# DC Array Behavior

# DC Field Behavior

This collection of functions computes the I(V) characteristics of an individual module given the module temperature and irradiance. The tool uses the 1-diode model. At this point in the simulation process, the irradiance seen by the PV module has been corrected for all optical losses.

# Module Temperature

Two models for estimating module temperature are supported by PlantPredict and selectable by the user in the Simulation Settings: the Sandia method and the static heat-balance model.

# Sandia Module Temperature

This routine empirically computes the module temperature from the air temperature, the wind speed, and the in-plane irradiation. This model uses two empirically-determined coefficients: *a _{t}* establishes the upper limit for module temperature at low wind speeds and high solar irradiance,

*b*establishes the rate at which module temperature drops as wind speed increases.

_{t}### Figure 34. Empirical Module Temperature Coefficients

## Inputs

## Outputs

1.) Compute the module temperature as a function of the available insolation, wind speed, and air temperature:

*G*of 1000 W/m

_{Ref}^{2}.

## Reference

King, D. L., Kratochvil, J.A., Boyson, W.L., Photovoltaic Array Performance Model, Report SAND2004-3535, Sandia National Laboratories, Albuquerque, NM, September 1997.

# Heat Balance Temperature Model

The thermal behavior of the field, which strongly influences the electrical performances, is determined by a thermal balance between the ambient temperature and the cell’s temperature, which is elevated by incident irradiation. In this simple model, α_{T} is the absorption coefficient of solar irradiation, and η_{m} is the PV efficiency (related to the Module area), i.e. the removed energy from the module.

## Inputs

## Outputs

## Algorithm

1.) Compute the module temperature as a function of the available insolation, wind speed, and air temperature:

The usual value of the absorption coefficient α is 0.9. The PV efficiency is taken from the Module File.

The thermal behavior is characterized by a thermal loss factor designated here by *k*, which can be split into a constant component *k _{c}* and a factor proportional to the wind velocity

*k*. These factors depend on the mounting mode of the modules (sheds, roofing, facade, etc.).

_{v}

2.) Convert the cell temperature to a module surface temperature.

## Reference

Mermoud, A., Conception et dimensionnement de systèmes photovoltaïques : Introduction des modules PV en couches minces dans le Logiciel PVsyst. Université de Genève, 2005.

# Module I(V) Curve

A complete module I(V) curve is computed under ambient conditions for the current time step. The “One-Diode” parameters are defined in the module file. The one-diode model uses an equivalent electrical circuit model and the governing diode equation to generate an explicit, continuous current vs. voltage function.

# One-Diode Model

Given the 1-diode model parameters for the module, this routine computes the module’s current-voltage characteristics. **Figure 35** shows the equivalent circuit of a PV cell. The following algorithm includes the recombination current, I_{rec}, a refinement in the model that can be set to zero if the parameters required to determine the current (μ, τ_{eff}, *di*, V_{bi}) are not provided in the module definition input file.

### Figure 35. One-Diode Equivalent Electric Circuit Model of a PV Module

## Inputs

## Outputs

## Algorithm

1.) Solve the following transcendental equation that relates the module’s output current *i* and output voltage υ to the photocurrent *I*_{ph}, and the 1-diode parameters corrected for the temperature and irradiance level (I_{0}, *R*_{sh}, *R*_{s}, γ).

The parameter b, given in the module parameter definition, combines the following terms:

Expanded, the transcendental function for the output current becomes:

2.) The current should be solved for diode voltages in the interval [0, ν_{oc,g}], where ν_{oc,g} is the guess for the open-circuit voltage, and can be found by setting the I to zero in the above equation (neglecting *I*_{rec}):

*i*and ν to 0, respectively.

4.) Find the maximum power point diode voltage using the Newton-Raphson method by iteratively solving the following equation until the difference between *V*_{d,n+1} and ν_{d,n} is arbitrarily small.

## Reference

Mermoud, A., Conception et dimensionnement de systèmes photovoltaïques : Introduction des modules PV en couches minces dans le Logiciel PVsyst. Université de Genève, 2005.

# One-Diode Model Temperature Correction

Given the 1-diode model parameters defined for the module under STC conditions (25 °C, 1000 W/m^{2}), compute the temperature and irradiance-corrected parameters at the actual module temperature *T _{m}* and the available solar energy

*G*. The default temperature correction of all 1-diode parameters is linear. If a non-linear temperature correction of the diode ideality factor is desired, the then a set of additional parameters (polynomial coefficients) is available to affect this correction.

_{T,Eff}This non-linear response is illustrated in **Figure 36** for a family of I(V) curves with a module temperature range of 8 °C to 75 °C. This data was measured in the laboratory using the module temperature control system and a Spire long-pulse solar simulator.

### Figure 36. Non-Linear Module Power Temperature Response using a Third-Order Polynomial Correction to the Diode Ideality Factor γ

## Inputs

## Outputs

## Algorithm

1.) Given the effective available insolation *G*_{T,Eff} and module temperature *T*_{m}, find the corrected 1-diode shunt and series resistance.

2.) If the linear correction to the diode ideality factor is desired, or all non-linear coefficients *a _{y},b_{y},c_{y},d_{y}* = 0, then correct it for the module temperature as follows:

*R*

_{s}. As a result, the junction still experiences a voltage bias and there are currents flowing through both the recombination path and through

*R*

_{sh}when the terminal voltage of the device is zero, i.e. the recombination term does not go to zero in the short circuit condition. Therefore the photocurrent,

*I*

_{ph}must be greater than

*I*

_{sc}in order to supply parasitic currents as well as the external current which has the rated

*I*

_{SC,ref}value. Find the actual

*I*

_{ph}as follows:

The temperature dependence on voltage is also computed, but not used elsewhere in the simulation. Note that μΙ_{sc} ≅ α_{Iph}, given in 1/°C.

## Reference

Schwieger, M., Michalksi, S., Non-Linearity of Temperature Coefficients, Equivalent Cell Temperature and Temperature Behaviour of Different PV-Module Technologies. TÜV Rheinland Energie und Umwelt, Cologne, Germany. Proceedings from the 28th EU PVSEC.

# Mismatch

Compute the power lost due to module I(V) curve characteristic mismatches within one module bin. This is an empirical constant and suppresses the maximum power point on the I(V) curve. The mismatch can be applied to voltages within the vicinity of the I(V) curve’s knee, but is undefined for the array’s *I*_{sc} and *V*_{oc}.

## Inputs

## Outputs

## Algorithm (Provisional)

1.) The mismatch factor is applied directly to the effective irradiance, prior to the evaluation of the 1-diode or Sandia models:

## Algorithm (Future)

Mismatch may in future be embedded as part of a DC degradation algorithm. A simple solution can include the following.

1.) The depressed maximum power point is found as follows:

# DC Field I(V) Curve

This routine aggregates the I(V) curves from individual modules, and then sums them in parallel and in series to yield an I(V) curve profile that is representative of the entire DC Field. **Figure 37** shows a general schematic of such parallel-series connection.

### Figure 37. Array Comprised of N Series x M Parallel PV Modules

Currents add for parallel-connected modules and voltages add in series-connected modules.

## Inputs

## Outputs

## Algorithm

1.) For combination of identical modules in series and parallel, the points on the I(V) curves or current and voltages can be scaled as follows:

a.) Compute the aggregate I(V) curve for a series string comprised of *N*_{ms} identical modules. The voltages can be multiplied by the number of modules in the series string; the current remains unchanged:

b.) Compute the I(V) curve for a parallel connection of *N*_{sp} identical series strings in parallel modules. The currents can be multiplied by the number of strings in parallel; the voltage remains unchanged:

a.) **Parallel connection:** Combine heterogeneous modules or strings connected in parallel by interpolating the I(V) array at common voltage points so that the sampled currents are aligned; then sum the individual currents.

b.) **Series connection:** Combine heterogeneous modules or strings connected in series by interpolating the I(V) array at common current points so that the sampled voltages are aligned; then sum the individual voltages.

# DC Wiring

This routine computes the power lost due to Joule heating losses within the DC portion of the array. The resulting current and voltage is that seen at the inverter terminals. The resistance can also be estimated if a percent Joule loss (*P*_{JL}) at STC is provided instead of the equivalent bulk resistance. The resistance from a percent Joule loss is estimated as follows from the module’s rated STC current and voltage and number of series & parallel connections in the DC Field:

## Inputs

## Outputs

## Algorithm (Delta Series Resistance)

1.) Given an equivalent bulk resistance *R*_{DC}, find the delta series resistance seen by one module as follows:

*R*

_{S}is then added to the series resistance in the one-diode model.

3.) An approximation of the power dissipated by the bulk resistance can be obtained from the DC Field power seen by the inverter is as follows:

# Degradation and Plant Construction Schedule

Large power plants are not constructed and commissioned at one point in time, but come on line on a Block-by-Block basis. Therefore, different sections of the plant will start contributing energy staggered in time as each comes on line, and accounting for energization times lasts until the last section is energized and the full capacity of the plant is realized.

In addition, the long term power plant production reduces continuously over time due to environmental exposure, component aging, and accrued module defects. Three mutually-exclusive models are supported, and can be selected by the user on simulation setup:

### Figure 38. List of Degradation Models

### Figure 39. The 3 Degradation Models at 0.5% per year Rate for 4 years. The Linear AC and DC Degradation Follow the Same Line.

# DC Linear

Linear DC degradation function reduces the DC power input to the inverter as a linear function of time by the specified rate from the user.

## Inputs

## Outputs

## Algorithm

1.) Loop for all simulation steps for each Block

a.) The continuous-time function in hours is as follows:

b.) Find the system power due to energization time and degradation for time *t*.

# AC Linear

This function will reduce the AC power output as a linear function of time.

## Inputs

## Outputs

## Algorithm

1.) Loop for all simulation steps for each Block.

a.) The continuous-time function in hours is as follows:

b.) Find the system power due to energization time and degradation for time *t*.

# AC Stepped

This function will reduce the AC power output as a stepped function of time. The AC output of the block is reduced by the linear annual degradation rate at a discrete point in time.

## Inputs

## Outputs

## Algorithm

1.) Loop for all simulation steps for each Block

a.) The discrete-time function in hours is as follows:

b.) Find the system power due to energization time and degradation for time *t*.