Real-time Control of Stormwater Networks using Horizon Planning

Ann Arbor's Stormwater Network

State of Stormwater

Larger Infrastructure

Chicago Sewers (4 Billion USD)

Urban Flooding

Smarter Stormwater Network

Real-time Control using Horizon Planning

1. Model of the network
2. Algorithms
   • Centralized Control
   • Distributed Control
3. Results
4. Future work

Network Model

Network ~ $G(N,A)$

• Nodes $(N)$: Storage elements in the network
  • Max Capacity for $i^{th}$ node : $C_i$
  • Volume at time $t$ in $i^{th}$ node : $V^t_i$
  • Outflow at time $t$ from $i^{th}$ node : $x^t_i$
  • Inflow at time $t$ to $i^{th}$ node from rain : $q^t_i$
• Links $(A)$: Routing Elements
  • Max flow carrying capacity in $i^{th}$ link : $u_i$
  • Total flow through $i^{th}$ link for $T$ horizon : ${Vx}^T_i$

Network dynamics

• Mass balance $$ V_i^t = V_i^{t-1} + q_i^{t} + \sum_{j} x_{ji}^{t-\delta_{ji}} - \sum_{j} x_{ij}^{t}$$

$\delta_{ji}$ is the travel time between nodes

• Physical constraints on outflows from nodes $$ q_{out} \leq \sqrt{2 \times g \times water\ depth} \times C_D $$

## Linearizing square root
height  = np.linspace(0,1000,100)/100    # (volume/area)
outflow = np.sqrt(2*9.81*height)         # Generate the outflows 
coeffws = np.polyfit(height, outflow, 1) # Get the first order polyfit 
plt.plot(height, outflow, label = "True")
plt.plot(height, coeffws[0]*height + coeffws[1], label = "Linearized")
plt.ylabel("Outflows (cu.m per sec)");plt.xlabel("Water Level (m)");plt.legend()

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Linearized constraint $$ q^t_{out} \leq 1.13047751 \times V^t_i/A_i + 3.65949363 $$

Control Algorithm

1. Centralized Control : Single Master Problem
2. Distributed Control : Master Problem and a set of sub-problems

Centralized Control Algorithm

Objective Function $$ \min \sum^N_{i} \sum^T_t V_i^t $$ Constraints

Mass Balance: $$ V_i^t = V_i^{t-1} + q_i^{t} + \sum_{j} x_{ji}^{t-\delta_{ji}} - \sum_{j} x_{ij}^{t}\ \text{for}\ \forall t\ \text{and}\ \forall i \in N$$ Flow thresholds: $$ x_{ij}^t \leq u_{ij}\ \text{for}\ \forall t\ \text{and}\ \forall ij \in A$$

Outflow limitation: $$ x_{ij}^t \leq f(V_{ij}^{t-1}) \ \text{for}\ \forall t\ \text{and}\ \forall ij \in A$$

Distributed Control Algorithm

Master Problem Objective $$ \max \sum^N_i V_{ij}^i $$ Constraints

Mass Balance: $$ V_i = V_i^0 + \sum_{j} V_{ji} - \sum_{i} V_{ij} $$ Maximum Volume Movement: $$ V_{ij} \leq T \times u_{ij} $$

Sub Problem $$ \max \sum^T_t x_{ij}^t $$ Constraints

Mass Balance: $$ V_i^t = V_i^{t-1} + q_{i}^t - \sum_j x_{ij}^t $$ Physical Constraints: $$ x_{ij}^t \leq 1.13047751 \times V^t_i/A_i + 3.65949363 $$ Constraint from Master Problem: $$ \sum_t x_{ij}^t \leq V_{ij} $$

$ q_i^t $ inflows into the node

Results :

Centralized Controller

Centralized Controller : Coordination

Distributed Controller

Issues to be addressed

1. Outflow is limited
2. Spikes in the flows
3. Horizon time

Thank you.