Warp Core Power Distribution
One possible method of resolving the conflict between "True Q", "Masterpiece Society", "Dauphin", and "Battle Lines" is to recognize the importance of the location of the power output measurement point. Although most Federation cultists do not appear to recognize the fact that power levels will drop throughout the energy-conversion process due to normal losses, this is not only true but it is in fact necessitated by the Second Law of Thermodynamics.
There are several stages in the energy conversion process, each of which involves a significant power loss. In "True Q" Data quantified the power output inside the warp core, because Amanda wanted to know how much power was "in there". In "Masterpiece Society" Geordi Laforge discussed the plasma coolant, which is the method used to transport energy to ships' systems like weapons, shields, and propulsion. In "Dauphin" Riker was talking about the actual power output of the ship's powered systems, excluding the warp nacelles (this exclusion shall be explained later). And finally, in "Battle Lines" the runabout crew was talking about a specific system (weapons) which is at the end of the generation/conversion/transmission/conversion process. Each of them in turn was discussing power output farther along the energy conversion process.
Stage One: Conversion of matter into energy
Let us imagine that the warp core was reacting roughly 71kg of matter with 71kg of antimatter per second at the time Data quantified its output in "True Q". This seems high in light of their maximum theoretical 450-ton antideuterium fuel storage capacity (and extremely high if they weren't doing anything, which would suggest horrifically low efficiency), but they may have been charging certain high-powered ship's systems in preparation for their critical upcoming environmental rescue mission, or running unusual experiments. The result would be roughly 1.28E19 watts of power generation. For the sake of order-of-magnitude rough calculations such as these, this figure will be rounded to 1E19 W. Keep in mind that although Data's dialogue was cut off in the actual episode, the original, complete line of dialogue from the screenplay was actually "12.75 billion gigawatts per second" (ref. Star Trek Scriptbooks Book One), so this entire line of dialogue is seriously suspect. Regardless of Data's vaunted positronic brain, if he can state power generation in meaningless units of watts per second, he is simply not reliable.
Nevertheless, the most conservative method of analysis is to ignore this enormous error and assume that he means "gigajoules per second" rather than "gigawatts per second." Of this total, probably 70% will be in the form of charged pions and 30% will be in the form of gamma rays. Although charged pions rapidly decay into neutrinos, their velocity is such that they must first pass through the surrounding EPS plasma before they can escape into the environment and decay. Their chances of passing through the EPS plasma are dependent on mean-free-path considerations and the usual eletromagnetic repulsion and attraction concerns, but their tiny size and high velocity means that most of them will pass through the plasma without interacting with any matter. The opposite is true of gamma rays; some will pass through, but the majority will probably interact with the plasma.
Some will undoubtedly assume that charged pions are highly interactive with EPS plasma, but they should note the behaviour of 20th century neutral-beam plasma heating systems used in fusion reactors:
Neutral-beam injection devices accelerate positively charged deuterium ions to velocities of as much as 150 keV (kinetic energy, which corresponds to roughly 0.01c) through a negatively charged neutralization chamber in which charge-reversal with the ambient gases occurs. Ions which are not successfully neutralized are redirected to a dump by an electromagnetic shunt, and the remaining stream of neutral deuterium particles will continue through the magnetic confinement fields of the Tokomak to transfer its energy to the plasma. However, this stream of particles will not immediately transfer its energy to the plasma. Instead, some of the particles become ionized and they get trapped in plasma by the magnetic confinement field, while others pass through. Once there, they transfer their energy to the surrounding plasma through the usual Coulomb collisions, in a process which takes several seconds. All information is from Fusion Energy, by R.A. Gross.
The ramifications are important: the charged particles which are released by a matter/antimatter reaction are travelling at speeds in the range of 0.9c, and they are several orders of magnitude smaller than a proton. They are therefore several orders of magnitude faster and several orders of magnitude smaller than the neutral-beam particles injected into a fusion plasma volume by a neutral-beam injection device. If the neutral-beam particles need to be confined with the plasma for several seconds to transfer their energy, then the much smaller, much faster charged pions will easily pass through the plasma without interacting.
We can therefore assume that only a miniscule fraction of the charged pions will actually heat the plasma, so that only the gamma rays will be effective in heating the warp plasma and contributing energy to a Federation power generation system. This means that of the original 1E19 watts, roughly 3E18 watts remain.
Stage Two: Plasma coolant heating
The efficiency of the gamma-heating process will depend on the density, thickness, and volume coverage of the plasma surrounding the reaction site- some (perhaps most) of the gamma rays will pass through the plasma and strike the reactor wall, heating it and requiring cooling action. Some of the radiation will even pass through the reactor wall and irradiate personnel standing near the reactor, as seen in "The Adversary" where the radiation escaping from the warp core was sufficient to kill a shapeshifter within seconds when he got too close! In fact, it totally disintegrated his body, a feat which required dozens of direct hits from Klingon disruptor weapons in "Apocalypse Rising".
The fact that such large intensities of radiation are passing through a thick reactor wall are an obvious indicator that second-stage energy conversion is not 100% efficient and in fact is far from 100% efficient. The fact that humans are safe at relatively short ranges of <10m is somewhat strange in light of the quick death of the changeling in "The Adversary", but there may be some kind of device (dampening field, etc) which removes the radiation once it has travelled a short distance outside the armored shell. By coming in contact with the reactor wall, the changeling in "The Adversary" presumably was inside the region where the device has not yet removed all of the radiation.
It is difficult to quantify the actual power loss associated with this stage due to a lack of data, but it would not be unreasonable to estimate that the power loss is sufficient to add up to a full order-of-magnitude power drop, when combined with the 70% drop in stage one.
Stage Three: Plasma evacuation from reaction chamber
Geordi Laforge, the ship's chief engineer, stated that the warp core "kicked plasma up into the terawatt range", which suggests that the overall power output has dropped from the 1E18 watt range to the 1E12 to 1E15 watt range by the time it has exited the warp core. If it exceeded 1E15 watts it would have been into the next SI unit prefix, and Laforge would have stated that the plasma was "kicked up" into the petawatt range. Therefore, the third stage accounts for another three order-of-magnitude drop in power output, in addition to the order-of-magnitude drop sustained in stage one and two.
Does this seem excessive? It certainly sounds excessive, doesn't it? However, the temperature at the reaction site is 2E12K according to the TM, so the plasma in the immediate vicinity of the reaction should emit very large amounts of radiation. The gas plama in the photosphere of the Earth's sun is only at 5800K and it emits roughly 62 MW per square metre of the sun's surface- the radiative output of plasma at 2E12K will therefore be many tens of thousands of times greater than the radiative output of Sol's surface, if not more. In fact, if the plasma acts as a black-body radiator, we could expect 2E12 K plasma to dump energy into its environment at a rate of 1E41 W/m²!
Obviously, the rate of power loss is actually an instantaneous rate and will drop off exponentially almost immediately, as the plasma sheds energy. At these rates, it is easy to see how the plasma will quickly waste most of its enthalpy in the form of radiative emissions to its environment- it is therefore possible that it might indeed lose most of its power during this process. The TM states that the temperature at the reaction site is 2E12K but the operating temperature of the plasma may be as low as 2.5E6K, so the massive power loss in this stage seems to be substantiated by the TM as well as Geordi Laforge's dialogue.
The use of extreme high-temperature conditions is actually a very poor engineering decision on the part of the Federation, who should have known better. As early as their 20th century, their nuclear fission reactors used lower coolant temperatures than their less-efficient low-powered coal-fired brethren (based on the measured coolant temperatures in Ontario Hydro's CANDU nuclear and conventional coal-fired generating stations). They produced higher power output anyway, which again defies the conclusions reached by intuitive methods (another reason not to rely on unaided intuition when discussing matters involving science and technology). Extreme high-temperature coolants are not suitable for transmission purposes because they quickly lose energy to their environments (the rate of energy loss through convective or radiative heat transfer processes is proportional to relative temperatures).
Stage Four: Plasma transportation through the ship
The first thing that happens to the plasma is that most of it gets drawn away and sent to the warp nacelles. The warp nacelles always draw at least 89% of the plasma for themselves even when the ship isn't at warp (presumably, the nacelles use this plasma to produce subspace "mass-lightening" fields), and it is obvious that as power levels increase, the warp nacelles will draw a proportionally larger share of the plasma. We know this because a Federation starship can maintain all of its usual weapon, shield, and autonomous support systems even when the warp core is off-line, and in the case of DS9, we know that it can do so very effectively, even with no warp core at all. Therefore, those systems would probably be overloaded or destroyed in the event that they continued to draw 11% of the warp core's output as it ramped up to full output (except for the deflector dish which is apparently capable of handling it in extreme situations).
The warp nacelles' consumption of plasma is important because their output is not measurable in watts. They produce a space-time distortion (or a subspace distortion) for which I cannot imagine a quantification method involving watts. Therefore, any time someone discusses wattage for starship power output, he is probably excluding the power that goes to the warp nacelles. This means that if someone is talking about the power output of the warp core as it relates to its use in downstream systems like weapons, shields, or autonomous support systems, he is probably talking about the power after losing more than 90% to the warp nacelles. This means we are looking at another order-of-magnitude drop in measured power output, again simply because of the location where we measure the power.
Furthermore, even at a reduced temperature of 2.5E6K, the plasma should still be radiating extreme amounts of energy to its environment! As it flows through hundreds of metres of plasma conduits, power couplings, etc. it will lose more of its energy. In fact, as the point-of-use gets farther and farther away from the warp core the available power will be lower and lower. In light of this fact, it is highly perplexing why the GCS would place its phaser banks so far from its warp core- their most effective combat vessels, the Defiant-class vessels, do not incorporate such a poor design decision. Unfortunately, it is difficult to quantify the power drop associated with transmission through the long conduits snaking through the ship, but it would not be surprising if this drop involved yet another order-of-magnitude power loss. This would suggest that stage four involves another two order-of-magnitude power drop, down to the 1E13 watt range.
Stage Five: Conversion into useful energy
At this point, the entire system has still not yet produced any energy that is actually useful to the ship. It has produced large numbers of charged subatomic particles, irradiated the reactor walls with deadly amounts of gamma radiation, placed severe demands on the ship's heat dissipation systems, and travelled through the entire vessel bleeding energy as it goes, but it has not yet performed any useful work. It cannot perform useful work until it is converted into another form of energy at the point of use, eg. a phaser bank or communications system.
How efficient is this final energy conversion stage? That would depend on the application. A phaser bank may have less efficiency than a communications array or a replicator, for example. Since Commander Riker stated that their entire ship could not generate a single TW, we can conclude that one final order-of-magnitude drop in power efficiency is involved during the actual usage stage, so that the actual quantifiable output of the vessel's various powered systems (excluding its warp nacelles) is roughly 1 TW or below. Some have argued that Riker was talking about their communications system when he said "our entire ship can't generate that much power" but if he were, there is no conceivable reason to specifically use the words "entire ship".
If he had said "we can't generate that much power" he might have been talking about a particular subsystem, but by saying "entire ship" he clearly indicated that he was talking about the sum total of all the powered systems on the vessel. This is consistent with the various SI unit quantifications in the TM, which are invariably in the MW or GW range rather than the TW range (such as the phaser banks, shields, etc). Remember that this would exclude the warp nacelles which produce an output that, as mentioned previously, cannot be quantified in conventional energy and power terms.
The low Stage Five power output estimates drawn from "Dauphin" are supported in the TM (where phaser bank total output is described as 1.02GW) and in "Battle Lines" where a 40MW power surge in a small defense satellite's weapons systems was so worrisome to the crew of a runabout that they retreated rather than face the ensuing weapon discharge.
It is doubtful that Federation cultists will accept any of these conclusions, but to date they have not even attempted to reconcile the disparate statements seen in "True Q", "Dauphin", "Masterpiece Society", and "Battle Lines"! This is to date, the first known attempt to reconcile those disparate quantifications other than the Federation cultists' ludicrous claim that a starship's chief engineer might be a half-dozen orders of magnitude off when discussing the power output of his vessel. Any engineer exhibiting such incompetence would have been relieved of duty even during their primitive 20th century.