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Constraints

1. Mutual exclusion constraints

Most constraints ensure that at most one measure from a group can be implemented on the same field. These are essential because many measures occupy the same land or are agronomically incompatible.

Core field exclusion (mutexc1)

mutexc1(i)..  Sum(mem(j), x(i,j)) ≤ 1
Only one measure from the mem set can be selected per field. mem includes nearly all field measures except LRl.

Core field exclusion with LRl (mutexc3)

mutexc3(i)..  Sum(mem2(j), x(i,j)) + Sum(VP2_Perm(j), x_VP2_p(i,j)) ≤ 1
mem2 is the same as mem but replaces LRh with LRl. Also accounts for any permanent VP2 measures already on the field (x_VP2_p).

Land use accounting (mutexc4, mutexc5, mutexc6)

These define and enforce me(i), an indicator that counts whether a land-retirement type measure is in use: 

mutexc4(i)..  me(i) =e= Sum(lr1(j), x(i,j) + x_VP2_p(i,j)) + Sum(lr2(j), x(i,j) + x_VP2_p(i,j)) / 2
mutexc5(i)..  Sum(lm(j), x(i,j)) + me(i) ≤ 1    [land management measures cannot co-exist with land retirement]
mutexc6(i)..  Sum(bz(j), x(i,j)) + me(i) ≤ 1    [buffer zones cannot co-exist with land retirement]

lr1 = WL, FO (full land retirement); lr2 = LRl, LRh (land retirement counted at 50% in me).

WWT mutual exclusion (mutexc8)

mutexc8(p)..  Sum(j$ww(j), WWTx(p,j)) ≤ 1
Only one WWT upgrade level per plant.

IBZ-specific exclusions (mutexc10–mutexc17)

IBZ (in-stream buffer zone) has the most exclusions of any measure. It cannot co-exist with: - BZ10 or BZ20 (10, 11) - LRh or LRl (12, 13) - WL (16) - FO (17) - VP2 WL or FO (14, 15) - VP2 LRl (7)

Stream measure exclusion (mutexc18)

mutexc18(w)..  v_w(w,"re_meandering") + v_w(w,"raising") ≤ 1
A watercourse cannot have both re-meandering and bed raising.

VP2 LRl double-counting prevention (LRl_VP2)

LRl_VP2(i)..  x(i,"LRl") + x_VP2_p(i,"LRl") ≤ 1

2. Tripartite Agreement area floors

These constraints encode the 2021 Danish Tripartite Agreement minimum implementation areas. They are always active in the current model. exceed_lav is a penalty variable allowing the lavbund floor to be soft when physical potential is insufficient. The MW, FO, WL, LRH, and SA floors are hard constraints (exceed_MW was removed from the model — confirmed 2026-04-06):

Lavbund_eq..      Sum(i, PotV(i,"LRL") × x(i,"LRL")) + exceed_lav ≥ 51,000  [ha]
Afforestation_eq.. Sum(k$(n_targets(k)>0), Sum(i$setki(k,i), PotV(i,"FO") × x(i,"FO"))) ≥ 28,000  [ha]
Wetland_eq..      Sum(k$(n_targets(k)>0), Sum(i$setki(k,i), PotV(i,"WL") × x(i,"WL"))) ≥ 7,500   [ha]
MW_eq..           [sum of MW areas in catchments with N targets] ≥ 40,000  [ha equivalent]
LRH_eq..          Sum(k$(n_targets(k)>0), Sum(i$setki(k,i), PotV(i,"LRH") × x(i,"LRH"))) ≥ 17,700 [ha]
SA_eq..           Sum(k$(n_targets(k)>0), Sum(i$setki(k,i), PotV(i,"SA") × x(i,"SA"))) ≥ 84,000  [ha]

Scope: Afforestation, Wetland, MW, LRH, and SA floors apply only to catchments with N targets (n_targets(k)>0). The lavbund (LRl) floor is nationwide.

See Tripartite Agreement for policy background.

3. P wetland vs. stream measure constraints

A P wetland (PWET) on field i and a stream restoration measure (re-meandering or raising) on adjacent watercourse w cannot both be implemented. This is encoded via the binary indicator s(w):

Pwet1_eq(w)..              s(w) ≥ Sum(w_Pwet(w_m), v_w(w,w_m))      [s = 1 if stream measure chosen]
Pwet2_eq(w,i)$wi_pwet..    x(i,"Pwet") + s(w) ≤ 1                   [PWET excluded if stream measure]
Pwet3_eq(w)..              Sum(w_Pwet(w_m), v_w(w,w_m)) ≥ s(w)      [s = 0 if no stream measure]

Additionally, total P reduction from P wetlands is capped at total P loss in the upstream lake catchment: 

P_Wet_limit_eq(up_lakecatch)..
    Sum(i$up_lake_i, P_Effects_Total(i,"Pwet") × x(i,"Pwet")) ≤ Total_P_loss(up_lakecatch)

4. VP2 legacy constraints

Results from Water Action Plan 2 (VP2) are loaded as fixed parameters: - x_VP2_p(i,j) — binary, 1 if field i had measure j implemented in VP2 - MW1x_VP2_p, MW2x_VP2_p, MW3x_VP2_p — VP2 mini-wetland decisions

These are used in: - mutexc3: VP2 permanent measures block re-use of the same field - LRl_VP2: prevents double implementation of LRl - mutexc7, 14, 15: IBZ exclusions with VP2 WL, FO, LRl - MW potential cap: VP2_adapt(ret) = max(MW_precalc2(ret), VP2MW_per_ret(ret)) ensures VP3 MW does not overclaim relative to VP2 baseline

See VP2 & VP3.

5. Mini-wetland potential cap

CatchmentMW(ret)..  MW_precalc1(ret) ≤ VP2_adapt(ret)

Where: - MW_precalc1(ret) = total MW area chosen in VP3 (including VP2 MW) - VP2_adapt(ret) = max(MW_precalc2(ret), VP2MW_per_ret(ret)) - MW_precalc2(ret) = natural potential (from MiniPot files: dark green + 10% light green + 10% dark yellow) - VP2MW_per_ret(ret) = VP2 implemented area

Only VP3 new MW is counted against the potential for additonality: MW_precalc1_noVP2 tracks new VP3 decisions only.

Commented-out scenario constraints

The model code contains several commented-out constraint blocks that represent alternative scenario configurations. These are important design history:

* x.UP(i,j) = 0;  [turning off all measures, then selectively enabling by P-loss percentile]
* Scenario_pho_eq: limit P measures to 75% of potential in each lake catchment
* Scenario_ochre_eq, sand_eq, etc.: limit stream measures to 75% of count
* Effectiveness_limit(i,j): exclude field-measure combos with cost-effectiveness > 50,000 DKK/kg P

These represent scenarios that were tested but are not the default. Each could be re-activated for specific scenario runs.

Open questions

  1. ✅ exceed_MW confirmed removed from model (2026-04-06). MW_eq is a hard constraint. Only exceed_lav remains as a penalty variable (lavbund floor can be soft if physical potential is insufficient).
  2. The scope condition n_targets(k)>0 on tripartite constraints — does this mean the area floors only apply in catchments that have N targets? Is this correct policy-wise?
  3. The 75% potential scenario constraints (Scenario_pho_eq, etc.) — under what policy assumption would these be activated?