Power Curve Analysis - Concentrating Solar Thermal

Introduction

Here we establish a power curve for transforming solar radiation into power output from Concentrating Solar Thermal (CST) plant. That is, we want to model power output given input solar radiation data. This is analogous to the wind power curves, except in this case we do not (so far at least) have reference data to compare against. Consideration of photovoltaics is here.

As general background, here is a very readable general overview on Concentrating Solar Power Systems.

Pre-existing Software Tools: Modelling and analysis tools (such as SAM) have been developed to solve the sort of problem we have here. However, two factors prevent looking into these tools in detail now: (i) the need for a fully understood baseline before employing 'black boxes', and (ii) time constraints, especially in relation to establishing a specific solar plant design. At a later time OzEA may choose and use a sophisticated package to estimate power output from solar plant.

For now [Feb 2011] we have a solid and generic first pass, which in turn has been used (with the data linked above) to give the first pass analysis of simulated CST Farms. The work here remains open for revision in light of any significant issues that arise.

Solar Thermal

Here the specific problem is obtaining a reasonable power-curve to proceed with. By considering the physics in overview and establishing parameter values we arrive at an informed but rough estimate. The goal is to be as simple, reasonable and conservative as can be managed.

What follows is based on a tower system. That is, a collector tower surrounded by heliostats, being reflectors that track the sun so as to reflect the direct sunlight onto a receiver, whence the concentrated light is absorbed and the resultant heat used to power a 'heat engine' that in turn runs a generator to produce electricity. We confine our attention to the heat engine being a conventional steam turbine (much as found within a coal power plant).

The problem breaks down into a number of component parts:
- the amount of solar radiation captured for concentration;
- the net heat energy flux captured at the receiver;
- the flow of heat energy through the 'engine', and that converted into mechanical work;
- the efficiency of the generator in producing electricity.

Considering each of these in turn:

The amount of solar radiation captured for concentration is the product of the direct solar radiation strength in W / m2, with the area intersected. Because the reflectors are working to direct the suns rays at the receiver on the tower, they will be at some angle from perpendicular to the rays, and this angle will change according to the time of day, day of the year, and the position of each reflector in relation to the tower. There is also the issue of heliostats shadowing each other, especially early and late in the day. These geometrical issues complicate a generalised analysis, but can be resolved in a routine manner. We assume for now that the collective behaviour of all the heliostats is designed to even out; clarification on this is being sought.

So, we take the intercepted power from the sun as: P_in = A_cap.P_sol where A_cap = A_cap(t) is the capture area and P_sol is the flux of direct solar radiation at each time point (e.g. 800 W/m²).

The net heat energy flux captured at the receiver (per m² of intercepted rays) can be written as:

P_cap = 1/A_cap . ( A_cap η_opt α P_sol  -  A_rec P_SB )

where first A_rec is the receiver area, and we define the concentration, C = A_cap / A_rec, to give:

P_cap = η_opt α P_sol  -  P_SB / C              (W / m² of intersected direct solar radiation)

where η_opt is the efficiency of the collecting optics / reflectors (0.9 for now), α is the collector absorptivity or fraction of incident radiation absorbed (0.95 for now), P_sol is the (Direct) solar radiation, and P_SB is the radiative losses as per the Stefan Boltzmann equation (detailed shortly). Also we have introduced C as the concentration factor, which we set conservatively at around 600. Of course this concentration factor depends on the ratio of the above intercepted cross-section of direct sunlight over the area of the receiver, and thus may vary in time with A_cap. We keep this understanding in the margin and proceed with C rather than any explicit area values.

The next issue is the radiative losses from the receiver, as per the Stefan Boltzmann equation: P_SB = ε σ T_H ⁴

Here σ is the Stefan-Boltzmann constant (5.67e-8 J / s / m² / K⁴), and ε is the emissivity of the collector. Emissivity is a physical, temperature dependent, property that describes how closely a material corresponds to a theoretical "blackbody" in the radiation it emmits at a given wavelength. Heat anything up and it will glow. Emissivity is the flip side of absorptivity (used above), which describes the fraction of incident radiation absorbed, and the two are related. For now we keep things simple and set α = ε = 0.95 for the reciever (see comment #3).

Also need a value of T_h, which we set at 900 K (as explained shortly).

The flow of heat energy through the engine, and that converted into mechanical work is simple for the first part and more complex for the second. If no energy were to flow, including ignoring any conductive loss flows, then the temperature of the receiver would increase until the radiative losses (as defined by the Stefan-Boltzmann equation) equaled the incoming energy. With ~600 x concentration, this temperature would reach 900 K when P_SB = ε σ 900⁴ = C η_opt α P_sol, which works out as P_sol = 69 W / m2, which is a low value for the direct radiation. Considerably higher receiver temperatures are posible at this level of concentration -- as usefully laid out on this Wikipedia page with relevant section snapped: part1, part2 -- however, standard steam turbines operate at around 800 K; the extra hundred is included as an allowance for temperature drop between the receiver and the steam boiler.

Now; the topic of heat engines and thermal efficiency needs a little discussion. In a heat engine some sort of working fluid is heated and expands, pushing a turbine or piston. This only happens because the higher temperature (and thus pressure) on the hot side is equalising with the cooler, lower pressure, on the other side. The bigger the difference between the temperatures (pressures) at either end, the more 'push' can be extracted. Generally the working fluid (water / steam in our case) is cycled through heating, expansion, heat-rejection (i.e. cooling) and compression phases.

Theory puts a clear upper bound on the fraction of the energy flowing through a heat engine that can be extracted as useful work (mechanical energy for turning a generator in this case). It is called the Carnot cycle, or limit. An idealised heat engine working with hot end temperature T_h and a cold end at T_c (in absolute temperature, i.e. degrees Kelvin) has a Carnot efficiency of (T_h - T_c) / T_h as the maximum possible. The rest of the energy must be dissipated into the environment as waste heat at temperature T_c. In practice, real heat engines are doing well to obtain 2/3 of what Carnot shows can be possible.

The key issue in relation to theoretical vs. actual efficiencies is the heat exchange at either end. In a theoretical environment this can be done slowly and carefully and throughly, whereas in a real engine the process is fast and imperfect, with cost-benefit considerations in the engineering to consider.

A steam turbine engine, as a specific type of heat engine, has its own well described theory; the Rankine cycle. The efficiency of a Rankine cycle depends on particular details, and is not so useful to us here. A way forward is that real (as opposed to ideal) heat engines are sometimes well described by Curzon-Ahlborn Engine Theory, the important result being the efficiency as: η_thermal = 1 - sqrt( T_c / T_h ), and (for now at least) we proceed with this instead of Carnot or a Rankine cycle parameterisation. Using T_h = 900 K (as above) and T_c = 400 K (~130 deg. C), gives a thermal efficiency of ~ 33%. Future concentrating solar thermal plant can be expected to do better, and perhaps operate at higher temperatures, but for now we take this as a reasonable starting point.

And finally, the efficiency of the generator in producing electricity is taken to be 95% (unless and until we know better).

So, the parameters we are working with, for now, are:

And so we can write the power output as:

P_out = η_gen η_thermal P_cap
        = 0.32 ( η_opt α P_sol  -  ε σ T_H ⁴ / C )
        = 0.32 ( 0.86 P_sol  -  59 )              (W / m² of intersected direct solar radiation)

Here is a plot of this curve:

The reality of engineering the infrastructure will involve tuning the design to be optimal at some level of input power, and from inspection of the solar data (distributions for raw solar data) it is seen that P_sol ~ 800 W / m² is a likely value for design optimisation.. At lower and higher values the operation may be sub-optimal.

In the above table of parameter values, most are simply constants, whereas in reality they will vary somewhat depending on the system state. With advice from people specialising in this area we may be able to capture the principle functionality, but for now this is accounted crudely to give the magenta power curve above, which we adopt as the OzEA round one power curve for a concentrating solar thermal power plant.

The reduction from the blue curve has been orchestrated to achieve a desired result, otherwise known as applying a fudge-factor. The 'reasoning' is to look at the minus 59 as a penalty (for overcoming the radiative losses) and to include a further penalty here by subtracting another term, 0.2 x | P_sol - 800 | , (i.e. abs( P_sol - 800 )), in order to achieve reduced power output away from the engineering optimum.

Concluding Remarks

The goal here was to obtain a reasonable first pass power curve, and to be conservative in doing so. I'm comfortable this has been achieved.

Knowledgable commenters are encouraged to address refining the analysis. Of particular interest is refining and justifying the various parameters, including reasonable ranges. In addition, there are three particular issues: (i) how much does the total intersected area of sunlight change through the day and year; (ii) does the 'fudge' applied here (magenta from blue curve) adequately account for the reduced efficiency at low input; and (iii) are ignored losses, such as conductive losses in the receiver tower, larger than supposed.

We have focused here on the heat engine being a steam turbine, so as to focus on a system that could be built tomorrow (for a price). This is appropriate for now, and for a while. In time we may consider CST with gas turbine style heat engines (Drayton Cycle) operating at higher temperatures and efficiencies, and with the potential to have direct gas backup.