1. Optimization methods in water allocation Sergei Schreider School of Mathematical and Geospatial Sciences Royal Melbourne Institute of Technology

2. Slide 2 Starting point Single objective optimization: one agent, one objective, one time horizon Dynamic games: 2+ agents, 1+ objectives, evolution in time. Dynamic optimization: one agent, one objective, evolution in time. Static game: 2+ agents, 1+ objectives, one time horizon Multi-objective optimization: one agent, 2+ objective, one time horizon

3. Slide 3 The model uses (cost minimization) network optimisation, genetically it is related to REALM LP model developed by Diment (1991) Physical elements of the model Ρ = the set of reservoirs, i є Ρ, i is a reservoir Δ = the set of demand centres, i є Δ, i is a demand centre N ij = the number of carriers from i to j Κ = the set of carriers, (i, j, n) є Κ, (i, j, n) is the n-th carrier from i to j n є {1,...,N ij } InputsOutputs initial state of the systemcumulative irrigation deliveries and revised allocations irrigational and urban demandsurban restrictions and deliveries preferenced reservoir volume distributionssystem water movements for the month system inflowslosses and spills evaporation dataupdates the state of the system irrigational and urban restrictions data Part 1. REALM and GMS Policy via allocation level (from 0 to 220%): Volume available=Entitlement × Allocation level

4. The Goulburn system

5. Slide 5 The nodal-carrier structure of the system

6. Slide 6 Variables, penalties, OF Variables: x[i,j,n,t] = the flow along n-th carrier from node i to node j during time-step t d[i,t] = the volume of shortfall in demand for demand centre node i in time-step t v[i,t] = the volume in reservoir node i in time-step t Penalty functions: c[i,j,n,t](x[i,j,n,t]) = penalty for flow x[i,j,n,t] along carrier (i, j, n) at time t s[i,t](d[i,t]) = penalty for (restricted) demand shortfall d[i,t] in demand centre node i at time t r[i,t](v[i,t]) = penalty for end reservoir volume v[i,t] in reservoir node i at time t So streamflows, shortfalls and reservoir volumes are chosen x[i,j,n,t], d[i,t], v[i,t] so as to minimise the following Objective function:

7. Slide 7 Nonlinearity Carrier penalty function, Demand centre shortfall penalty function, and Reservoir penalty function

8. Slide 8 Constraints Where exogenous to the time-step parameters, C[i,j,n,t] = maximum capacity of carrier (i, j, n) in time-step t D[i,t] = (restricted) demand for demand centre node i in time-step t Let is maximum volume of reservoir node i.

9. Slide 9 Constraints Define, E[i,t] = system evaporation to reservoir i in time-step t I[i,t] = system inflow to reservoir i in time-step t λ[i,j,n] = percentage of flow lost in transmission from carrier (i, j, n) Then for each reservoir i water balance implies, In words it means: end volume (inc. spills) = start volume - evaporation + system inflows - carrier outflows + carrier inflows Secondly the water balance conditions for demand nodes are: In words it means: carrier inflows + demand shortfall = (restricted) demand

10. Slide 10 Practical applications: irrigation demand and water distribution cooperation between 8 provinces (with Reza Roozbahani) LP formulation: Maximisation of total agricultural income, subject to delivery constraints in the gravity driven system (the Safidrud basin, Iran). Decision variables: water realocated from node i to j: x ij and water used in node i: x ii. Agents: 8 provinces, 24 nodes in total. Part 2. The Sefidrud Basin, Iran

11. Slide 11

12. Slide 12

13. Formulation of a single OF LP problem Slide 13 Maximize WP i = the water profit of agricultural sector in node i EP i = the environment shortage penalty coefficient (a big enough number) in node i and the decision variables are: x s ii = the supplied surface water for agricultural demand in node i x g ii = the supplied ground water for agricultural demand in node i x es i = the environment shortage in supply node i (MCM) Transferred capacity Ground water availability Satisfying environment demand Water-balance

14. Slide 14 Scenarios ScenarioDescription S1Considering current agricultural water demand in the basin S2Considering total agricultural demand (current plus expected agricultural development) in the basin Results S1 S2

15. Slide 15 Solutions S1 S2

16. Slide 16 Compromise programming The compromise programming is a mathematical programming technique that allows researchers to aggregate the effects of various conflicting criteria in a multi-criteria decision problem Firstly, it identifies solution that is closest to the optimal solution of single objective models The solution that gives the optimal values for each objective is generally infeasible called “ideal” The measure of proximity in the solution space is determined by some metrics in the space of objective functions referred as the distance metrics. This metrics (distance), which is calculated for each alternative solution, is a function of the criteria values themselves, the relative importance of the various criteria to the decision makers, and the importance of the maximal deviation from the ideal solution All alternative solutions are ranked according to their respective distance metric values. The alternative with the smallest distance metric is typically selected as the “best compromise solution”.

17. Mathematically, in the generic form, compromise programming distance metric can be presented as follows: Slide 17

18. Three objective functions: maximizing the revenue of the Basin: Slide 18 minimizing the water shortage of environment and minimizing transferred water from upstream to downstream areas

19. Slide 19 Some results

20. Slide 20 Some results The average supplied water and supplied water in 90% of times for the Qazvin and Gilan provinces given by the compromised model (Unit: MCM)

21. Slide 21 Some results The reliability of supplied water to the environment by the compromised model (in percent)

22. Slide 22 Some results Comparing the percentage of supplied water to the Kordestan and Hamedan provinces between the compromised model for weights w 1 =0.3, w 2 =0.4, w 3 =0.3 and the single objective models (in percent)

23. Slide 23 Economic modelling (classic PE approach) Land and water use Crop yields Economic returns Optimisation (LP, NLP) model Household type Climate Land Resource Water availability Input Prices Crop Prices Labour Constraints Shocks: Policy instruments Investments Demographic change Basic assumption: irrigators behaviour is driven by revenue maximisation Part 3. Water trading

24. Slide 24 Water allocation modelling Land and water use Allocation level Regulated discharge Optimisation (LP, NLP) model Areas under different crops Crop coefficients Transmission losses Carrier capacities Reservoir capacities Environmental and political regulations Shocks: Climate variations Changes in infrastructure Environmental regulations Upstream/inflow changes Basic assumption: behaviour of water authorities is driven by trade-off between satisfying all irrigators demands and a set of social and environmental constraints

25. Slide 25 Acknowledgements to coauthors Vladimir Gurvich, Endre Boros – Rutgers Business School Panlop Zeephongsekul and Babak Abassi – RMIT University Peter Dixon, Glyn Wittwer, Marnie Griffith – Centre for Policy Studies, Monash University Erwin Weinmann – Department of Civil Engineering, Monash University Daniel McInnes and Boros Miller (Math Dept, Monash University) Barry James, Mark Eigenraam and Kes Kesari – Department of Sustainable Environment, Victoria Matt Fernandes, Jonathan Plummer, Jin Cui (PhD and Masters Students)