From 12142608b66ae467741767427b5fe580ca10bdaf Mon Sep 17 00:00:00 2001 From: dlcole3 Date: Sat, 4 Jan 2025 18:17:50 -0600 Subject: [PATCH 1/2] Added HVAC example --- docs/src/assets/HMPC_sols.png | Bin 0 -> 229534 bytes docs/src/assets/MPC_Example2_Graph.png | Bin 0 -> 86161 bytes docs/src/assets/MPC_Ulayer.png | Bin 0 -> 21520 bytes docs/src/assets/MPC_example1.png | Bin 0 -> 32174 bytes docs/src/assets/room_diagram.png | Bin 0 -> 28071 bytes docs/src/tutorials/HVAC.md | 461 +++++++++++++++++++++++++ 6 files changed, 461 insertions(+) create mode 100644 docs/src/assets/HMPC_sols.png create mode 100644 docs/src/assets/MPC_Example2_Graph.png create mode 100644 docs/src/assets/MPC_Ulayer.png create mode 100644 docs/src/assets/MPC_example1.png create mode 100644 docs/src/assets/room_diagram.png create mode 100644 docs/src/tutorials/HVAC.md diff --git a/docs/src/assets/HMPC_sols.png b/docs/src/assets/HMPC_sols.png new file mode 100644 index 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Note that the `value` function requires first specifying the graph that has been optimized. This is because Plasmo supports having hierarchical, nested graphs that can contain the same variable objects, so we have to specify which graph to query the variable value from. + +```julia +T_room_1 = [value(graph, node[:T][1]) for node in nodes] +T_room_2 = [value(graph, node[:T][2]) for node in nodes] +T_room_3 = [value(graph, node[:T][3]) for node in nodes] + +u_room_1 = [value(graph, node[:u][1]) for node in nodes] +u_room_2 = [value(graph, node[:u][2]) for node in nodes] +u_room_3 = [value(graph, node[:u][3]) for node in nodes] + +t_range = 0:dtu:(NU*dtu - dtu) +plot(t_range, T_room_1, label = "Room 1", linewidth = 2, color = "blue") +plot!(t_range, T_dU[:, 1], label = "Set Point Room 1", linewidth = 1, color = "blue", linestyle = :dash) +plot!(t_range, T_room_2, label = "Room 2", linewidth = 2, color = "red") +plot!(t_range, T_dU[:, 2], label = "Set Point Room 2", linewidth = 1, color = "red", linestyle = :dash) +plot!(t_range, T_room_3, label = "Room 3", linewidth = 2, color = "black") +plot!(t_range, T_dU[:, 3], label = "Set Point Room 3", linewidth = 1, color = "black", linestyle = :dash) +xlabel!("Time") +ylabel!("Temperature (K)") + +plot(t_range, u_room_1, label = "Room 1", linewidth = 2, color = "blue") +plot!(t_range, u_room_2, label = "Room 2", linewidth = 2, color = "red") +plot!(t_range, u_room_3, label = "Room 3", linewidth = 2, color = "black") +xlabel!("Time") +ylabel!("Air Flowrate") +``` + +The resulting figures can be seen below: + +![ULayer_Results_T](../assets/MPC_example1.png) + +![ULayer_Results_u](../assets/MPC_example1_u.png) + +## Hierarchical Implementation + +We can now extend our work above to the hierarchical instance. The hierarchical formulation includes an upper layer and a lower layer. The upper layer solves the MPC formulation with a coarse time resolution. One challenge of the MPC formulation is that each room is linked to the room next to it, so each room cannot be solved independently, and the computational requirements grow with the number of rooms. To address this challenge, Long et al. propose using the solution of the coarse resolution problem to estimate the temperature of the adjacent rooms, and then solve each room independently with a finer time resolution. In other words, the finer resolution problem is solved for each room $i$, and for the expression $\sum_{j \in \mathcal{R}_i} (T_{j,t} - T_{i, t})/R_{i,j}$, the terms $T_{j, t}$ are set as constants based on the solution of the coarse upper level problem. + +We will build this problem in Plasmo.jl, but, to highlight the modeling capabilities of Plasmo.jl, we will not fully make each lower level graph independent of the upper layer (though they will not be linked to other lower layer graphs). The upper layer will be represented by a single subgraph, and Instead, we will use a linking constraint between the upper and lower layers and directly link the upper layer variables to the lower layers. This captures the structure of the different layers with links between them, and it is a structure that could be potenially exploited by different decomposition approaches (e.g., Generalized Benders Decomposition). In this case, we will use one subgraph for the upper layer and then we will create a subgraph for each room on the lower layer (i.e., we will have one subgraph representing the upper layer and three subgraphs representing the lower layer). This allows us to reuse some of the functions we have defined above. However, we will have to define a function for linking the lower layers to the upper layers. + +To begin, we will first define a function to map the index of the lower level to the upper level since they use different temporal resolutions. + +```julia +function map_L_to_U(idx, NU, NL) + nl_per_nu = Int(NL / NU) + + return Int(ceil(idx / nl_per_nu)) +end +``` + +Next, we will define a new function that will operate similar to `add_room_linking!`. The upper and lower level problem subgraphs are "owned" by an overall graph, which is what will "own" the linking constraints. Since each lower level graph represents a single room (and the upper level graph includes all three rooms), this function will take the lower level nodes (`nodes`), the upper level nodes (`ref_nodes`), the set point values (`Tdl`), the time resolution of the lower level (`dtl`), and the room number (`room_num`). Like `add_room_linking!`, this function links across time points, but for the term using the temperature of the adjacent room, it uses the temperature from the upper level. This function also sets the objective of the lower level. We call this new function `add_hierarchical_room_linking!`. + +```julia +function add_hierarchical_room_linking!(overall_graph, nodes, ref_nodes, NU, NL, TdL, dtl, room_num) + c1 = 10 + c2 = 0.5 + + for (i, node) in enumerate(nodes) + if node != nodes[NL] + T_var = nodes[i + 1][:T][1] + T_var_next = nodes[i + 1][:T][1] + node_ref_idx = map_L_to_U(i, NU, NL) + + if room_num == 1 # if in the first room of the system + T_ref = ref_nodes[node_ref_idx][:T][room_num + 1] + @linkconstraint(overall_graph, Ci * (T_var_next - T_var) / dtl == + node[:u_cp_T][1] + + (T_ref − T_var) / Rij + + (Toa − T_var) / Rioa + Pid + ) # relationship between room temps + + elseif room_num == num_rooms # if in the last last room of the system + T_ref = ref_nodes[node_ref_idx][:T][room_num - 1] + @linkconstraint(overall_graph, Ci * (T_var_next - T_var) / dtl == + node[:u_cp_T][1] + + (T_ref − T_var) / Rij + + (Toa − T_var) / Rioa + Pid + ) # relationship between room temps + + else # if in a room surrounded by 2 rooms + T_ref1 = ref_nodes[node_ref_idx][:T][room_num - 1] + T_ref2 = ref_nodes[node_ref_idx][:T][room_num + 1] + @linkconstraint(overall_graph, Ci * (T_var_next - T_var) / dtl == + node[:u_cp_T][1] + + (T_ref1 − T_var) / Rij + + (T_ref2 − T_var) / Rij + + (Toa − T_var) / Rioa + Pid + )# relationship between room temps + end + end + @objective(node, Min, + (c1 * (sum((node[:T][1] − TdL[i, room_num]) .^ 2)) + + (c2*sum(node[:u] .^ 2))).*dtl) + end +end +``` + +Lastly, Long et al. propose that the temperature of the lower layers should be within some value of the upper layers (that is, the lower layer should not deviate too much from the upper layer). As the time horizon increases, the deviation from the upper layer is allowed to increase as well. We add this constraint in the function `add_deviation_constraints` below + +```julia +function add_deviation_constraints!(graph, nodes, ref_nodes, NU, NL, room_num, epsilon, r) + for (i, node) in enumerate(nodes) + node_ref_idx = map_L_to_U(i, NU, NL) + + for room_num in 1:num_rooms # The absolute value of the deviation from the upper layer remains within some bound + @linkconstraint(graph, nodes[i][:T][1] - ref_nodes[node_ref_idx][:T][room_num] <= epsilon * (1 + i * r)) + @linkconstraint(graph, - nodes[i][:T][1] + ref_nodes[node_ref_idx][:T][room_num] <= epsilon * (1 + i * r)) + end + end +end +``` + +Note that each of the above functions could be revised to use the solution value of the upper layer by calling `value(upper_graph, ref_nodes[node_ref_idx][:T][room_num])` where the variable appears in the link constraints. + +### Example Simulation + +We can now run an example problem of this hierarchical problem. To begin, we define the data for the upper and lower levels. Here, we will use 10 time points for the upper layer and 40 time points for the lower layer. + +```julia +NL = 40 +NU = 10 +dtu = 10 +dtl = 2.5 +num_rooms = 3 +T_init = 300 # Kelvin + +T_dU = zeros(NU, num_rooms) +T_dU[:, 1] .= [292 - i * .02 for i in 1:NU] +T_dU[:, 2] .= [291 - i * .01 for i in 1:NU] +T_dU[:, 3] .= [290 - i * .02 for i in 1:NU] + +T_dL = zeros(NL, num_rooms) +T_dL[:, 1] .= [292 - i * .02 * .25 for i in 1:NL] +T_dL[:, 2] .= [291 - i * .01 * .25 for i in 1:NL] +T_dL[:, 3] .= [290 - i * .02 * .25 for i in 1:NL] +``` + +We can next define the upper and lower level graphs. Since each lower level graphs only contain one room, we set the `num_rooms` argument of the `MPC_HVAC` function to be 1. We then add each of these subgraphs to an overall graph. + +```julia +Ugraph, Unodes = MPC_HVAC(NU, num_rooms, T_init); + +Lgraph1, Lnodes1 = MPC_HVAC(NL, 1, T_init); +Lgraph2, Lnodes2 = MPC_HVAC(NL, 1, T_init); +Lgraph3, Lnodes3 = MPC_HVAC(NL, 1, T_init); + +graph = OptiGraph() +add_subgraph!(graph, Ugraph) +add_subgraph!(graph, Lgraph1) +add_subgraph!(graph, Lgraph2) +add_subgraph!(graph, Lgraph3) +``` + +Next, we can add the linking constraints across layers (and within the upper level graph). + +```julia +add_room_linking!(Ugraph, Unodes, T_dU, dtu) # add links to upper layer + +# Link lower levels to upper levels +add_hierarchical_room_linking!(graph, Lnodes1, Unodes, NU, NL, T_dL, dtl, 1) +add_hierarchical_room_linking!(graph, Lnodes2, Unodes, NU, NL, T_dL, dtl, 2) +add_hierarchical_room_linking!(graph, Lnodes3, Unodes, NU, NL, T_dL, dtl, 3) + +add_deviation_constraints!(graph, Lnodes1, Unodes, NU, NL, 1, 1, .2) +add_deviation_constraints!(graph, Lnodes2, Unodes, NU, NL, 2, 1, .2) +add_deviation_constraints!(graph, Lnodes3, Unodes, NU, NL, 3, 1, .2) +``` + +We can visualize the structure of this problem by using PlasmoPlots.jl. Here, we will omit node labels and we will color each subgraph its own color. + +```julia +plt_graph = layout_plot( + graph, + node_labels=false, + markersize=5, + labelsize=8, + linewidth=2, + subgraph_colors = true, + layout_options=Dict( + :tol=>.01, + :iterations=>500, + :C => 50, + :K => 1 + ), + plt_options=Dict( + :legend=>false, + :framestyle=>:box, + :grid=>false, + :size=>(600,600), + :axis => nothing + ) +) +``` + +The resulting structure can be seen below. + +![HMPC_Structure](../assets/MPC_Example2_Graph.png) + +With those linking constraints added, we can now solve the problem. + +```julia +set_to_node_objectives(graph) +set_optimizer(graph, Ipopt.Optimizer) +set_optimizer_attribute(graph, "max_iter", 200) + +optimize!(graph) +``` + +After optimizing the graph, we can query and plot the solutions. + +```julia +# Query solutions of Temperatures +T_room1L = [value(graph, Lgraph1[:nodes][i][:T][1]) for i in 1:Plasmo.num_nodes(Lgraph)] +T_room1U = [value(graph, Ugraph[:nodes][i][:T][1]) for i in 1:Plasmo.num_nodes(Ugraph)] +T_room2L = [value(graph, Lgraph2[:nodes][i][:T][1]) for i in 1:Plasmo.num_nodes(Lgraph)] +T_room2U = [value(graph, Ugraph[:nodes][i][:T][2]) for i in 1:Plasmo.num_nodes(Ugraph)] +T_room3L = [value(graph, Lgraph3[:nodes][i][:T][1]) for i in 1:Plasmo.num_nodes(Lgraph)] +T_room3U = [value(graph, Ugraph[:nodes][i][:T][3]) for i in 1:Plasmo.num_nodes(Ugraph)] + +# Query solutions of flowrates +u_room1L = [value(graph, Lgraph1[:nodes][i][:u][1]) for i in 1:Plasmo.num_nodes(Lgraph)] +u_room1U = [value(graph, Ugraph[:nodes][i][:u][1]) for i in 1:Plasmo.num_nodes(Ugraph)] +u_room2L = [value(graph, Lgraph2[:nodes][i][:u][1]) for i in 1:Plasmo.num_nodes(Lgraph)] +u_room2U = [value(graph, Ugraph[:nodes][i][:u][2]) for i in 1:Plasmo.num_nodes(Ugraph)] +u_room3L = [value(graph, Lgraph3[:nodes][i][:u][1]) for i in 1:Plasmo.num_nodes(Lgraph)] +u_room3U = [value(graph, Ugraph[:nodes][i][:u][3]) for i in 1:Plasmo.num_nodes(Ugraph)] + +# Set time ranges +t_rangeU = (0:dtu:(NU*dtu - dtu)) +t_rangeL = (0:dtl:(NL*dtl - dtl)) + +# Plot solutions +plot(t_rangeU, T_room1U, color = "black", label = "Upper Level", linewidth = 2 ) +plot!(t_rangeL, T_room1L, color = "gray", label = "Lower Level", linewidth = 2, ls = :dash ) +plot!(t_rangeL, T_dL[:, 1], color = "darkgray", label = "Setpoint", linewidth = 1) +xlabel!("Time") +ylabel!("Temperature") +title!("Room 1") + +plot(t_rangeU, T_room2U, color = "black", label = "Upper Level", linewidth = 2 ) +plot!(t_rangeL, T_room2L, color = "gray", label = "Lower Level", linewidth = 2, ls = :dash ) +plot!(t_rangeL, T_dL[:, 2], color = "darkgray", label = "Setpoint", linewidth = 1) +xlabel!("Time") +ylabel!("Temperature") +title!("Room 2") + +plot(t_rangeU, T_room3U, color = "black", label = "Upper Level", linewidth = 2 ) +plot!(t_rangeL, T_room3L, color = "gray", label = "Lower Level", linewidth = 2, ls = :dash ) +plot!(t_rangeL, T_dL[:, 3], color = "darkgray", label = "Setpoint", linewidth = 1) +xlabel!("Time") +ylabel!("Temperature") +title!("Room 3") + +plot(t_rangeU, u_room1U, color = "black", label = "Upper Level", linewidth = 2 ) +plot!(t_rangeL, u_room1L, color = "gray", label = "Lower Level", linewidth = 2, ls = :dash) +xlabel!("Time") +ylabel!("Air Flowrate") + +plot(t_rangeU, u_room2U, color = "black", label = "Upper Level", linewidth = 2 ) +plot!(t_rangeL, u_room2L, color = "gray", label = "Lower Level", linewidth = 2, ls = :dash) +xlabel!("Time") +ylabel!("Air Flowrate") + +plot(t_rangeU, u_room3U, color = "black", label = "Upper Level", linewidth = 2 ) +plot!(t_rangeL, u_room3L, color = "gray", label = "Lower Level", linewidth = 2, ls = :dash) +xlabel!("Time") +ylabel!("Air Flowrate") +``` + +The plots of the solution are shown below + +![HMPC_Solutions](../assets/HMPC_sols.png) From 076b00e86d2f190b89ec08a213b1a884137c3885 Mon Sep 17 00:00:00 2001 From: dlcole3 Date: Sat, 4 Jan 2025 18:19:47 -0600 Subject: [PATCH 2/2] Added missing figure --- 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