Patent Publication Number: US-2011066401-A1

Title: System for and method of monitoring and diagnosing the performance of photovoltaic or other renewable power plants

Description:
RELATED APPLICATION 
     This application claims priority under 35 U.S.C. §119(e) from the co-pending U.S. provisional patent application Ser. No. 61/241,523, filed Sep. 11, 2009, and titled “Diagnostic System for a Photovoltaic (Renewable) Power Plant,” which is hereby incorporated by reference. 
    
    
     FIELD OF THE INVENTION 
     This invention is related to energy converters. More particularly, this invention is related to monitoring the performance and diagnosing any underperformance of energy converters, such as photovoltaic arrays. 
     BACKGROUND OF THE INVENTION 
     Because they rely on a freely available and renewable energy sources, are environmentally friendly, and pay for themselves by reducing energy costs, photovoltaic (PV) modules are used on an increasing number of homes and businesses. When PV modules are combined in a PV power plant, they can power entire communities. When these PV modules in a PV power plant are not operating at optimum efficiency, however, their underperformance is felt on a larger scale: Entire communities can be affected by lower power production. By some estimates, underperforming modules in PV power plants reduce productivity and resulting profits of the PV power plant operators by up to 20%. 
     Some PV power plants are monitored using “Symmetry Analysis,” a method that compares the currents through different strings of a PV module. When any two currents differ by a predetermined amount, the monitoring system determines that the string with the smaller current is underperforming and generates an alarm message. Other monitoring systems use a “day-before” comparison, in which the day&#39;s current through each string is compared to the previous day&#39;s current through the same string. Large enough differences again indicate a malfunctioning string. 
     Whatever abnormality-discovering method is used, staff are required to monitor the performance of the PV modules around the clock. This type of monitoring is only as effective as the staff are diligent and the measuring equipment is accurate. Even then, most staff members are not trained to determine whether any underperformance is truly indicative of a malfunctioning PV module and, if so, the cause. Even fewer staff are qualified to determine how to remedy the underperformance. By the time a problem is found and a remedy is applied, the accumulated lost productivity can be significant. 
     BRIEF SUMMARY OF THE INVENTION 
     In accordance with embodiments of the invention, a system monitors one or more energy converters, such as a photovoltaic array or wind turbine, to ensure that they are operating at acceptable levels. The system compares the actual output of the energy converter to a predicted output, generated using a mathematical model of the energy conversion unit. When the system determines that the energy converter is underperforming, it determines possible reasons for the underperformance, schemes to diagnose the underperformance, and remedial actions for increasing the performance to acceptable levels. All of this information can be displayed to personnel monitoring the output generated by energy converters. This information, or a subset of it, is then assembled into messages transmitted to personnel to service the energy converters. 
     In one aspect, a system for monitoring an efficiency or health status of an energy converter includes a module that determines an amount an output of the energy converter differs from a predicted output (an underperformance value), a possible cause of underperformance, a strategy for diagnosing the possible cause of the underperformance, a corresponding remedial action, or any combination thereof. The predicted output is based on operating conditions of the energy converter, such as a current time of day, a current month, or both. Alternatively, the operating conditions correspond to a microclimate surrounding the energy converter. 
     The system also includes a monitor for measuring the output of the energy converter and a transmission module for notifying an agent (e.g., a staff member or dedicated service personnel) when an underperformance metric of the energy converter exceeds a predetermined threshold. 
     The predicted output, the possible cause of the underperformance, a strategy for diagnosing the possible cause of underperformance, the corresponding remedial action, or any combination thereof are automatically determined using a learning algorithm. 
     In a second aspect, a system for monitoring an efficiency of a photovoltaic array includes a monitor that measures an output of the photovoltaic array and a first module that determines an amount the output differs from a predicted output of the photovoltaic array. The first module also determines a possible cause of underperformance for the photovoltaic array, a strategy for diagnosing the possible cause of underperformance, a corresponding remedial action, or any combination thereof. 
     Possible causes of underperformance include theft or vandalism of a component of the photovoltaic array, a fault in the photovoltaic array, a presence of an object blocking illumination to the photovoltaic array, or any combination thereof. 
     In one embodiment, the predicted output is determined from an amount of irradiation striking the photovoltaic array, an incidence angle of irradiation striking the photovoltaic array, a temperature of the photovoltaic array, or any combination thereof. Alternatively, or additionally, the predicated output is based on predetermined operating characteristics of the photovoltaic array. 
     In a third aspect, a system includes a first module that determines underperformance values of multiple energy conversion units in multiple different geographic locations and a second module that determines for each of the underperformance values a possible cause of underperformance, a strategy for diagnosing a cause of the underperformance, and a corresponding remedial action. The first module monitors outputs from each of the multiple energy conversion units to determine the underperformance values. The second module monitors current microclimates surrounding each of the multiple energy conversion units to determine the underperformance values. 
     In a fourth aspect, a device includes multiple light detectors aimed in different directions. The device is configured to determine irradiance impinging on the multiple light detectors. A portion of the multiple light detectors are directed outwardly at different angles about a central axis. The different directions include a first direction along a first vector and second directions at one or more angles to the first vector. 
     In one embodiment, the multiple light detectors include a pyranometer directed in the first direction and multiple photosensors directed in the second directions. Preferably, the device also includes an opaque shield between the pyranometer and the multiple photosensors. The shield is arranged, in size and location, to shadow the multiple light detectors from sunlight traversing an arc through a normal to the pyranometer. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
         FIG. 1  illustrates monitoring the efficiency of a PV module mounted to a roof of a house in accordance with embodiments of the invention. 
         FIGS. 2A and 2B  are displays of underperformance information and suggested remedial actions, in accordance with embodiments of the invention. 
         FIG. 3  shows the components of the PV module of  FIG. 1  in more detail. 
         FIG. 4  is a high-level diagram of components for remotely monitoring the performance of and dispatching service personnel to a PV module in accordance with embodiments of the invention. 
         FIG. 5  is a block diagram of components of a module for measuring performance of a PV module in accordance with embodiments of the invention. 
         FIG. 6  shows components of a data warehouse in accordance with embodiments of the invention. 
         FIG. 7  shows the functional relationship between a fault diagnostics inference engine and a data warehouse in accordance with embodiments of the invention. 
         FIG. 8  shows a table storing performance data in accordance with embodiments of the invention. 
         FIGS. 9A-C  show tables storing information used to predict performance data for a PV module in accordance with embodiments of the invention. 
         FIG. 10  is an insolation map used to predict performance of PV modules in accordance with embodiments of the invention. 
         FIG. 11  shows an a posteriori probability matrix in accordance with embodiments of the invention. 
         FIG. 12  shows a fault dictionary in accordance with embodiments of the invention. 
         FIGS. 13A and 13B  are perspective and top views, respectively, of a light detection module in accordance with one embodiment of the invention. 
         FIGS. 14A and 14B  are perspective and top views, respectively, of a light detection module in accordance with one embodiment of the invention. 
         FIG. 15  shows a table containing parameters for predicting an instantaneous power output from a PV module in accordance with embodiments of the invention. 
         FIG. 16  shows graphs of estimated current versus estimated voltage and estimated power versus estimated voltage, both used in accordance with embodiments of the invention. 
         FIG. 17  is a Web page displaying information about a PV site, sun, and PV module operating performance in accordance with embodiments of the invention. 
         FIG. 18  is a flow chart of steps to monitor and service underperforming PV modules in accordance with embodiments of the invention. 
         FIG. 19  depicts multiple locations containing PV modules and components for monitoring and servicing them in accordance with embodiments of the invention. 
         FIGS. 20A-C  show alternative energy converters monitored in accordance with embodiments of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In accordance with embodiments of the invention, energy converters, such as photovoltaic (PV) cells, wind turbines, and water turbines, are monitored in real time to ensure that they are performing at acceptable, pre-determined levels. Performance metrics, such as power generated for the day, are displayed on a Web or other page. When an energy converter underperforms, the amount of underperformance is automatically calculated and used to determine the cause of underperformance and possible remedial actions. The remedial actions are included in instructions to service personnel, who can then service the energy converter. By taking steps to quickly return the energy converter to acceptable levels of operation, the overall output of the energy converter is maximized, critical in large-scale energy conversion systems such as PV power plants. In this way, the overall health status of an energy converter (or multiple energy converters) can be monitored and maintained. 
     Underperformance can be determined in any number of ways. As one example, underperformance is measured as the difference between a predicted power output of the energy converter and its actual power output. When the energy converter is a PV module composed of multiple PV arrays (solar panels), the output is predicted by generating a mathematical model that characterizes an optimal output based on parameters such as solar radiation, temperature, time of day, orientation of the PV module to the sun, and PV module ratings, to name only a few such parameters. The actual output, whether measured in power, current, or voltage, is compared to this benchmark to determine an underperformance metric. The model can be refined over time to increase its accuracy. 
       FIG. 1  shows an energy conversion unit  100 , a photovoltaic (PV) array mounted atop a building  150 . The PV array  100  contains, among other things, an array of PV cells or panels and associated circuitry. Two displays, one display  155  inside the building  150  and another display  170  coupled to the PV module  100  over the Internet  160 , display the operating efficiency of the PV module  100 . The display  170  also indicates a possible cause of the underperformance, a broken cell of the PV module  100 . A technician viewing the message on the display  170  can be dispatched to repair the PV array  100 . 
     It will be appreciated that the displays  155  and  170  can be on any type of device. As only some examples, the display  155  is on a personal computer, a smart phone, or a personal digital assistant, and the display  170  is on a smart phone or a pager capable of displaying short message service (SMS) messages. 
     Many conditions can cause the PV module  100  to underperform, with corresponding different remedial actions.  FIGS. 2A and 2B , for example, show two messages  200  and  210  generated on different occasions when the PV module  100  underperforms by different amounts. As explained below, the messages  200  and  210  can be displayed on the display  155 , the display  170 , or on other displays at other locations. The “warning” message  200  in  FIG. 2A  was generated seconds after a leaf dropped onto a surface of the PV module  100 , covering much of one of its solar panels. The message  200  was generated based on the following parameters: One minute before the message  200  was generated, the measured output was 2,300 W, the predicted (benchmark) output was 2,650 W, and the irradiance was 3% higher than the present value. The message  200  states that an unexpected shading across the PV module  100  caused a loss of power. The message  200  includes a remedial action for remedying the loss: using a pole grabber to remove an obstacle from a surface of the PV module  100 . 
     The “alert” message  210  in  FIG. 2B  was generated after dust had been gathering on a surface of the PV module  100  over the last 3 months. The message  210  states that a possible layer of dust on the PV module  100  caused a loss of power. The message  210  also includes a remedial action: using a power washer to spray away the layer of dust. The message  210  was generated based on the following parameters: A maintenance log indicated that it had been 3 months since the last power wash, and no rain was recorded since then. The output is 3.4% below the benchmark output. A message is entered into a batch advisory to schedule a power wash for the PV module  100 . 
     The possible causes of underperformance and corresponding remedial actions can be determined from a number of factors, including the amount of underperformance, the rate of change of underperformance, current weather conditions, and the current season, to name only a few factors. For example, quick but large changes in performance during calm summer months can indicate a component failure, requiring the replacement of the component. Small but quick changes during windy months can indicate the falling of a branch or leaves onto a surface of the array of PV cells, requiring service personnel to bring a pole grabber. Small but gradual changes during dry months can indicate the accumulation of dust or soot on a panel of a PV module, requiring service personnel to bring a spray washer. Small but gradual changes during cloudy moments followed by a return to normal performance levels can indicate the movement of overhanging clouds, requiring no action by service personnel. 
       FIG. 3  shows some of the components of the PV module  100 , used to explain the subject of performance messages generated in accordance with embodiments of the invention. As in all the figures, identical labels refer to identical or similar elements. The PV module  100  includes separate PV cells or panels  101 A-P (collectively,  101 ) coupled to a module  110 . The module  110  includes combiner box instrumentation, a data acquisition controller (discussed below), and an Internet Gateway micro Web server. The module  110  is coupled over the line  140  to a local area network, which is coupled to the Internet  160  ( FIG. 1 ) to transmit performance metrics of (e.g., actual power generated by) the PV module  100 . The module  110  is also coupled to a load/grid (not shown) through an inverter  130 . The inverter  130  transforms DC power generated by the array of PV modules  101  into AC power available at the power sockets of the building  150 . The PV module  100  also includes an irradiance sensor  145  mounted adjacent to the PV cells  101 . 
       FIG. 4  is a high-level diagram illustrating how the performance of a PV module  300  is monitored in accordance with embodiments of the invention. As shown in  FIG. 4 , the PV module  300  contains a solar array with sensors  305  coupled to a combiner box with circuit modules and a Gateway  310 . The PV module  300  receives solar energy and translates it into any combination of AC current, voltage, and power (collectively referred to as AC power). The Gateway transmits information indicating the generated AC power over the Internet  330  to a Data and Web Server  315 , which includes a Knowledge Base and Algorithms, described in more detail below. The Data and Web Server  315  uses the information to determine whether the PV module  300  is underperforming and, if so, the amount of underperformance, reasons for the underperformance, and any remedial actions that can be taken to increase performance, collectively referred to as the “performance and service data.” The Data and Web Server  315  then transmits text or other data used by a system  325  to display the performance and service data, such as to service personnel. 
       FIG. 5  shows some of the components of the module  110  in accordance with one embodiment of the invention. The module  110  includes one or more (as shown by the overlapping rectangles) Signal Conditioning Multiplexers (SCM)  111 , one or more Custom Data Acquisition Modules (CDAQ)  113 , one or more Theft-Vandal Detectors (TVD)  115 , one or more Component Fault Detectors (CFD)  117 , and an Internet Gateway module  119 . (To simplify the following discussion, one or more of an element in  FIG. 5  will be referred to in the singular.) The SCM  111  accommodates a number of different sensor signals such as various brand and models of irradiance, temperature, voltage and current sensing transducers. The SCM  111  achieves “Galvanic” isolation, such that the circuitry inside the SCM  111  and its interconnected members are protected from the hazardously high DC output of the modules and the strings of the PV module  100 . The SCM  111  can be configured in the field for the specific needs of the PV module  100  being monitored, selecting the parameters transmitted to a database server (e.g., Data &amp; Web server  315  in  FIG. 4 ), discussed below. 
     The CDAQ  113  receives sensor signals from the SCM  111  and transforms the signals into computer-understandable words that are manipulated, compared, analyzed, and stored in database tables for future retrieval, processing, and display. The CDAQ  113  also transmits performance and other information, such as over the Internet, through the Internet Gateway module  119 , for display on a Web page. 
     Normally, the CDAQ  113  samples performance metrics at a rate of about 3 samples per second. In a hardware diagnostic mode, during which the CDAQ  113  performs burst sampling at a rate of up to 100,000 samples per second, the CDAQ  113  is also referred to as a high-speed digital signal processor because it brings signals from points at the interface between the PV cells and the inverter  130  ( FIG. 3 ). In the hardware diagnostic mode, the CDAQ  113  functions like an oscilloscope and is thus referred to as a “virtual scope signal processor.” The CDAQ  113  is used to generate what is alternatively called an “XY-display,” a “virtual oscilloscope displaying electrical waveforms,” or parameters with a time or frequency scale reference. 
     The TVD  115  is used to detect the theft or vandalism of components of the PV module  100 . The TVD  115  functions by monitoring “always-on” electrical signals, generated when the components of the PV module  100  are in place and working properly. When any component of the PV module  100  is disconnected or vandalized, a corresponding “always-on” electrical signal is turned off. This condition is transmitted to a database server and supervisory program for validation. An alert message is then generated and transmitted to designated personnel. 
     The CFD  117  monitors the PV module  100  for specific malfunctioning components, such as a leaky capacitor, shorted or open diodes, or other physical damage to the PV module  100  or inverter  130 . This condition is transmitted to the database server and the supervisory program for validation, and an appropriate alert message is transmitted to the designated personnel. 
     The module  110  includes computer-readable media containing the algorithms for performing the steps executed by the SCM  111 , the CDAQ  113 , the TVD  115 , the CFD  117 , and the Internet Gateway module  119 . The module  110  also includes one or more processors configured to execute those steps. The number of instances of each of the modules  111 ,  113 ,  115 , and  117  depends on the size of the entire energy converter. Each of the modules  111  and  113  is capable of accommodating up to 8 sensor signals, while each of the modules  115  and  17  works with one string of solar panels. 
     In accordance with embodiments of the invention, a data warehouse stores, among other things, performance and service data used to track underperformance and to determine remedial actions. In one embodiment, a data warehouse is remote from the module  110  ( FIG. 4 ). Those skilled in the art will recognize other locations on which a data warehouse can be housed. 
       FIG. 6  shows a data warehouse (DWD)  400  in accordance with embodiments of the invention. In one embodiment, the DWD  400  forms part of the Data and Web server  315  of  FIG. 4 . The DWD  400  stores records corresponding to performance and other data transmitted from the SCM  111 , the CDAQ  113 , and Gateway module  119  of  FIG. 5 . The data include the archived performance data for a PV module. The data are processed and transformed into a form that is suitable for storage on and retrieval from the Data and Web server  315  and that allows for easy access and processing by classification and diagnostic algorithms. Preferably, the Gateway module  119  has a non-volatile memory configured to store performance data for several days, in case an Internet or other connection is broken, preventing data from being transferred for remote storage and display. 
     Referring to  FIGS. 4 and 6 , the DWD  400  contains a performance record database (PRD)  401  that stores records of performance data for the PV module  300 , a Fault Dictionary (FDx)  410 , a conditional probability or a posteriori probability matrix (APM)  420 , and a “Knowledge Base System” (KBS)  420 . All data in the PRD  401  are date and time stamped. The FDx  410  correlates possible faults in the PV module  300  with a list of observables, a description of symptoms, and possible schemes or methodologies for diagnosing each fault. When underperformance is detected, the FDx  410  is queried to associate the symptoms with the schemes or methodologies to determine one or more possible remedial actions. 
     The APM  420  expresses the conditional relationship between various underperformance conditions and an array of possible fault sources. The APM  420  can be accessed and manipulated by algorithms to sort out the most likely faulty conditions from among a list of candidates to be selected and reported. The APM  420  is also affected by the “personality” characteristics as captured by “PV site attribute database elements,” such as shown in  FIGS. 9A-C  below. While  FIG. 6  shows a single APM  420 , it will be appreciated that the DWD  400  can contain any number of APMs, such as to simplify storage and updating, speed up processing to determine possible faults, or for any other reason. 
     The KBS  430  contains entries that correlate causes of underperformance with remedial actions. The KBS  430  stores underperformance information, remedial actions, and related information in a “knowledge representation” format that allows for manipulation by processing algorithms. As one example, the KBS  430  stores data, rules that indicate knowledge, and deduction rules, all manipulated by an induction engine that correlates symptoms, underperformance metrics, and remedial actions. The correlations can be updated and fine-tuned using learning algorithms. For example, it may be determined that a cause of underperformance is more likely than previously thought; the KBS is  430  is updated to reflect this. The underperformance values and corresponding remedial actions can later be stored in different formats, such as in a relational database, that allows for easy storage, retrieval, and display. 
     The DWD  400  also includes three software programs, a Knowledge Acquisition Module (KAM)  440 , a Knowledge Discovery Module (KDM)  450 , and a Fault Model Programming (FMP) module  460 . The KAM  440  and the KDM  450  function with the KBS  430  to acquire or discover new elements so as to increase or refine the knowledge as it relates to performance issues and characteristics for fault conditions in a solar power plant. The FMP module  460 , uses algorithms to develop one or more mathematical models used to receive any combination of measures of underperformance, current weather conditions, and operating characteristics of the PV module  110  and, from them, determine possible causes of underperformance and corresponding remedial actions. The FMP module  460  cooperates with the KBS  430  to account for conditions and to grow and evolve knowledge or algorithms for diagnosing faults. Using heuristics or other learning algorithms, these mathematical models are refined to more accurately predict the possible causes of underperformance, the remedial actions, or both. 
     The data in the DWD  400  are accessed by a Fault-Diagnostic Inference Engine (FIE), a supervisory program that takes symptoms of underperformance and returns possible causes of underperformance, corresponding remedial actions, or both.  FIG. 7  illustrates how an FIE  490  interacts with the DWD  400  in accordance with one embodiment of the invention. The FIE  490  receives as input symptoms of underperformance (e.g., a value of underperformance, such as ΔP, discussed below), uses the input to access the DWD  400 , and returns possible causes of underperformance and their remedial actions. The FIE  490 , which runs in the Data and Web server  315  ( FIG. 4 ), is always active, triggered whenever any underperformance is detected. 
     It will be appreciated that the elements described above are only illustrative of one embodiment and that any of the elements can be replaced with a similarly functioning element. For example, the FDx  410  can be replaced with any element that lists faulty components or external factors with attendant symptoms associated with each condition or failure. Similarly, the APM  420  can be replaced with any element that captures conditional probabilities associated with a given symptom of various faulty conditions due to internal failure or external conditions or factors, as accumulated from operational experience, or electrical or physical relationships. 
     It will also be appreciated that in accordance with embodiments, performance and other data can be collected independently of their analysis. Thus, for example, data can be collected periodically but analyzed in response to specific commands, for particular purposes. When current data is needed, for whatever purpose, a new data collection process can be initiated independently of, and thus without disturbing, any ongoing data processing. The data collection and data analysis components can thus be modular, operating independently of each other. 
       FIGS. 8-11  show tables used to determine underperformance values and corresponding remedial actions, all in accordance with embodiments of the invention.  FIG. 8  shows a performance database  500 . Each entry (row) includes, in sequential columns  501 A-E, a PV module identifier, a date, a time, a measured power output, and a measured current output. Thus, for example, row  1  shows that PV module  1  (column  501 A) generated 88,000 W (column  501 D) and  100 A (column  501 E) on Jan. 1, 2010, (column  501 B), at 12:00:00 p.m. (column  501 C). The next row shows similar information corresponding to 3 seconds later. In one embodiment, the performance data are generated by a CDAQ, such as the CDAQ  113  of  FIG. 5 . 
       FIGS. 9A-C  show tables in a relational database system used to predict power generated by a PV module in accordance with embodiments of the invention.  FIG. 9A  shows a table  510  that correlates PV modules with locations. Thus, for example, row  1  of table  510  indicates that PV module  1  is at Location  1 . In one embodiment, locations are represented by (longitude, latitude) pairs, or indirectly by city name or zip code.  FIG. 9B  shows a table  520  that correlates PV modules with manufacturer specifications and mount angles. The specifications for a particular PV module (e.g., power produced for a specific temperature and irradiance) and mount angle (e.g., angle that a PV module is positioned on a roof with respect to the sun at the azimuth) all factor into the power generated by a particular PV module at any particular time of day.  FIG. 9C  shows a table  530  that correlates locations to information about the sun (such as its angle to the azimuth, its intensity, and total irradiance) and temperature. Again, this information factors into the power generated by a PV module. The table  9 C can be populated periodically, such as every minute. 
     Referring to  FIGS. 9A-C , as one example, when a PV module is mounted on a rooftop, its latitude and longitude are entered into table  510 , and its manufacturer specifications and mounting angle are entered into table  520 . Periodically, such as once a minute, the sun information is updated in table  530 . When, as discussed below, a power output is predicted for a PV module, the identifier for the PV module is used to access tables  510  and  520  to determine the location and mount angle of the PV module. The location is then used as a key into table  530  to determine the sun information. The manufacturer&#39;s specifications, mount angle, and sun information are all used to determine a predicted output, referred to below as P opt . 
     In one embodiment, sun information is determined from an insolation map such as the insolation map  600  in  FIG. 10 . The insolation map  600  is derived from heat-sensing satellite instruments, indicating the intensity of solar energy impinging on the earth&#39;s surface. Insolation maps are available from a number of government and private entities, some free, or with minimal cost. Insolation maps are the basis of PV-Watts, a program developed by National Renewable Energy Laboratories (NREL) to assist with site analysis or partial performance analysis for generic solar sites. Alternatively, site information is determined by querying a weather service, such as over the Internet. 
     While  FIGS. 9A-C  show information for multiple PV modules, such as when multiple PV modules are monitored from a central location, in other embodiments information for only a single PV module is stored. 
       FIGS. 11 and 12  show, respectively, an a priori probability matrix (APM)  650  and a Fault Dictionary (FDx)  700  in accordance with embodiments of the invention. When a measured output of a PV module is smaller than the predicted output by a threshold amount, the amount of underperformance is used to query the APM  650  to determine a possible cause of underperformance, which in turn is used to query the FDx  700  to determine one or more corresponding remedial actions. 
     Referring to  FIG. 11 , APM  650  contains multiple entries (rows), each of which includes an amount (metric) of underperformance (e.g., ΔP=P opt −P measured )(column  651 ), a time interval over which ΔP occurred (column  652 ), and a probability (column  653 ) that a specific reason (column  654 ) causes the underperformance. For example, the first entry in table  650  indicates that a ΔP value of 10 W (column  651 ) over a 1 second interval (column  652 ) has a 60% probability (column  653 ) of being caused by a leaky capacitor (column  654 ). The second entry in table  650  shows that the same ΔP and ΔT have a 25% probability of being caused by the presence of a leaf on a surface of the PV module. The remaining entries in table  650  are similarly explained. 
     The entries in table  650  can be input in any number of ways, such as by an operator with statistics about causes of underperformance. Later, the entries can be updated automatically by learning algorithms known to those skilled in the art. The entries, or information corresponding to them, can be stored in a knowledge based system (KBS), such as KBS  430  in  FIG. 6 , which stores the information in a form suitable for knowledge processing. That information can then be translated into elements in the table  650 , suitable for quick retrieval and display. 
     The FDx  700  of  FIG. 12  contains multiple entries (rows), each of which includes a cause of underperformance (column  701 ), a method of diagnosing the cause (column  702 ), and a remedial action (column  703 ). For example, the first entry in FDx  700  indicates that a leaky capacitor (column  701 ) can be diagnosed by shunting the capacitor leads (column  702 ). If the capacitor is truly leaking, it should be replaced (column  703 ). The second entry in FDx  700  indicates that leafs on the panel (column  701 ) can be detected by visually inspecting the surface of the PV array of the underperforming PV module (column  702 ). If leafs truly are on the surface, the underperformance can be remedied by spray washing the surface (column  703 ). The remaining entries in FDx  700  are similarly explained. 
     Still referring to  FIGS. 11 and 12 , in operation, when underperformance of a PV module is detected, APM  650  is queried to determine the most likely cause. Using the cause, FDx  700  is queried to determine methods to diagnose the cause and corresponding remedial actions. In one embodiment, the diagnostic methods and remedial actions are assembled in a message displayed to monitoring personnel, transmitted to service personnel, or both. 
     Examples of Underperformance 
     The term “underperformance” can refer to any value that reflects a level of operating inefficiency of a PV module. For example, the term can refer to a percentage that the actual (e.g., measured) power (P A ) differs from the predicted power. The term can refer to the difference (ΔP), measured in Watts, between an optimal power output (P opt ) for a PV unit and P A . The term can refer to a normalized value, such as 1−(P opt −P A )/P opt . Those skilled in the art will recognize other values that can be used to measure the operating efficiency or inefficiency of a PV unit. 
     As used herein, “performance” can be refer to a measure of voltage, current, or power output from a PV module. Those skilled in the art will recognize other measurable parameters that can be used to indicate the performance of a PV module. 
     Mathematical Models 
     In accordance with different embodiments, one or more mathematical models are derived to determine what is variously referred to as an “optimal,” “predicted,” or “benchmark” performance value, such as power output (e.g., P opt , discussed above). 
     Applying the equivalent circuit theory by Thevenin and Norton, every PV array can be represented by an equivalent circuit for optimally operating the array, nominally derived from a datasheet of every brand and model of PV modules—namely open circuit voltage, short circuit current, maximum power voltage, and maximum power current, all by applying the series and parallel configuration of a PV array. Thus, an equivalent circuit of a well-functioning PV array can be characterized by a region in an IV-Chart, driven continuously by its environmental conditions, but nevertheless quantified mathematically. This dynamically changing region can be referred to as the “sweet spot” for a PV array or power plant. A set of entries in a family of database tables will fully characterize the generic, as well as unique, aspects of a PV site. 
     Equation (1) below is a mathematical model derived using characteristics for predicting the performance in accordance with one embodiment of the invention, used to determine “underperformance.” When P A  varies significantly from the mathematically computed “sweet spot” in Equation (1), the system is considered underperforming. 
         P   opt   =S *Cos(Φ)* D *Area(1−( K *( T− 25)))  Equation (1)
         where   S=irradiance   Φ=the incidence angle between an array of PV cells and the position of the sun   D=panel efficiency (usually between 14 and 18 percent, as derived from the manufacturers&#39; datasheet)   Area=total active area of the array of PV cells   K=temperature coefficient of the solar module per datasheet (e.g., 0.5/° C.)   T=temperature of the array of PV cells       

     The values S, Φ, and T can be measured in any number of ways. As one example, S is measured by a pyranometer mounted alongside the array of PV cells on top of a roof, Φ is determined by the time of day and current month, and T is measured by a thermocouple mounted alongside the array of PV cells. D is a rating, determined for each array of PV cells identified by manufacturer and part number. 
     Equation (1) estimates P opt  by sensing the direct normal component of sunlight. It has been determined that diffused components of sunlight also strike the surface of PV cells. This is especially pronounced on cloudy days, when a larger percentage of light striking a PV array is reflected or diffused light. In accordance with embodiments of the invention, an irradiance sensor is arranged to sense not only the direct normal component of sunlight but also directional diffused components, thereby more accurately detecting more of the energy striking the PV cells and thus more accurately predicting P opt  for the solar array. 
       FIG. 13A  is a perspective view of an irradiance sensor  750  in accordance with one embodiment of the invention. The irradiance sensor  750  has a housing that includes a base  770  supporting a funnel mount  770 . A rod  775  extends along a central axis (labeled z) of the funnel mount  770  and is topped by a pyranometer  751 . Eight photosensors  760 - 767  are uniformly spaced along the outer surface of the funnel mount  770 . As discussed more fully below, the pyranometer  751  and the photosensors  760 - 767  are all aimed in different directions, outwardly from the surface of the funnel mount  770 , arranged to capture direct sunlight and sunlight reflected from clouds, buildings, and other locations. 
     The pyranometer  751  and photosensors  760 - 767  all generate signals corresponding to the irradiance striking them. These signals are transmitted along the cables  751 A and  760 A- 767 A coupling the pyranometer  751  and photosensors  760 - 767 , respectively, to a processing module (not shown) that translates the signals into a combined irradiance metric for measuring a performance of a PV array. 
     Referring to the x-y-z coordinate system shown in  FIG. 13A , the pyranometer  751  is oriented (e.g., aimed or directed) along the z-axis, and each of the photosensors  760 - 767  is oriented to make the same angle Θ to the z-axis. Preferably, when installed on a rooftop or other location, the irradiance sensor  750  is mounted so that the z-axis is directed to the sun at its azimuth. In one embodiment, Θ is 45 degrees, but other values of Θ can be used. While each of the photosensors  760 - 767  is oriented to make the same angle Θ to the z-axis, it will be appreciated that any or all of the photosensors  760 - 767  can be oriented to make different angles Θ 0 , Θ 1 , . . . , Θ 7  to the z-axis. 
     When the z-axis is directed to the sun at its azimuth and the x-y plane is aligned with the horizontal, the angle that a particular photosensor  760 - 767  makes with the horizontal is referred to as the “elevation” or “tilt” angle. (This angle equals 90−Φ.) 
     It will be appreciated that the x-y-z coordinate system is shown only for explanation. Other reference systems, oriented in different ways, can also be used to describe the embodiments. 
       FIG. 13B  is a top view of the irradiance sensor  750 , taken along the z-axis. Each of the photosensors  760 - 767  is removed from an adjacent sensor by a 45 degree rotation (angular increment) about the z-axis, such that the angular difference between any two of the photosensors  760 - 767  is any multiple of 45 degrees between 0 and 345 degrees. In other embodiments, the photosensors  760 - 767  are spaced in non-uniform angular increments about the z-axis. 
     The angular rotation about the z-axis for a particular photosensor  760 - 767 , relative to a reference point, is referred to as the “pan angle” (Ω). Together, the tilt and pan angles define a direction. 
     Preferably, each of the photosensors  760 - 767  has operational characteristics similar to those of the junction materials in the PV array whose performance is being monitored. In this way, the photosensors  760 - 767  mimic and thus more accurately track the performance of the PV array. In one embodiment, the photosensors  760 - 767  are mono crystalline silicon sensors, though other types of sensors can also be used. 
     It will be appreciated that the photosensors  760 - 767  can be arranged in any number of ways to capture sunlight reflected from different directions. Furthermore, while the funnel mount  770  has a frusto-conical shape, it will be appreciated that mounts with other shapes configured to direct or aim the photosensors  760 - 767  outwardly, at different directions, can also be used. In other embodiments, at least some of the photosensors  760 - 767  are spaced non-uniformly along the outer surface of the funnel mount  770 . 
     In accordance with one embodiment, P opt  calculated for a PV array using the irradiance sensor  750  is determined by Equation (2): 
         P   opt =IrrEff*Cos(Φ)* D *Area*(1 −K *( Tc− 25))*FaultSources  Equation (2)
         where   IrrEff=Sum[Irr(i)*Cos(Ωi)]   Irr(i)=i-th sensor (e.g.,  760 - 767 ) mounted at the Ωi incidence angle (i=0 to 7), where Ωi (any one or more of which can be complex) varies from 0 to 359 degrees, as needed, to capture the commonly missed energy components   Φ=the incidence angle between an array of PV cells and the position of the sun   D=panel efficiency (usually between 14 and 18 percent, as derived from the manufacturers&#39; datasheet)   Area=total area of the array of PV cells   K=temperature coefficient of the solar module per datasheet (e.g., 0.5)   T=temperature of the array of PV cells   FaultSources=All known sources of external factors that impact the array output (e.g., sources stored in the Table  650  of  FIG. 11 )       

     It has been determined that the accuracy of irradiance measurements is increased by substantially limiting one set of light sensors to measure direct normal sunlight and another set to measure indirect, diffused light. In accordance with one embodiment,  FIG. 14A  is a perspective view of an irradiance sensor  790 , and  FIG. 14B  is a top view taken along the z-axis. The irradiance sensor  790  includes all the components of the irradiance sensor  750  but also has a collar (e.g., light shield or light shade)  785  positioned between the pyranometer  751  and the photosensors  760 - 767 . (For clarity, the labels  751 A and  760 A- 767 A are not included in  FIGS. 14A and 14B .) The light shield  785  is opaque and arranged to substantially shield the photosensors  760 - 767  from sunlight as the sun traces an arc that includes a normal to the pyranometer  751 . In one embodiment, the arc spans 45 degrees. Thus, when the sun is within this arc, the sunlight falls almost exclusively upon the pyranometer  751 . Diffuse sunlight outside that range, such as that reflected from clouds and buildings, falls upon the photosensors  760 - 767 . 
     It will be appreciated that the light shield  785  can have different configurations and still achieve the principles of the invention. In the embodiment of  FIGS. 14A and 14B , the horizontal surface of the light shield  785  is substantially perpendicular to the rod  775 . In one embodiment, the light shield  785  has a radius of 4 inches, a circle  771  centered on the z-axis and delimited by the photosensors  760 - 767  has a radius of 2 inches, and the circle  771  and the light shield  785  are 1.25 inches apart. As shown in  FIG. 14B , the light shield  785  entirely overlies the circle  771 . It will be appreciated that the components can have other dimensions and can be arranged in different ways. For example, the surface of the light shield  785  can make other angles with the rod  775  and can have other shapes, so long as it substantially shields the photosensors  760 - 767  from direct sunlight in the manner discussed here. 
     Preferably, the light shield  785  includes an opaque material. Also preferably, the light shield  785  is constructed to withstand temperature extremes, precipitation, wind, and other outdoor conditions. As one example, the light shield  785  comprises stainless steel with an anti-reflective coating. Those skilled in the art will recognize other suitable materials for the light shield  785 . 
     The light shield  785  can be temporarily removed for calibration or during troubleshooting or maintenance operations. 
     In different embodiments, the irradiance sensor  750  or the irradiance sensor  790  replaces the irradiance sensor  145  shown in  FIG. 3 . 
     The irradiance sensors  750  and  790  leverage the power of embedded computing and intelligent server resources to capture direct and diffused energies from the sun. Preferably, the irradiance sensors  750  and  790  contain no moving parts and thus are low-cost approaches for sensing light energy. 
     Model Parameters 
     Every PV power plant site is uniquely defined by a set of characteristics such as location-latitude and longitude, mounting of the individual PV modules, brand and model of the PV modules, and micro-climate of the site, to name only a few characteristics. This “personality,” sometimes characterized qualitatively, other times quantitatively, is used to determine any operating abnormalities. 
       FIG. 15  shows a table  800  that includes mathematical model parameters for predicting instantaneous power output (e.g., P opt ) for a PV module using Equation (1), in accordance with embodiments of the invention. (While some of the following examples discuss Equation (1), the principles apply equally to Equation (2).) Table  800  shows, in columns  801 - 804  respectively, (1) attributes, (2) symbol terms or modules, (3) nominal ranges, and (4) modifiers or deviations from the norm. Referring to each entry (row) in turn, table  800  includes the attribute “Power Output,” which has a nominal range of 0 to 110% of Standard Test Conditions (STC) rating; an irradiance, which has a nominal range of 0 to 1,350 W/m 2 ; a UV index, which has a nominal range of 0 to 13; a smog index, which has a nominal range of 0 to 200; a cell temperature, which has a nominal range of −20° C. to 100° C.; an ambient temperature, which has a nominal range of −40° C. to 50° C.; an incidence angle (sun&#39;s ray to normal), which has a nominal range of 0° to 90°; an azimuth (degrees from North), which has a nominal range of 90° to 270°; a tilt angle (degrees from the horizon), which has a nominal range of 0° to 90°; a latitude and longitude; a wind speed; a dust and soot accumulation value, which has a nominal range of 0 to 20% by millimeter; a shading; a system aging degradation value, which has a nominal range of 0 to 1.5% per year; a component defect value; and a wiring-connection value. 
     The entries in table  800  are all taken into account when modeling Equation (1).  FIG. 16  shows two graphs, the first graph  850  plotting estimated current (on the y-axis) versus estimated voltage (on the x-axis), the second graph  860  plotting estimated power (on the y-axis) versus estimated voltage (on the x-axis), both generated using Equation (1). The two graphs  850  and  860  are used to compare the ideal macro-IV Chart and Fault Condition. Superimposed on the graph  850  are points () showing the actual current and voltage measured on a PV module. Superimposed on the graph  860  are points (▪) showing the actual power measured on the PV module. The smaller the distances between (1) the points () and the graph  850  and (2) the points (▪) and the graph  860 , the more accurate Equation (1). The accuracy of Equation (1) can be increased by adjusting its parameters, thereby refining components that rely on it, such as the fault-detection, fault-modeling, and fault-diagnosing programs used in accordance with embodiments of the invention. 
     The benchmark output in power, voltage, or current (and thus Equations (1) and (2) above) is based on different parameters, such as the materials from which the PV module is made, the test conditions used to rate the performance of the PV module, and other factors, all of which are discussed below. 
     Materials and Composition 
     Among other things, the performance of a PV module depends on the module material. The conversion efficiency of amorphous silicon modules varies from 6% to 8%. Modules of multi-crystalline silicon modules have a conversion efficiency of about 15%. Mono-crystalline silicon modules are the most efficient, with a conversion efficiency of about 16% to 24%. Modules are roughly 1 m 2  to 1.5 m 2  in area, and getting larger, and typically include between 36 and 72 individual PV cells. 
     Standard or Practical Test Conditions 
     The DC output of solar modules is rated by manufacturers under Standard Test Conditions (STC). These conditions are easily recreated in a name-plate and allow for consistent comparisons of products, but they must be modified to estimate output under common ambient operating conditions. STC conditions include a solar module temperature of 25° C.; a solar irradiance (intensity) of 1,000 W/m 2  (often referred to as peak sunlight intensity, comparable to clear summer noon-time intensity); and a solar spectrum as filtered by passing through 1.5 times normal of atmosphere (ASTM Standard Spectrum). A manufacturer can rate a particular solar module output at 200 Watts of power under STC and call the product a “200-watt solar module.” This module will often have a production tolerance of +/−5% of the rating, which means that the module can produce 190 Watts and still be called a “200-watt module.” 
       FIG. 16 , a graphical presentation of the current versus the voltage (I-V curve) from a photovoltaic module, was generated by rapidly sampling of array voltage and current values. The shape of the curve characterizes module performance; this can be called “name-plate performance” or performance of a PV module under specified operating conditions. 
     Light Energy Spectrum Response 
     The electrical current generated by photovoltaic devices is also influenced by the spectral distribution (spectrum) of sunlight. It is also commonly understood that the spectral distribution of sunlight varies during the day, being “redder” at sunrise and sunset and “bluer” at noon. The magnitude of the influence that the changing spectrum has on performance can vary significantly, depending on the PV technology being considered. In any case, spectral variation introduces a systematic influence on performance that varies by time-of-day. Similarly, the optical characteristics of PV modules or pyranometers can result in a systematic influence on their performance related to the solar incidence angle. 
     Cell Temperature 
     Since roughly 80% of the sun&#39;s energy is dissipated into heat, PV module output power reduces as the module temperature increases. When operating on a roof, a solar module will heat up substantially, reaching inner temperatures of 50° C. to 75° C. For crystalline modules, a typical temperature reduction factor recommended by the California Energy Commission is 89% or 0.89. Therefore, the 200-Watt solar module will typically operate at about 170 Watts (190 Watts*0.89=170 Watts) in the middle of a spring or fall day, under full sunlight conditions. To ensure that PV modules do not overheat, they must be mounted in such a way as to allow air to move freely around them. This is particularly important in locations that are prone to extremely hot midday temperatures. The ideal PV module operating conditions are cold, bright, sunny days. 
     Dust or Soot 
     Dust or soot can accumulate on the PV module surface, blocking some of the sunlight and thus degrading output. Although typical dust is washed away during rainy seasons, it is more practical to estimate system output taking into account the reduction due to dust buildup in the dry season. A typical annual dust reduction factor is approximately 5% or 0.95. Therefore, a 200-Watt solar module operating with some accumulated dust may operate, on average, at about 79 Watts (170 Watts*0.93/2=158 Watts/2). 
     A 1.6 GW STC group of grid-tied solar arrays (as specified under STC conditions) located on the Googleplex in Mountain View, Calif., U.S.A., was studied by a team at Google and publicized. As confirmed by the study, layers of dust or soot that accumulate over time may degrade the PV module&#39;s output by as much as 7%. The mathematical models of Equations (1) and (2) can thus be enhanced with an element that represents the accumulated layer of dust, with modifiers for a region&#39;s dust and rainfall characteristics, which can be tracked and modified by the occurrence of rainfall or cleaning. A nominal 0.1% degradation may be used as baseline model, for every week that goes by without any intervening event, such as rain or high winds. 
     Mismatch and Wiring Losses 
     The maximum power output of a total PV module is always less than the arithmetic sum of the maximum output of the individual modules. This difference is a result of inconsistencies in performance among modules, and is called “module mismatch,” which can result in roughly 2% loss in system power. Power is also lost due to resistance in the system wiring. These losses should be kept low with proper wire-sizing and good workmanship, but it is often difficult to keep them below 3%. A common derating factor for these losses is 95%. 
     DC-to-AC Conversion Efficiency 
     The DC power generated by the solar module must be converted into common household AC power using an inverter. Some power is lost in the conversion process, and there are additional losses in the wires from the rooftop array, down to the inverter, and out to the house panel. Modern inverters commonly used in residential PV power systems have peak efficiencies of 92% to 94%, as indicated by their manufacturers, but these again are measured under well-controlled name-plate conditions. Actual field conditions usually result in overall DC-to-AC conversion efficiencies of about 88% to 92%, with 90% or 0.90 a reasonable compromise. Thus, a 200-Watt solar module output, reduced by production tolerance, heat, dust, wiring, AC conversion, and other losses should translate into about 136 Watts of AC power delivered to the house panel during the middle of a clear day (200 Watts*0.95*0.89*0.93*0.95*0.90=134 Watts). 
     Calculating System Power Output 
     The PV module should be positioned and mounted to absorb the most energy from the sun. If the photovoltaic modules have a fixed position, their orientation with respect to the south (northern hemisphere), and tilt angle, with respect to the horizontal plane, should be optimized. For grid-connected PV systems in the U.S., for instance, the optimum tilt angle is about 25 degrees. For regions nearer to the equator, this tilt angle will be smaller, and for regions nearer the poles, it will be larger. The output from the array will rise gradually from 0, during dawn hours, increase with the sun angle to its peak output at solar noon, and then gradually decrease into the afternoon and back down to 0 at night. While this variation is due in part to the changing intensity of the sun, the changing incidence angle also has an effect. The pitch of the roof or tilt angle or structural frame will affect the sun angle on the PV module plane (e.g., angle θ in  FIG. 1 ), as will the azimuth orientation of the roof. These effects and others are all taken into account by the mathematical model in table  800  in  FIG. 15 . 
     Display Mechanism 
     Performance information, remedial actions, and other types of data measured and generated in the embodiments can be displayed in any number of ways. Messages can be transmitted for display to the building to which the PV module is mounted, to a central location used to monitor multiple PV modules at geographically dispersed locations, to a repair person making rounds, or to any other person or location. The information can be transmitted over local area networks, over the Internet, using wireless transmissions such as WiFi or cellular, to a cell phone or personal digital assistant, or by any other means. 
     Preferably, messages are categorized according to the amount that the output is degraded, the amount of underperformance. As one example, a message is categorized as an “alert” when system performance is 10% below the norm, as a “warning” when system performance is 20% below the norm, and as an “alarm” when system performance is 30% or more below the norm. With these categorizations, service personnel can quickly determine in what order and how quickly sites must be serviced. When the system performance is within acceptable limits, such as no more than 5% below the norm, an “OK” message, along with a relative percentage of the benchmark level, is transmitted, thereby letting operators know that the notification system is functioning. Of course, other thresholds based on other percentages of degraded output can also be used. 
     In one embodiment, one or more Web pages or other electronic textual elements display various parameters used to track the performance of a PV site such as:
         solar irradiance   cell temperature, such as measured using one or more sensors attached to a back of a solar module   time-of-day   photovoltaic power production in kW   photovoltaic power (kW) relative to utility-provided electrical power plants   photovoltaic power (kW) on a time scale, total photovoltaic power production   daily power production (kW), power production relative to utility power consumption   solar power production (kW) over the lifetime of the PV module   daily solar production relative to maximum possible production, and   a benchmark bar graph illustrating the current day&#39;s solar electricity production, hour-by-hour       

     Preferably, a user is presented with information that allows him to track the output of a PV module or PV power plant and understand why the PV module or PV power plant is not performing as expected.  FIG. 17  is a Web page  900  generated in accordance with one embodiment of the invention, showing Tables  900 A-C. The Table  900 A contains site and array information including the date, both Greenwich Mean Standard and local, a site identifier, a latitude and longitude, a panel tilt angle, a panel azimuth, a cell temperature, the number of panels at the PV site, the panel identifier by manufacturer and part number, the site area, the site efficiency, and the panel efficiency. The Table  900 B contains sun position information including azimuth, incidence angle, irradiance, and output. The Table  900 C is a PV Performance Lookup Table showing P opt  values calculated according to Equation (1), for a particular value of the irradiance angle Φ, 55° 30′ 04″. 
     Web Services 
     In some embodiments, customers can subscribe to Web services offered in accordance with the embodiments. With this service, a central site operator monitors PV arrays at a customer site and provides the customer with one-time or periodic reports detailing the performance of the PV arrays. The customer can select the type of performance data included in the reports. 
     Virtual Visit Solar Site Assessment Reporting 
     In some instances, the actual irradiance striking a PV array cannot be determined. For example, a location is too remote for personnel to install an irradiance sensor adjacent to PV arrays. In accordance with one embodiment, a mathematical model (e.g., Equations (1) and (2), above) is generated using parameters other than the irradiance, such as air temperature or other environmental data surrounding or sufficiently close to the location being monitored. The location is thus “virtually” visited. In one embodiment, agent-like programs are dispatched to harvest environmental data surrounding an area and used to approximate modeling parameters. 
     Examples of Determining Underperformance 
       FIG. 18  shows the steps  1000  of a process for detecting underperformance and determining corresponding remedial actions in accordance with embodiments of the invention. The process starts in the step  1001  in which any parameters are initialized. In the step  1003 , the process determines illumination parameters for a PV module by accessing table  510  (to read the location of the PV module) and table  530  (to retrieve the sun information), in  FIGS. 9A and 9C , respectively. Next, in the step  1005 , the process uses the illumination parameters and PV module characteristics from table  520  ( FIG. 9B ) to predict the performance for the PV module (P opt ) using Equation (1) or (2) above. In the step  1007 , the process determines the actual (measured) performance of the PV module (P A ) from Table  500  in  FIG. 8 . In the step  1009 , the process compares the actual performance and the predicted performance to determine any underperformance (ΔP=P opt −P A ). In the step  1011 , the process determines whether the underperformance (ΔP) is greater than a threshold level. As one example, the threshold level is a 5% difference. If the underperformance is greater than the threshold, then the process continues to the step  1013 ; otherwise, the process continues to the step  1021 . 
     In the step  1013 , the process accesses Table  650  in  FIG. 11  to determine one or more causes of underperformance. Preferably, the process selects the most likely cause of underperformance, such as determined from column  653  in  FIG. 11 . Alternatively, multiple causes, arranged from most likely to least likely, are selected. In the step  1015 , the process uses the one or more causes of underperformance to access the Table  700  in  FIG. 12  to determine one or more diagnostic tests, remedial actions, or both. In the step  1017 , the process transmits a description of the causes, diagnostic tests, remedial actions, or any combinations of these to a display. In the step  1019 , the one or more causes are stored in a history database. In the step  1021 , the process waits T time units and then returns to the step  1003 . As one example, T is 1 second, though any other time unit can be used. 
     In alternative embodiments, the step  1017  is supplemented with the step of automatically taking the remedial action. As one example, when the remedial action is spray washing a surface of the PV module, this action is taken automatically by triggering a rooftop sprinkler system to wash away leaves or other debris. Those skilled in the art will recognize other remedial actions that can be taken automatically. 
     It will be appreciated that the steps  1000  of  FIG. 18  are merely exemplary. Some of the steps can be deleted, other steps can be added, and the steps can be performed in different orders. 
     It will be appreciated that references to “cause of underperformance” and “remedial action” can refer to single or multiple causes and remedial actions. Each is referred to in the singular merely to simplify the discussion. 
     In one embodiment, the steps  1000  are performed by a processor executing instructions on a computer-readable medium. In different embodiments, the computer-readable medium is programmed using software, hardware, firmware, any other means for executing instructions, or any combination of these. It will be appreciated that the functionality shown in  FIG. 18  can be distributed among the different components and locations in any number of ways. For example, underperformance can be determined at the structure on which the PV module is mounted or at a central location. Considerations include the distribution of processing loads, the desire to reduce the amount of information that must be transmitted over the Internet, down Internet connections, and the like. 
     While the examples discussed above illustrate monitoring a single PV array at a single location, it will be appreciated that multiple PV modules at different geographic locations can be monitored at a one or more central locations.  FIG. 19  shows a system  1100  for centrally monitoring multiple PV modules at different sites  1110 ,  1115 , and  1120 , and dispatching service personnel to each, such as from a service distribution site  1150 . The system  1100  includes a Web page display unit  1101  and a database management system (DBMS)  1105  that includes a Data and Web server with KnowledgeBase &amp; Algorithms, such as discussed above. The system  1100  includes one or more processors and computer-readable media for performing the algorithms (e.g., the steps  1000  in  FIG. 18 ) discussed herein. The system  1100  is configured to receive actual performance values from a PV module (e.g., power, current, or voltage generated, or any combination of these) over the Internet, over a LAN, using wireless communications such as WiFi, or using any other communications means. The DBMS  1101  calculates any underperformance of PV modules  1 - 3 , and determines causes of underperformance and any remedial actions. 
     The Web page display unit  1101  shows information for the PV modules  1 - 3 , respectively, similar to that includes in the Web page  900  in  FIG. 17 . The system  1100  also transmits messages to service personnel at the distribution site  1150  or in the field. The messages, similar to the “alert,” “warning,” and “alarm” messages discussed above, include the location of a PV module, the amount of underperformance, a cause of underperformance, possible diagnostic schemes, and one or more remedial actions for the service personnel to take. 
     While the examples above describe PV modules, it will be appreciated that the principles of the invention are suitable for monitoring the output of other types of energy conversion units.  FIG. 20A , for example, shows a wind farm that includes multiple wind turbines  1201 A-D that provide electrical power to a building  1202 . Possible causes of underperformance are the presence of branches or other obstacles in the turbine blades, theft or vandalism of components of the wind turbines, dust on the blades, and the like.  FIG. 20B  shows a hydro-electric system  1210  that includes multiple micro-hydro turbines that provide electrical power to a building  1212 . Possible causes of underperformance are the presence of branches or rocks clogging the inlet or upstream obstructions to water flow.  FIG. 20C  shows a solar heating system that includes a collector  1220  and a storage tank  1222 . In accordance with embodiments of the invention, the wind turbine, hydro-electric, and solar water heating systems of  FIGS. 20A-C , respectively, each includes systems that determine underperformance, possible causes of underperformance, diagnostic schemes, corresponding remedial actions, and means for displaying or transmitting each of these elements, such as described throughout this application. 
     In accordance with embodiments of the invention, multiple PV modules are mounted to rooftops at sites at different geographic locations. For each PV array, information is stored at a central location, information such as location (e.g., latitude and longitude), operating specifications for the PV module, and orientation relative to the sun&#39;s azimuth angle for the location. A mathematical model used to predict performance (e.g., power output) for each PV module is generated. The predicted performance is based on the current time and the current weather conditions surrounding the PV module, including the intensity of the sun, cloud cover, wind speed, and the like. Preferably, the predicted performance is modeled, for example, using Equation (1) or Equation (2) above. The central location also houses a database populated with possible causes of underperformance, diagnostic methods, and remedial actions. 
     In operation, the output of each PV module is periodically measured and transmitted to the central location. At the central location, performance information for each PV array is displayed to monitoring personnel. The measured performance is compared to the predicted performance, and if the difference between the two is above a threshold value, the system determines that the PV module is underperforming. 
     The amount of underperformance is used, with other variables such as the current weather and output history of the PV module, to determine diagnostic strategies and remedial actions. The diagnostic strategies and remedial actions are explained in messages transmitted to service personnel who are dispatched to service the underperforming PV arrays. In this way, the output of the PV arrays can be maintained at optimal levels; preferably, any diminished output levels are restored. 
     In one embodiment, the predicted performance is based on irradiance measured at each site. The irradiance is determined by an irradiance sensor having a pyranometer directed to the sun at its azimuth and multiple photosensors directed at various angles relative to the pyranometer. An opaque light shield is located between the pyranometer and the multiple photosensors. 
     After service personnel have visited sites, they input data indicating the actual causes of underperformance, strategies they used to determine the actual causes of underperformance, and the actual diagnosed cause of underperformance. This updated data is used by learning systems and other artificial intelligence components to update and refine the mathematical models (e.g., the coefficients of the mathematical models) and the databases that correlate the underperformance metrics, the causes of underperformance, the diagnostic strategies, and the remedial actions. 
     After reading this application, those skilled in the art will recognize many possible variations within the spirit of the invention. For example, a table of the causes and remedial actions can be stored on a device carried by service personnel. In this way, rather than transmitting text describing the causes of underperformance and corresponding remedial actions to service personnel, only the indices corresponding to the table entries need to be transmitted. It will be readily apparent to one skilled in the art that other modifications may be made to the embodiments without departing from the spirit and scope of the invention as defined by the appended claims.