Powering 7 LEDs in series
Bad news: your design isn't quite realistic exactly as it is. But let's try to make it workable.

First, a basic information: you don't need the resistors. STP16CPP05 is a constant current driver, so it will monitor and limit the current itself. So forget the resistors, they'll just uselessly dissipate power and make your supply voltage requirements higher.

Now, why it doesn't work:


Using a 14.7V supply to power a 7 * 2.1V LED string is risky. Unless you bin the LEDS very precisely and maintain the temperature constant, the Vf drop of the LEDS will probably vary between 1.8V - 2.2V. Moreover, the driver itself has a voltage drop (~350mV, see table 11 of datasheet). So you need more margin to ensure you can drive the leds at 20mA. An absolute minimum of 2.2V * 7 + 0.35V is required, but I would advise at least 17V to account for the power supply tolerance.
A supply of 30V gets you above the STP16CPP05 maximum output voltage, specified at 20V.


So, now, suppose we use a supply of 17V and a 7 x 28 array. Let's see what will be the worst case power dissipation of the driver chip. If we take the min Vf of diodes (e.g. 1.8V), and max voltage of supply (e.g. 17.85V if we say it has 5% tolerance), it means the voltage drop of the driver must be 17.85 - 1.8*7 = 5.25V. Time 20mA times 16 strings per chip, it means each STP16CPP05 chip dissipates ~1.7W worst case. Wow. But this is workable with the TSSOP24 with exposed pad chip package. You better solder the pad, though.

Note that it would be even worse with the 14 x 14 array (if we could use higher voltages with this driver). Because what makes the sizing difficult is the Vf tolerance, so the more LEDs you have on your string, the higher the worst case voltage drop you'll have at the chip (and consequently the higher the dissipated power at each chip).

So what I would actually advise, to make it safe, is to actually keep your supply voltage at 12V. This removes the need for a boost, and spread the dissipated power on more drivers because you have less LEDs on each string. It means you'd have 5 LEDs per string (2.2V * 5 + 0.35V = 11.35V min supply). I know, this fits less well with the 196 LEDs requirements, because it's not even an integral number of strings (39.2 strings), and you now need 3 driver chips. But this actually makes it simpler (no boost, less power on drivers). To better spread the power, put only 14 strings per chip. Max power for a driver chip is now ~1W (with 12.6V supply max). Workable with SO-24 package. With a 14.4V supply max (lead-acid battery), it would make 1.5W. Much closer to the limit, but still acceptable with this package, if the airflow is sufficient in the case. If in doubt, pour in an additional driver chip and put only 10 strings per chip.

Note: on the third driver chip, you'll have 11 complete strings plus one that has only one LED. This single LED string will higher the dissipation on the chip. This is still fine in this case because you only have 11 complete strings, but you could put a resistor on that string to move some of this dissipation away from the chip to the resistor. Max resistor value is (min supply voltage - 2.2V - 0.35V) / 20mA) = 440&Omega;. Make it a 1/4W.