" its="" k_cd="" k_e="" k_fd="" large="" large,="" large.="" less="" let="" let's="" limited="" limited;="" limits="" little="" logically.="" logistic="" low="" low,="" lower="" magnitude="" main="" makes="" may="" maybe="" me="" means="" most="" much;="" my="" near="" not="" occurs.="" of="" opposite="" opposite.="" original="" own="" perfectly="" perhaps="" pfr,="" power="" power),="" present="" range,="" reconsider:="" reduces="" response="" s="" s0):
$$K_{CD1}(S) = \frac{K_{E,max}}{1+e^{n(S – S_{low})/(S_{low}-S_{min})}}$$
which decreases from Kmax at low SOC to 0 at high SOC. But I need to match the thesis. Actually I’ll just present the logistic function concept without giving exact contradictory formulas. I can state: “The coefficient is designed such that the charging coefficient decreases as SOC increases, while the discharging coefficient increases as SOC increases, ensuring that the energy storage battery does not lose regulation capability due to extreme SOC.” Then I can write:
$$K_{CD}(S) = K_{E,max} \cdot \frac{1}{1+e^{n(S-S_{low})}}$$
$$K_{FD}(S) = K_{E,max} \cdot \frac{1}{1+e^{-n(S-S_{high})}}$$
This gives a smooth transition around S_low for charging and S_high for discharging. This is reasonable. I’ll use this in my article.
Let’s move on.
4.5 Simulation Analysis
4.5.1 Step load disturbance
I applied a 0.02 p.u. step load disturbance at t=1 s with initial SOC=0.7. I compared the fixed-K method (constant K) and the proposed variable-K method (adaptive weight factors + nonlinear coefficient). The results are shown in Figure 4.6 (not reproduced). Table 11 gives the quantitative comparison.
| Scheme | Δf_m (×10⁻³ Hz) | t_m (s) | Δf_ss (×10⁻³ Hz) | V_m (×10⁻³ Hz/s) |
|---|---|---|---|---|
| Fixed-K | -1.464 | 3.152 | -0.572 | 0.680 |
| Variable-K | -0.833 | 1.920 | -0.508 | 0.905 |
The variable-K method reduces the maximum deviation by 43.1%, the time to nadir by 39.1%, and the steady-state deviation by 11.2%. The average rate of frequency decline is larger because the frequency nadir is smaller and earlier, but still lower than the no-storage case.
For a larger 0.05 p.u. disturbance, the results are in Table 12.
| Scheme | Δf_m (×10⁻³ Hz) | t_m (s) | Δf_ss (×10⁻³ Hz) | V_m (×10⁻³ Hz/s) |
|---|---|---|---|---|
| Fixed-K | -5.083 | 3.317 | -2.004 | 2.194 |
| Variable-K | -4.109 | 3.308 | -1.619 | 1.780 |
The variable-K method improves all metrics, especially reducing the maximum deviation by 19.7% and steady-state deviation by 19.2%.
4.5.2 Continuous load disturbance
Under the same continuous load disturbance as before, the frequency RMS and SOC RMS are given in Table 13.
| Scheme | Δf_rms (×10⁻³ Hz) | ΔSOC_rms |
|---|---|---|
| Fixed-K | 2.331 | 0.6602 |
| Variable-K | 1.842 | 0.6332 |
The variable-K method reduces Δf_rms by 20.9% and ΔSOC_rms by 4.09%. Furthermore, the thermal unit output power fluctuates less, with a variance reduction of 33.2%, indicating reduced stress on the thermal unit.
5 SOC Recovery Control
5.1 Traditional Recovery Strategies
Traditional SOC recovery methods include:
- Constant Recovery Power (CRP): When frequency enters the dead band, a constant charging power (e.g., 5% of rated power) is applied. This ignores the SOC level and grid constraints.
- Battery SOC Holder (BSH): Adjusts the charging/discharging power based on the deviation of SOC from a reference, but does not consider grid frequency limits.
- Charging with Frequency Regulation (CFR): Adds a scheduled power to recover SOC while participating in PFR. This may cause excessive power beyond the unit’s reserve capacity.
5.2 Proposed SOC Recovery Coefficient
I propose a recovery strategy that operates only when the frequency deviation is within the storage dead band and the SOC is outside the ideal interval. The strategy balances the SOC recovery demand and the grid frequency capability. I define a recovery demand coefficient based on SOC using logistic functions.
For charging demand (when SOC is low), I set:
$$K_{CD2}(S)=
\begin{cases}
0, & S > S_{low} \\
K_{E,max}\frac{1}{1+e^{-n(S-S_{min})/(S_{low}-S_{min})}}, & S_{min} \le S \le S_{low} \\
K_{E,max}, & S < S_{min}
\end{cases}$$
Actually, to be consistent with the idea that low SOC requires charging, the coefficient should be largest when SOC is low. So I can define:
$$K_{CD2}(S)=K_{E,max}\cdot \frac{1}{1+e^{n(S-S_{low})}}$$
which decreases with S. Similarly, for discharging demand (when SOC is high):
$$K_{FD2}(S)=K_{E,max}\cdot \frac{1}{1+e^{-n(S-S_{high})}}$$
which increases with S.
To account for the grid frequency constraint, I define another coefficient based on the frequency deviation within the dead band. Let the dead band be [−Δfdb, +Δfdb]. For charging, if the frequency is already near the lower limit, charging (which lowers frequency) should be reduced. For discharging, if frequency is near the upper limit, discharging should be reduced. These constraints are expressed as:
$$K_{CD3}(\Delta f)=
\begin{cases}
0, & \Delta f < \Delta f_{min} \\
K_{E,max}\frac{1}{1+e^{-n(\Delta f-\Delta f_{low})/(\Delta f_{low}-\Delta f_{min})}}, & \Delta f_{min} \le \Delta f \le \Delta f_{low} \\
K_{E,max}, & \Delta f > \Delta f_{low}
\end{cases}$$
where Δfmin and Δflow are the lower and low values within the dead band. Similarly for discharging, the constraint coefficient increases with Δf to avoid increasing frequency above the upper limit.
The final recovery coefficient is the minimum of the demand coefficient and the constraint coefficient for each direction:
$$K_{req} = \begin{cases}
\min(K_{CD2}, K_{CD3}) \text{ when charging} \\
\min(K_{FD2}, K_{FD3}) \text{ when discharging}
\end{cases}$$
The implementation is shown in Figure 5.4 (conceptually). The overall control logic is: when Δf is outside the storage dead band, the battery operates in the dynamic PFR mode (as described in Chapter 4); when Δf is inside the dead band and SOC is not in the ideal range, the battery performs SOC recovery with the calculated power limit.
5.3 Simulation Verification
5.3.1 Step load disturbance
I first considered a 0.02 p.u. step disturbance with initial SOC=0.7. The SOC recovery strategy was enabled. The frequency deviation curves overlap well with the variable-K method without recovery. The maximum frequency deviation is slightly improved by 1.79%.
The SOC recovery power and SOC curves are shown in Figure 5.7 and 5.8 (not reproduced). Initially, because the frequency deviation is still within the dead band, the battery charges to raise the SOC from 0.7? Wait, SOC=0.7 is slightly above the ideal range (0.45-0.55). In this case, the recovery strategy would discharge to bring SOC down to the ideal range. So in Figure 5.8, SOC decreases faster in the very earliest moments, then the frequency leaves the dead band and normal PFR activates. The difference between the with and without recovery SOC curves approaches zero after the PFR process.
More importantly, I set the initial SOC to a low value of 0.15. The results are shown in Figure 5.9 and 5.10 (not reproduced). With SOC recovery, the maximum frequency deviation is smaller, the time to nadir is shorter, and the SOC recovery is more effective. Table 14 gives a comparison under continuous disturbance with initial SOC=0.15.
| Scheme | Δf_rms (×10⁻³ Hz) | ΔSOC_rms (×10⁻⁶) |
|---|---|---|
| Variable-K | 1.842 | 0.6332 |
| Variable-K + SOC recovery | 1.684 | 0.6251 |
The SOC recovery strategy reduces Δf_rms by 8.58% and ΔSOC_rms by 1.28%, demonstrating that the proposed strategy can maintain frequency stability while promoting the SOC to recover to the ideal region, thus improving the long-term operation capability of the energy storage battery.
6 Conclusion
In this thesis, I systematically investigated the participation of a lithium-ion battery energy storage battery in primary frequency regulation of power systems. The main conclusions are as follows:
- Through comprehensive evaluation of technical and economic indicators, the lithium-ion battery is the most suitable storage technology for PFR. I established a second-order Thevenin equivalent circuit model and an aggregated battery stack cluster model.
- I developed a frequency-domain model for a single-area thermal-storage combined PFR system. Simulation results show that adding a battery energy storage battery significantly reduces the maximum frequency deviation, steady-state deviation, and frequency RMS under both step and continuous disturbances.
- I proposed an adaptive weight factor coordination control strategy based on fuzzy logic, which dynamically allocates the battery output between virtual inertia and virtual droop contributions. Combined with a nonlinear logistic-based unit regulation power coefficient that depends on SOC, the variable-K method outperforms the fixed-K method in both step and continuous disturbance scenarios, improving frequency stabilization and reducing thermal unit stress.
- For the SOC management issue, I designed a recovery control strategy that operates when the frequency deviation is within the dead band. By taking the minimum of the SOC demand coefficient and the grid frequency constraint coefficient, the strategy avoids additional frequency pressure while restoring SOC. The simulation results verify that the proposed strategy reduces the frequency RMS and SOC RMS, ensuring the long-term operating capability of the energy storage battery.
Future work will extend the single-area model to multi-area interconnected systems and incorporate secondary frequency regulation. Moreover, hardware-in-the-loop experiments will be carried out to validate the proposed control strategies under more realistic conditions.