Grid-Tied Solar Inverter Example¶

Single-phase photovoltaic inverter with Maximum Power Point Tracking (MPPT) and grid synchronization.

Overview¶

Grid-tied solar inverters convert DC power from PV arrays to AC power synchronized with the utility grid: - Maximum Power Point Tracking (MPPT) maximizes PV output - Phase-Locked Loop (PLL) for grid synchronization - Current injection in phase with grid voltage - Anti-islanding protection - Reactive power support capability - High efficiency (>97%)

Specifications¶

Circuit Files¶

• pv_inverter_basic.ipes - Single-phase full-bridge inverter
• pv_inverter_mppt.ips - With Perturb & Observe MPPT control
• pv_inverter_grid_sync.ipes - PLL-based grid synchronization
• pv_inverter_complete.ipes - Full system with all controls
• pv_array_model.ipes - PV cell and array characteristics

System Architecture¶

┌──────────────────┐
│   PV Array       │ V_pv ∈ [200, 450V]
│ (5 kW nominal)   │
└────────┬─────────┘
         │
         ▼
    ┌─────────────┐
    │ MPPT DC-DC  │  Boost/Buck converter
    │ Converter   │  Tracks V_mpp
    └────┬────────┘
         │
         ▼
    ┌─────────────┐       ┌──────────────┐
    │  400V DC    │──────►│ Full-Bridge  │
    │   Bus       │       │   Inverter   │
    └─────────────┘       └────┬─────────┘
                               │
                               ▼
                          ┌──────────┐
                          │ L-C      │
                          │ Filter   │
                          └─────┬────┘
                                │
                                ▼
                          ┌──────────┐
                          │  Grid    │
                          │ (230V AC)│
                          └──────────┘

Theory¶

Photovoltaic Cell Model¶

The single-diode PV cell model:

Where: - Iph: Photocurrent (proportional to irradiance) - I0: Diode reverse saturation current - q: Elementary charge (1.6×10⁻¹⁹ C) - n: Ideality factor (1-2) - kB: Boltzmann constant (1.38×10⁻²³ J/K) - T: Junction temperature (K) - Rs: Series resistance (Ω) - Rsh: Shunt resistance (Ω)

Simplified form (ignoring Rs and Rsh): $\(I = I_{ph} - I_0 e^{qV/(nk_B T)}\)$

Maximum Power Point (MPP)¶

Power output: P = V × I

At the MPP, the voltage-power curve has zero slope: $\(\frac{dP}{dV} = 0 \implies I + V\frac{dI}{dV} = 0\)$

Key characteristics: - MPP voltage ≈ 0.7–0.8 × Voc (open-circuit voltage) - Varies with temperature and irradiance - Typical PV module: Vmpp ≈ 30V (at STC: 1000 W/m², 25°C) - Array: Series/parallel connection to reach desired voltage

Temperature Dependence¶

MPP voltage decreases with temperature: $\(V_{mpp}(T) = V_{mpp,ref} - \alpha(T - T_{ref})\)$

Where α ≈ 0.5 mV/°C (temperature coefficient, negative).

MPPT Control Algorithms¶

Perturb and Observe (P&O)¶

Simplest and most common algorithm:

Loop at time interval Ts:
  1. Measure V(k), I(k), P(k) = V(k) × I(k)
  2. Calculate ΔP = P(k) - P(k-1), ΔV = V(k) - V(k-1)
  3. Update reference:
     if ΔP > 0
       if ΔV > 0
         V_ref = V_ref + ΔV_step  (moved right on curve)
       else
         V_ref = V_ref - ΔV_step  (moved left on curve)
     else
       if ΔV > 0
         V_ref = V_ref - ΔV_step  (moved away from MPP)
       else
         V_ref = V_ref + ΔV_step  (moved away from MPP)

Advantages: Simple, few computations Disadvantages: Oscillates around MPP (can be reduced with small step size)

Incremental Conductance (IncCond)¶

More sophisticated, detects MPP directly:

Loop at time interval Ts:
  1. Measure ΔI/ΔV (conductance change)
  2. Calculate I/V (instantaneous conductance)
  3. Compare:
     if ΔI/ΔV = -I/V       → At MPP, hold V_ref
     if ΔI/ΔV > -I/V       → Left of MPP, increase V_ref
     if ΔI/ΔV < -I/V       → Right of MPP, decrease V_ref

Advantages: Fast tracking, no steady-state oscillation Disadvantages: More computation, noise sensitivity

Grid Synchronization via PLL¶

The Phase-Locked Loop (PLL) detects the phase angle of the grid voltage:

                    ┌───────────────┐
vgrid ──────────────┤  Normalize &  │
                    │  Filter (LP)  │
                    └────────┬──────┘
                             │ sin(θ_grid)
                             ▼
                    ┌───────────────┐
                    │  Multiplier   │ × V_ref
                    │   (demod)     │
                    └────────┬──────┘
                             │ e_pll
                             ▼
                    ┌───────────────┐
                    │  PI Control   │
                    │               │
                    └────────┬──────┘
                             │ ω_pll
                             ▼
                    ┌───────────────┐
                    │  Integrator   │
                    │  θ_pll = ∫ω   │
                    └───────────────┘
                             │
                             └──► θ_grid estimate

PLL Equations:

Error signal: e_pll = vgrid × sin(θ_pll - θ_grid) ≈ Vgrid·(θ_pll - θ_grid)

Angular frequency: ω_pll = ω_grid + Kp·epll + Ki·∫epll

Tuning: Bandwidth ≈ 50-100 Hz (fast PLL locks to grid variations)

Current Control for Grid Injection¶

Reference Current Generation:

The inverter injects current in phase with grid voltage: $\(i_{ref} = I_{max} \sin(\omega_e t + \theta_pll)\)$

Where Imax is set by available PV power: $\(I_{max} = \frac{P_{pv}}{V_{grid,rms}}\)$

Reactive Power (Optional):

For grid support, inject reactive current: $\(i_{ref} = I_d \sin(\omega_e t + \theta_{pll}) + I_q \cos(\omega_e t + \theta_{pll})\)$

Where Id = active current, Iq = reactive current.

Current Control Loop:

         i_ref
           │
      ┌────┴────┐
      │    -    │
      └────┬────┘
           │ i_error
           ▼
      ┌─────────┐
      │   PI    │
      │ Control │
      └────┬────┘
           │ v_ref
           ▼
    ┌─────────────┐
    │    SPWM     │
    │ (carrier)   │
    └─────┬───────┘
          ▼
    ┌─────────────┐
    │  Inverter   │
    │  Switches   │
    └─────────────┘

Anti-Islanding Protection¶

Detects when the grid is disconnected (preventing island formation).

Passive Detection Methods¶

1. Frequency Drift: If grid disconnects, frequency drifts (no stiff voltage source)
2. Voltage Drift: RMS voltage deviates from nominal
3. Rate of Change: dV/dt or df/dt monitoring

Active Detection Methods¶

1. Frequency Shift (Frequency Bias): Inject small frequency offset; detect via grid coupling
2. Reactive Power Injection: Inject reactive power and observe voltage response
3. Voltage Shift: Modulate voltage injection level

Typical Thresholds (EN 50160 / IEC 61727)¶

Control Structure¶

         PV Array
            │
            ▼
      ┌──────────┐
      │  MPPT    │ Vpv_ref
      │ Control  │
      │ (P&O)    │
      └─────┬────┘
            │
            ▼
       ┌────────────┐
       │ DC-DC Buck │ I_mppt
       │ Converter  │
       └────┬───────┘
            │ (400V DC)
            ▼
      ┌─────────────┐
      │ Grid Voltage│ sin(ωt + θ)
      │ Sensing PLL │
      └────┬────────┘
           │ θ_grid, ω_grid
      ┌────┴──────────────┐
      │ Current Reference │ i_ref
      │ Generator         │
      └────┬──────────────┘
           │
      ┌────┴─────────┐
      │  - (subtraction)
      └────┬────────┘
           │ i_error
           ▼
      ┌────────────┐
      │ PI Current │
      │ Control    │
      └────┬───────┘
           │ v_ref
           ▼
      ┌────────────────┐
      │ Sinusoidal PWM │
      │ (SPWM) / SVM   │
      └────┬───────────┘
           │
           ▼
      ┌────────────┐
      │ Full-Bridge│
      │ Inverter   │
      └────┬───────┘
           │
           ▼
      ┌─────────────┐
      │  L-C Filter │
      └────┬────────┘
           │
           ▼
        ┌─────┐
        │Grid │
        └─────┘

Design Steps¶

Step 1: PV Array Configuration¶

For 200-450V DC input: - Typical module: 40V at MPP - Series: 5-10 modules (200-400V nominal) - Parallel: 2-3 strings (increases current)

Step 2: MPPT DC-DC Converter¶

Design a boost/buck converter: - Input: 200-450V DC (PV array) - Output: 400V DC (regulated) - Power: 5 kW

Equations: $\(D = 1 - \frac{V_{pv}}{V_{out}}\)$ $\(L = \frac{V_{pv}(1-D)}{f_{sw} \Delta I_L}, \quad \Delta I_L = 0.2 \times I_{pv}\)$ $\(C = \frac{I_{pv} D}{f_{sw} V_r}\)$

Step 3: Full-Bridge Inverter¶

Select switches for 400V DC bus: - IGBT or SiC MOSFET rated for 600V minimum - Switching frequency: 16 kHz (balance switching loss vs. filter size) - Deadtime: 200-500 ns

Gate driver PWM frequency: fsw = 16 kHz Modulation index: Ma = 0.9 (for linear region)

Step 4: Output L-C Filter¶

Reduce high-frequency switching ripple: $\(L = \frac{V_{dc}}{8 f_{sw} I_{max}}, \quad C = \frac{I_{max}}{8 f_{sw} V_r}\)$

Example: L = 5 mH (reduces di/dt), C = 5 μF (10% voltage ripple)

Step 5: PLL Tuning¶

For 50 Hz grid, typical PLL bandwidth 50-100 Hz: $\(K_p = 2 \times BW, \quad K_i = BW^2\)$

For BW = 100 Hz: Kp = 200, Ki = 10000

Step 6: MPPT Tuning¶

Sampling interval: 100 ms (slow update, stable) Step size: ΔVref = 5V (balance speed vs. oscillation)

Exercises¶

1. PV Curve Tracing: Measure I-V and P-V curves at different irradiance levels
2. MPPT Algorithm: Implement P&O or IncCond, verify tracking efficiency >99%
3. PLL Synchronization: Measure phase error; tune for fast lock (<100 ms)
4. Current Injection: Verify sine wave injection in phase with grid (PF > 0.98)
5. THD Measurement: Measure output voltage THD; design filter for <3%
6. Anti-Islanding: Simulate grid disconnection; verify detection within 2 seconds
7. Efficiency Mapping: Plot efficiency vs. load and irradiance
8. Reactive Power: Inject Q = ±2 kVAr for grid support if applicable

Related Resources¶

• Wind Converter
• PFC Converters
• Single-Phase Inverter
• Three-Phase Systems