3. Constraints¶

3.1. Potential restricition rule¶

In (3.1) you can see the potential of the jth heat generator defines the upper bound for the sum over all the thermal energy values. If $potential_j > 0$ , then (3.1)is used.

(3.1)¶\[ \sum_t (Cap_j - x_{{th}_{j,t}}) \le potential_j \]

3.2. Power restriction rule¶

If $pow_{cap_{j}} > 0$ , then (3.2) is used.

(3.2)¶\[ \sum_t (Cap_j - x_{{th}_{j,t}}) \le pow_{cap_{j}} \]

3.3. Electrical power generation¶

If j is valid or $n_{{th}_{j,t}} \neq 0$ , then (3.3)is used.

(3.3)¶\[ x_{{el}_{j,t}} = \frac{x_{{th}_{j,t}}}{n_{{th}_{j,t}}}\ \cdot n_{{el}_{j}} \]

3.4. Heating generation covers demand¶

At any time, the heating generation must cover the heating demand.

(3.4)¶\[ \sum_t (x_{{th}_{j,t}}) - \sum_{hs} (x_{{load}_{hs,t}}) = demand_{f} \cdot demand_{th_{t}} \]

3.5. Capacity restriction rule - maximum¶

If the inv_flag is set true, then the rule in (3.5) is valid, otherwise (3.7) is valid.

(3.5)¶\[ Cap_j \le demand_{f} \cdot max_{demand} \]

(3.7)¶\[ Cap_j = x_{{th}_{cap,j}} \]

3.6. Capacity restriction rule - minimum¶

(3.7)¶\[ Cap_j \ge x_{{th}_{cap,j}} \]

3.7. Generation restriction rule¶

The amount of heat energy generated must not exceed the installed capacities. $$ x_{{th}_{j,t}} \le \frac{Cap_{j}}{n_{{th}_{nom,jt}}}\ \cdot n_{{th}_{j,t}} \cdot Active_{j,t} \cdot restrictionfactor_{j,t} $$ (eq_seven)

3.8. Generation rule - minimum operating power¶

All generators have a minimum operating power.

(3.8)¶\[ x_{{th}_{j,t}} \ge \frac{Cap_{j}}{n_{{th}_{nom,jt}}}\ \cdot n_{{th}_{j,t}} \cdot Active_{j,t} \cdot min_{out_{factor,j}} \]

3.8.1. Must run rule¶

(3.9)¶\[ x_{{th}_{j,t}} \ge \frac{mr_{j} \cdot Cap_j}{n_{{th}_{nom,jt}}}\ \cdot n_{{th}_{j,t}} \cdot restrictionfactor_{j,t} \cdot Active3_{j,t} \]

3.8.2. Must run rule 2¶

(3.10)¶\[ x_{{th}_{j,t}} \ge demand_{th_{t}} \cdot Active2_{j,t} \]

3.8.3. Must run rule 3¶

(3.11)¶\[ Active2_{j,t} + Active3_{j,t} = 1 \]

3.9. Solar restriction rule¶

Solar gains depend on the installed capacity and the solar radiation. The value 1000 represents the radiation at wich the solar plant has maximal power, see (3.12).

(3.12)¶\[ x_{{th}_{j,t}} \le \frac{Cap_{j} \cdot radiation_t}{max(radiation_t)}\ \]

3.10. Waste incineration restriction rule¶

No investment in waste incineration plants possible, thus geneartion can’t exeed already installed capacities. $$ x_{{th}_{j,t}} \le x_{{th}_{cap,j}} $$ (eq_fourteen)

3.11. Power loss when full heat extraction¶

With full heat extraction, a loss of power can occur which, only reduces the maximum power.

3.11.1. CHP generation restriction rule 1¶

(3.13)¶\[ sv_{chp} = \frac{ratioPMaxFW - ratioPMax}{ratioPMax \cdot ratioPMaxFW}\ \]

(3.14)¶\[ x_{{el}_{j,t}} \le \frac{Cap_j}{ratioPMax}\ - (sv_{chp} \cdot x_{{th}_{j,t}}) \]

3.11.2. CHP generation restriction rule 2¶

(3.15)¶\[ P_{min_{el,chp}} \le x_{{el}_{j,t}} \]

3.11.3. CHP generation restriction rule 5¶

(3.16)¶\[ Q_{min_{th,chp}} \le x_{{th}_{j,t}} \]

3.12. Decrese of maximum heat decoupling¶

The ratio of the maximum heat decoupling to the maximum electrical power determines the decrease in the maximum heat decoupling with the produced electrical net power

3.12.1. CHP generation restriction rule 3¶

Depending on the value of j, (3.17) or (3.18) will be taken.

(3.17)¶\[ x_{{el}_{j,t}} \ge \frac{x_{{th}_{j,t}}}{ratioPMaxFW}\ \]

(3.18)¶\[ x_{{el}_{j,t}} == \frac{x_{{th}_{j,t}}}{n_{{th}_{j,t}}}\ \cdot n_{{el}_{j}} \]

3.12.2. CHP generation restriction rule 4¶

Setting capacity for chp generation.

(3.19)¶\[ x_{{th}_{j,t}} \le demand_{th_t} + \sum_{hs} x_{load_{hs,t}} \]

3.12.3. Storage state heat storage rule¶

Heat storage level = old_level + pumping - turbining Depending on the value of t there is one special condition rule. If t is equal to 1, (3.20) is taken, otherwise (3.21).

(3.20)¶\[ store_{level_{hs,t}}(hs,t) == store_{level_{hs,t}}(hs,8760) \]

(3.21)¶\[ store_{level_{hs,t}}(hs,t) == store_{level_{hs,t-1}}(hs,8760) \cdot (1- cap_{losse_{hs}}) + x_{load_{hs,t}} (hs, t-1) \cdot n_hs \]

3.12.4. Storage capacity restriction heat storage rule¶

Installed capcities must be greater or equal to the pre-installed capacities. If the inv_flag is set, then (3.22) is taken, otherwise (3.23).

(3.22)¶\[ Cap_{hs} \ge cap_hs \]

(3.23)¶\[ Cap_{hs} == cap_hs \]

3.12.5. Storage state capacity restriction heat storage rule¶

(3.24)¶\[ store_{level_{hs,t}}(hs,t) \le Cap_hs \]

3.12.6. Load heat storage restriction rule¶

The discharge and charge amounts are limited

(3.25)¶\[ x_{load_{hs,t}}(hs,t) \le load_{cap_{hs}} \]

3.12.7. Unload heat storage restriction rule¶

If the flag is set, then the (3.26) is taken, otherwise (3.27).

(3.26)¶\[ x_{load_{hs,t}}(hs,t) \ge -unload_{cap_{hs}} \]

(3.27)¶\[ x_{load_{hs,t}}(hs,t) \ge -store_{level_{hs,t}}(hs,t) \]

3.13. Ramp CHP rule¶

If t is equal to 1, (3.28) is taken, otherwise (3.29).

(3.28)¶\[ ramp_{j_{waste}} == 0 \]

(3.29)¶\[ ramp_{j_{waste}} \ge ( x_{{th}_{j,t}}(j,t) - x_{{th}_{j,t}}(j,t-1)) \]

3.14. Ramp waste rule¶

If t is equal to 1, (3.30) is taken, otherwise (3.21).

(3.30)¶\[ ramp_{j_{waste,t}} == 0 \]

(3.31)¶\[ ramp_{j_{waste,t}} \ge ( x_{{th}_{j,t}}(j,t) - x_{{th}_{j,t}}(j,t-1)) \]

3.15. Coldstart rule¶

If t is equal to 1, (3.32) is taken, otherwise (3.33).

(3.32)¶\[ coldstart_{j,t} == 1 \]

(3.33)¶\[ coldstart_{j,t} \ge ( activate_{j,t}(j,t) - activate_{j,t}(j,t-1)) \]

(3.33) has only a true output for coldstarts.

3.16. Renewable factor rule¶

If $ \sum_j rf_j \neq 0$, (3.34) is taken.

(3.34)¶\[ \sum_j (\sum_t (x_{{th}_{j,t}} + x_{{el}_{j,t}}) \cdot rf_j) \ge rf_{tot} \cdot \sum_j (\sum_t (x_{{th}_{j,t}} + x_{{el}_{j,t}}) \]

HotmapsDispatch - Model Description