PlasmaCracking

Introduction

The plasma cracking process consists in breaking the connection between the carbon and the hydrogen of the methane using high frequency microwaves. The reaction is the following:

$$CH_4 --> C + 2H_2$$

This process allows to extract solid carbon out of methane and if used with biomethane and electricity from renewable sources, it leads to negative $CO_2$ emissions.

Gradient computation - Summary

H2_price : techno price \ X : percentage resource \ PC_cost : Plasma Cracking cost

Formula:

$$H2_{price}= PC_{cost} * Margin * X$$

$$H2_{price}= PC_{cost}(invest, energy\_prices,energy\_CO2\_emission) * Margin$$ $$* X(invest, energy\_prices, all\_stream\_demand\_ratio, resources\_price)$$

Gradients to compute:

$$\dfrac {\partial H2_{price}}{\partial invest}= Margin * X * \dfrac {\partial PC_{cost}}{\partial invest} + PC_{cost} * Margin * \dfrac {\partial X}{\partial invest}$$

$$\dfrac {\partial H2_{price}}{\partial energy\_prices}= Margin * X * \dfrac {\partial PC_{cost}}{\partial energy\_prices} + PC_{cost} * Margin * \dfrac {\partial X}{\partial energy\_prices}$$

$$\dfrac {\partial H2_{price}}{\partial energy\_CO2\_emission}= Margin * X * \dfrac {\partial PC_{cost}}{\partial energy\_CO2\_emission} + 0$$

$$\dfrac {\partial H2_{price}}{\partial all\_stream\_demand\_ratio}= PC_{cost} * Margin * \dfrac {\partial X}{\partial all\_stream\_demand\_ratio}$$

$$\dfrac {\partial H2_{price}}{\partial resources\_price}= PC_{cost} * Margin * \dfrac {\partial X}{\partial resources\_price}$$

X computation:

$$X = \dfrac {H2\_revenue}{H2\_revenue + A }$$

with:

if Carbon_prod < Carbon_demand :

$$A = Carbon\_sold\_revenue$$

if Carbon_prod > Carbon_demand :

       Carbon_storage = Carbon_prod - Carbon_demand

$${\footnotesize A = [\dfrac {Carbon\_prod* Carbon\_mol * CO2\_credit}{CO2\_mol}]+ [Carbon\_demand * (Carbon\_price - \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol})]}$$

energy_prices gradient computation:

$$\dfrac {\partial X}{\partial energy\_prices} = \dfrac { \dfrac {\partial H2\_price}{\partial energy\_prices} * H2\_prod * A }{[H2\_revenue + A ]^2 }$$

with:

if Carbon_prod < Carbon_demand :

$$A = Carbon\_sold\_revenue$$

if Carbon_prod > Carbon_demand :

       Carbon_storage = Carbon_prod - Carbon_demand

$${\footnotesize A = [\dfrac {Carbon\_prod* Carbon\_mol * CO2\_credit}{CO2\_mol}]+ [Carbon\_demand * (Carbon\_price - \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol})]}$$ \

invest gradient computation:

$$\dfrac {\partial X}{\partial invest} = \dfrac { [ \dfrac {\partial H2\_prod}{\partial invest} * H2\_price * A ] - [\dfrac {\partial Carbon\_prod} {\partial invest} * B * H2\_revenue] }{[H2\_revenue + A]^2}$$

with:

if Carbon_prod < Carbon_demand :

$$A = Carbon\_sold\_revenue$$

$$B = Carbon\_price$$

if Carbon_prod > Carbon_demand :

       Carbon_storage = Carbon_prod - Carbon_demand

$${\footnotesize A = [\dfrac {Carbon\_prod* Carbon\_mol * CO2\_credit}{CO2\_mol}]+ [Carbon\_demand * (Carbon\_price - \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol})]}$$

$$B = \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol}$$ \

all stream demand ratio computation:

$${\footnotesize\dfrac {\partial X}{\partial all\_stream\_demand\_ratio} = \dfrac { [ \dfrac {\partial H2\_prod}{\partial all\_stream\_demand\_ratio} * H2\_price * A ] - [\dfrac {\partial Carbon\_prod} {\partial all\_stream\_demand\_ratio} * B * H2\_revenue] }{[H2\_revenue + A]^2}}$$

with:

if Carbon_prod < Carbon_demand :

$$A = Carbon\_sold\_revenue$$

$$B = Carbon\_price$$

if Carbon_prod > Carbon_demand :

       Carbon_storage = Carbon_prod - Carbon_demand

$${\footnotesize A = [\dfrac {Carbon\_prod* Carbon\_mol * CO2\_credit}{CO2\_mol}]+ [Carbon\_demand * (Carbon\_price - \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol})]}$$

$$B = \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol}$$

resources price computation:

$${\footnotesize\dfrac {\partial X}{\partial resources\_price} = \dfrac { - [\dfrac {\partial Carbon\_price} {\partial resources\_price} * B * H2\_revenue] }{[H2\_revenue + A]^2}}$$

with:

if Carbon_prod < Carbon_demand :

$$A = Carbon\_sold\_revenue$$

$$B = Carbon\_sold\_revenue$$

if Carbon_prod > Carbon_demand :

       Carbon_storage = Carbon_prod - Carbon_demand

$${\footnotesize A = [\dfrac {Carbon\_prod* Carbon\_mol * CO2\_credit}{CO2\_mol}]+ [Carbon\_demand * (Carbon\_price - \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol})]}$$

$$B = Carbon\_demand$$

Gradient computation - More details

$$H2_{price}= PC_{cost} * Margin * X$$

Gradients to compute:

$$\dfrac {\partial H2_{price}}{\partial invest}= Margin * X * \dfrac {\partial PC_{cost}}{\partial invest}+ PC_{cost} * Margin * \dfrac {\partial X}{\partial invest}$$

$$\dfrac {\partial H2_{price}}{\partial energy\_prices}= Margin * X * \dfrac {\partial PC_{cost}}{\partial energy\_prices}+ PC_{cost} * Margin * \dfrac {\partial X}{\partial energy\_prices}$$

$$\dfrac {\partial H2_{price}}{\partial energy\_CO2\_emission}= Margin * X * \dfrac {\partial PC_{cost}}{\partial energy\_CO2\_emission}+ 0$$

X computation:

$$X = \dfrac {H2\_revenue}{SUM\_revenues}$$

$$= \dfrac {H2\_revenue}{H2\_revenue+ Carbon\_sold\_revenue+Carbon\_storage\_revenue}$$

if Carbon_prod - Carbon_demand < 0:

$$X = \dfrac {H2\_revenue}{H2\_revenue+ Carbon\_sold\_revenue+ 0}$$

$$= \dfrac {H2\_prod * H2\_price}{[H2\_prod * H2\_price ]+ [Carbon\_prod * Carbon\_price]}$$

if Carbon_prod > Carbon_demand :

       Carbon_storage = Carbon_prod - Carbon_demand

$${\scriptsize X = \dfrac {H2\_prod * H2\_price}{[H2\_prod * H2\_price ]+[Carbon\_demand * Carbon\_price]+ [\dfrac {(Carbon\_prod -Carbon\_demand)* Carbon\_mol * CO2\_credit}{CO2\_mol}]}}$$

$${\scriptsize = \dfrac {H2\_prod * H2\_price}{[H2\_prod * H2\_price ] + [\dfrac {Carbon\_prod* Carbon\_mol * CO2\_credit}{CO2\_mol}]+ [Carbon\_demand * (Carbon\_price- \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol})]}}$$

$$= \dfrac {H2\_revenue}{H2\_revenue + A}$$

Gradient computation:

if Carbon_prod < Carbon_demand :

$$X = \dfrac {H2\_prod * H2\_price}{[H2\_prod * H2\_price ]+ [Carbon\_prod * Carbon\_price]}$$

energy_prices:

$$\dfrac {\partial X}{\partial energy\_prices} =\dfrac {[ \dfrac {\partial H2\_price}{\partial energy\_prices} * H2\_prod * [H2\_prod * H2\_price+ Carbon\_prod * Carbon\_price]}{[H2\_prod * H2\_price+ Carbon\_prod * Carbon\_price]^2}$$

$$\dfrac { -[ \dfrac {\partial H2\_price} {\partial energy\_prices} * H2\_prod * (H2\_prod * H2\_price)]}{[H2\_prod * H2\_price+ Carbon\_prod * Carbon\_price]^2}$$

$$= \dfrac {\dfrac {\partial H2\_price}{\partial energy\_prices} * H2\_prod * Carbon\_prod * Carbon\_price]}{[H2\_prod * H2\_price+ Carbon\_prod * Carbon\_price]^2}$$

$$= \dfrac {\dfrac {\partial H2\_price}{\partial energy\_prices} * H2\_prod * Carbon\_sold\_revenue]}{[H2\_revenue+ Carbon\_sold\_revenue]^2}$$

invest:

$$\dfrac {\partial X}{\partial invest} =\dfrac {[ \dfrac {\partial H2\_prod}{\partial invest} * H2\_price * [H2\_prod * H2\_price+ Carbon\_prod * Carbon\_price]]}{[H2\_prod * H2\_price+ Carbon\_prod * Carbon\_price]^2}$$

$$\dfrac { -[ \dfrac {\partial H2\_prod} {\partial invest} * H2\_price + \dfrac {\partial Carbon\_prod} {\partial invest} * Carbon\_price] *(H2\_prod * H2\_price)}{[H2\_prod * H2\_price+ Carbon\_prod * Carbon\_price]^2}$$

$$=\dfrac {[ \dfrac {\partial H2\_prod}{\partial invest} * H2\_price * ( Carbon\_prod * Carbon\_price) ]}{[H2\_prod * H2\_price+ Carbon\_prod * Carbon\_price]^2}$$

$$\dfrac {-[\dfrac {\partial Carbon\_prod} {\partial invest} * Carbon\_price *(H2\_prod * H2\_price)]}{[H2\_prod * H2\_price+ Carbon\_prod * Carbon\_price]^2}$$

$$=\dfrac {[ \dfrac {\partial H2\_prod}{\partial invest} * H2\_price * Carbon\_sold\_revenue ]-[\dfrac {\partial Carbon\_prod} {\partial invest} * Carbon\_price *H2\_revenue]}{[H2\_revenue+ Carbon\_sold\_revenue]^2}$$

if Carbon_prod > Carbon_demand :

$${\scriptsize X = \dfrac {H2\_prod * H2\_price}{[H2\_prod * H2\_price ] + [\dfrac {Carbon\_prod* Carbon\_mol * CO2\_credit}{CO2\_mol}]+ [Carbon\_demand * (Carbon\_price- \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol})]} }$$

$$= \dfrac {H2\_prod * H2\_price}{H2\_revenue + A}$$

with:

       Carbon_storage = Carbon_prod - Carbon_demand

$${\small A = [\dfrac {Carbon\_prod* Carbon\_mol * CO2\_credit}{CO2\_mol}]+ [Carbon\_demand * (Carbon\_price- \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol})]}$$

$${ = [\dfrac {Carbon\_prod* Carbon\_mol * CO2\_credit}{CO2\_mol}]+ [Carbon\_demand * (Carbon\_price- \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol})]}$$

energy_prices:

$$\dfrac {\partial X}{\partial energy\_prices} =\dfrac {\dfrac {\partial H2\_price}{\partial energy\_prices} * H2\_prod *A}{[H2\_revenue + A ]^2}$$

invest:

$$\dfrac {\partial X}{\partial invest} =\dfrac {[ \dfrac {\partial H2\_prod}{\partial invest} * H2\_price * [H2\_revenue + A]}{[H2\_revenue + A ]^2}$$

$$\dfrac { -[ \dfrac {\partial H2\_prod} {\partial invest} * H2\_price + \dfrac {\partial Carbon\_prod} {\partial invest} * \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol}] *H2\_revenue}{[H2\_revenue + A ]^2}$$

$$=\dfrac {[ \dfrac {\partial H2\_prod}{\partial invest} * H2\_price * A ] - [ \dfrac {\partial Carbon\_prod} {\partial invest} * \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol} * H2\_revenue]}{[H2\_revenue + A ]^2}$$

with:

$${\small A = [\dfrac {Carbon\_prod* Carbon\_mol * CO2\_credit}{CO2\_mol}]+ [Carbon\_demand * (Carbon\_price- \dfrac {Carbon\_mol * CO2\_credit}{CO2\_mol})]}$$