# Sandia Inverter Model

The Sandia Inverter Model provides a means to predict AC output power ($$P_{AC}$$) from DC input power ($$P{_{DC}}$$).

The form of the model is as follows:

$$P_{AC}=\left \{ \frac{P_{AC0}}{A-B} -C\left ( A-B \right )\right \}\left ( P_{DC}-B \right )+C\left ( P_{DC}-B \right )^{2}$$

where

$$A=P_{DC0}\left \{ 1+C_{1}\left ( V_{DC}-V_{DC0} \right ) \right \}$$,

$$B=P_{s0}\left \{ 1+C_{2}\left ( V_{DC}-V_{DC0} \right ) \right \}$$, and

$$C=C_{0}\left \{ 1+C_{3}\left ( V_{DC}-V_{DC0} \right ) \right \}$$

Parameters:

• $$V_{DC}$$ : DC input voltage (V).  This is typically assumed to be the array’s maximum power voltage.
• $$V_{DC0}$$ : DC voltage level (V) at which the AC power rating is achieved at reference operating conditions.
• $$P_{AC}$$ : AC output power (W)
• $$P_{AC0}$$ : Maximum AC power rating for inverter at reference conditions (W).  Assumed to be an upper limit.
• $$P_{DC0}$$ : DC power level (W) at which the AC power rating is achieved at reference operating conditions.
• $$P_{s0}$$ : DC power required to start the inversion process (W)
• $$C_{0}$$ : Parameter defining the curvature of the relationship between AC output power and DC input power
• $$C_{1}$$ : empirical coefficient allowing $$P_{DC0}$$ to vary linearly with DC-voltage input, default value
is zero, (1/V)
• $$C_{2}$$ : empirical coefficient allowing $$P_{s0}$$ to vary linearly with DC-voltage input, default value
is zero, (1/V)
• $$C_{3}$$ : empirical coefficient allowing $$C_0$$ to vary linearly with dc-voltage input, default value is
zero, (1/V)

#### Algorithm to estimate model parameters from inverter efficiency curves

The Sandia inverter model requires eight parameters: $$P_{AC0}, P_{DC0}, P_{s0}, V_{DC0}, C_0, C_1, C_2, C_3$$.

Given measurements of an inverter’s AC power, DC voltage and efficiency, parameters for the Sandia inverter model are determined by the following algorithm. Denote the AC power measurements by $$P_{AC, d, r, k}$$ where $$d \in {min, nom, max}$$ is the DC voltage level, $$r \in {0, 0.2, 0.3, 0.5, 0.75, 1.0}$$ is the power level, and $$k$$ indexes the replicated measurements. Similarly denote the DC voltage measurements as $$V_{DC, d, r, k}$$ and measured efficiency as $$\eta_{r, d, k}$$.

Step 1: Set $$P_{AC0}$$ equal to the inverter’s AC rating.

Step 2: Calculate $$P_{DC, d, r, k} = \frac{P_{AC, d, r, k}}{\eta_{d, r, k}}$$

Step 3: For $$d \in {min, nom, max}$$ :

Step 3a: Calculate $$\bar{V}_{d} = \mathrm{Average}_{r, k}( V_{DC, d, r, k})$$

Step 3b: Obtain coefficients $$a_d, b_d, c_d$$ for a quadratic fit by linear regression using the data indexed by $$r,k$$ : $$P_{AC, d, r, k} = a_d P^2_{DC, d, r, k} + b_d P_{DC, d, r, k} + c_d$$

Step 3c: Set $$C_{0,d} = a_d$$

Step 3d: Solve each of the following equations to obtain $$P_{DC0, d}, P_{s0, d}$$:

$$a_d P^2_{DC0, d} + b_d P_{DC0, d} + c_d – P_{AC0} = 0$$

$$a_d P^2_{s0, d} + b_d P_{s0, d} + c_d = 0$$

Step 3e: Denote $$X_d = \bar{V}_{d} – \bar{V}_{nom}$$ and use linear regression on the data indexed by $$d$$  to find coefficients $$\beta_1, \beta_0$$  for each of

$$P_{DC0, d} = \beta_{DC, 1} X_d + \beta_{DC, 0}$$

$$P_{s0, d} = \beta_{s, 1} X_d + \beta_{s, 0}$$

$$C_{0,d}=\beta_{C,1}X_d+\beta_{C,0}$$

Step 4: Extract parameters as:

$$P_{DC0} = \beta_{DC, 0}$$

$$V_{DC0} = \bar{V}_{nom}$$

$$P_{s0} = \beta_{s, 0}$$

$$C_0 = \beta_{C, 0}$$

$$C_1 = \frac{\beta_{DC, 1}}{\beta_{DC, 0}}$$

$$C_2 = \frac{\beta_{s, 1}}{\beta_{s, 0}}$$

$$C_3 = \frac{\beta_{C, 1}}{\beta_{C, 0}}$$