Inequality constraints (for each time-step $k$ ) 1. Linear power consumption model inequality constraint power_model_ineq_constraint.m $$\begin{equation}
(n^j(k)-1) \, U_{power} \le \left( m^j_q \, q^j(k) + m^j_s \,s^j(k) + c^j \right) - p^j(k) \le (1-n^j(k)) \, U_{power}
\end{equation}$$
The above LHS and RHS inequalities can be represented as:
$$\begin{equation}
\left\{
\begin{array}{lll}
-m^j_q \, q^j(k) - m^j_s \,s^j(k) + p^j(k)& \le & U_{power} + c^j \\\
+ m^j_q \, q^j(k) + m^j_s \,s^j(k) - p^j(k) & \le & U_{power} - c^j
\end{array}
\right. \quad \forall j \in \mathbf{I}_{pumps}
\end{equation}$$
2. Binary linearization of the power consumption variable with respect to pump status - power_ineq_constraint.m $$\begin{equation}
0 \le p^j(k) \le n^j(k) \, U_{power}
\end{equation}$$
which translate into the following two single-sided inequalities:
$$\begin{equation}
\left\{
\begin{array}{lll}
-p^j(k) & \le & 0 \\\
+p^j(k) - U_{power} \, n^j(k) & \le & 0
\end{array}
\right. \quad \forall j \in \mathbf{I}_{pumps}
\end{equation}$$
3. Boundary constraints for pipe segment flows - pipe_flow_segment_constraints.m $$\begin{equation}
bb_i^j(k) \, q^j_{i,min} \le ww_i^j(k) \le bb_i^j(k) \, q^j_{i,max}
\end{equation}$$
which translate into the following two single-sided inequalities:
$$\begin{equation}
\left\{
\begin{array}{lll}
-ww_i^j(k) + q^j_{i,min} \, bb_i^j(k) & \le & 0 \\\
+ww_i^j(k) - q^j_{i,max} \, bb_i^j(k) & \le & 0
\end{array}
\right. \quad \forall j \in \mathbf{I}_{pipes} \quad \forall i \in \mathbf{I}_{pipesegments}
\end{equation}$$
4. Linearized pump speed inequality constraint - s_box_constraints.m $$\begin{equation}
n^j(k) \, s^j_{min} \le s^j(k) \le n^j(k) \, s^j_{max}
\end{equation}$$
which translate into the following two single-sided inequalities:
$$\begin{equation}
\left\{
\begin{array}{lll}
-s^j(k) + s^j_{min} \, n^j(k) & \le & 0 \\\
+s^j(k) - s^j_{max} \, n^j(k) & \le & 0
\end{array}
\right. \quad \forall j \in \mathbf{I}_{pumps}
\end{equation}$$
5. Linearized pump flow inequality constraint - q_box_constraints.m $$\begin{equation}
0 \le q^j(k) \le n^j(k) \, q^j_{intcpt}(s^j_{max})
\end{equation}$$
which translate into the following two single-sided inequalities:
$$\begin{equation}
\left\{
\begin{array}{lll}
-q^j(k) & \le & 0 \\\
+q^j(k) - q^j_{intcpt}(s^j_{max}) \, n^j(k) & \le & 0
\end{array}
\right. \quad \forall j \in \mathbf{I}_{pumps}
\end{equation}$$
6. Linearized pump hydraulics constraints - pump_equation_constraints.m $$\begin{equation}
-(1-n^j(k)) \, U_{pump} \le \Delta h(k) - \Delta h_{pump}(k) \le (1-n^j(k)) \, U_{pump}
\end{equation}$$
where
$$\begin{equation}
\Delta h_{pump}(k) = \sum_{i=1}^{n_{s,pump}} \left( dd^j_i \, ss^j_i(k) + ee^j_i \, qq^j_i(k) + ff^j_i \, aa^j_i(k) \right)
\end{equation}$$
which translate into the following two single-sided inequalities:
$$\begin{equation}
\left\{
\begin{array}{lll}
-\Delta h(k) + \Delta h_{pump}(k) + U_{pump} \, n^j(k) & \le & U_{pump} \\\
+\Delta h(k) - \Delta h_{pump}(k) + U_{pump} \, n^j(k) & \le & U_{pump}
\end{array}
\right. \quad \forall j \in \mathbf{I}_{pumps}
\end{equation}$$
7. Linearized pump characteristic domain constraints - pump_domain_constraints.m $$\begin{equation}
m_{ss,i}^n \, ss_i^j(k) + m_{qq,i}^n \, qq_i^j(k) + c_i^n \le 0 \quad \forall i \in \mathbf{I}_{s,pump} \quad \forall j \in \mathbf{I}_{pumps} \quad \forall n \in \{1 \ldots 4\}
\end{equation}$$
8. Setting preferences for pump selection within groups of equal pumps to break problem symmetry - pump_symmetry_breaking.m $$\begin{equation}
-n^{j+1}j(k) + n^j(k) \le 0 \quad \forall j \in \{1 \ldots (n_{pumps}-1)\} \quad \textrm{for each pump group with equal pumps}
\end{equation}$$