# Electrical Mismatch

In the preceding sections, we have modelled the performance of individual modules on the assumption that they operate independently from one another, and hence can all operate at their Maximum Power Point, and we have assumed that the Maximum Power Point is determined by the average irradiance over the submodule. However, modules are connected in series strings, and cells are connected in strings within a module, forcing all cells in each string to operate at the same current. Additionally strings can be connected in parallel to form arrays, forcing all parallel strings to operate at the same voltage. Finally, arrays are connected to an inverter, which will adjust its input voltage (i.e. the voltage across the parallel strings) to extract the maximum amount of power from the array at all times (within the inverter performance envelope).

The consequence of this is that modules in the plant array will largely not be operating at their individual Maximum Power Points as we have previously assumed, and we need to evaluate the impact of this on the overall power output of the plant array.

To establish the operating point of an array, it is necessary to combine the I-V curves of all submodules in the array into a single I-V curve representing the entire array.

When calculating the electrical mismatch we first calculate I-V curves for each submodule by passing the *mismatch irradiance*, \(G_{\text{msh}}\), to the Module Performance Model (as described in Module Performance Model), and applying the adjustments to the curve based as derived in preceding sections.The *mismatch irradiance* is a more accurate indicator of module behaviour than the *effective irradiance" because most shaded cell determines how much current can pass through the whole submodule as all the cells are in series.

As the first step will be to add voltages from the submodules in each string at each of a set of discrete current values chosen to represent the entire IV curve, we need to obtain values for submodule voltage at each of these currents. If \(I_{\text{n}}\) is the n^{th} value of current in the current vector, then to obtain the corresponding voltage \(V_{\text{n}}\) we need to query the IV curve under the specific irradiance and temperature conditions for the submodule operating at a current equal to:

$$I_{n}^{'} = \frac{I_{n}}{\eta_{submod,mm} \bullet \eta_{submod,other} \bullet \eta_{submod,LID} \bullet \eta_{submod,binning} \bullet \eta_{submod,npmc}}$$

$$V_{j,n} = V(IV_{\text{submod}}\left( G_{\text{msh}}(j,t),T_{c}\left( j,t \right) \right),I_{n}^{'})$$

**Submodule voltages summed to give string IV curves.**The voltages from all submodules in each string are summed for each distinct current value to give an array of string voltages \(V_{\text{string,n}}\) corresponding to currents \(I_{\text{n}}\).

$$V_{string,n}\left( t \right) = \sum_{j}^{}{V_{j,n}(t)}$$

$$IV_{\text{string}}\left( t \right) = IPointwiseVSum_{\text{submodule}}\left( IV_{submod,mpc}(t) \right)$$

**String IV curves summed to give array IV curves.**The individual string I-V curves are combined by summing the string current values at each of a set of voltage values. The string I-V curves must first be interpolated to a common set of voltage values.

$$IV_{\text{array}}\left( t \right) = VPointwiseISum_{\text{string}}\left( IV_{\text{string}}\left( t \right) \right)$$

**Identify the Maximum Power Point for the array.**At this stage, we are ignoring any inverter thresholds and limits; the array is assumed to be operating at the maximum power point of the curve, representing the power output of the array considering electrical mismatch:

$$P_{array,\text{emm}}\left( t \right) = \text{MPP}\left( IV_{\text{array}}\left( t \right) \right)$$

The Electrical Mismatch Effect at time t is calculated as:

$$\Delta_{\text{emm}}\left( t \right) = \left( \frac{\sum_{k = 1}^{N_{\text{arrays}}}{P_{array,emm}\left( t \right)}}{P_{Parray,mm}\left( t \right)} - 1 \right) \bullet 100\%$$

Powers can be summed over desired periods to get monthly or annual Electrical Mismatch Effects. As for other effects, it will be necessary to de-season if the input data is not a single complete year.