2.2. Constraints
The optimal design problem includes the following constraints on the WDS side, which constitute node mass conservation (1), tank continuity (2), tank closure (3), pump curve (4), link energy conservation (5), node head bounds (6), pump rotation speed bounds (7), and non-negative pump flow (8), and head (9):
$${A}^{T}\bullet {Q}^{t}={d}^{t} \forall t\in \mathcal{T}$$
(1)
$${H}_{tank,s}^{t}={H}_{tank,s}^{t-1}+\frac{{Q}_{tank,s}^{t}\bullet \varDelta t}{{\alpha }_{tank,s}} \forall t\in \mathcal{T} \forall s\in S$$
(2)
$${H}_{tank,s}^{24}\ge 0.95\bullet {H}_{tank,s}^{0}\forall s\in S$$
(3)
$${H}_{p,l}^{t}={a}_{pumps,l}\bullet {\left(\frac{{n}_{p,l}^{t}}{{n}_{max}}\right)}^{2}-{b}_{pumps,l}\bullet {\left({Q}_{p,l}^{t}\right)}^{2} \forall l\in L \forall t\in \mathcal{T}$$
(4)
(5) \(A\bullet {H}^{t}=-R{Q}^{t}{\left|{Q}^{t}\right|}^{0.852}-{A}_{0}\bullet {H}_{0}+{A}_{pumps}\bullet {H}_{p}^{t} \forall t\in \mathcal{T}\)
$${H}_{min}^{t}\le {H}^{t}\le {H}_{max}^{t} \forall t\in \mathcal{T}$$
(6)
$${n}_{min}^{t}\le {n}_{p,l}^{t}\le {n}_{max}^{t} \forall t\in \mathcal{T} \forall l\in L$$
(7)
$${Q}_{p,l}^{t}\ge 0 \forall t\in \mathcal{T} \forall l\in L$$
(8)
$${H}_{p,l}^{t}\ge 0 \forall t\in \mathcal{T} \forall l\in L$$
(9)
where: \({A}^{T}\) is the network's incidence matrix; \(d\) is vector of nodal demands; \({H}_{tank,s}\) is the total head of water in tank \(s\) [m]; \({Q}_{tank,s}\) is the flow of water in the pipe connected to tank s [m3/hr]; \(\varDelta t\) is the length of a timestep [hr]; \({\alpha }_{tank,s}\) is the cross-section area of tank \(s\) [m2]; \({H}_{tank,s}^{0}\) is a given initial condition for the head of tank \(s\); \({H}_{tank,s}^{24}\) is the elevation of tank \(s\) at the end of the simulation period [m]; \(R\) is a column vector of pipe resistances; \({A}_{0}\) is a matrix with elements {0,1}, which defines the columns corresponding to the sources in the graph matrix; \({H}_{0}\) is a column vector containing the values of constant heads at the sources [m]; \({A}_{pumps}\) is a matrix containing pump links indicators inside the WDS, used to adjust the dimensions of matrix \({H}_{p}^{t}\); \(\) is the Hadamard product (elementwise multiplication); \({a}_{pumps,l}\) and \({b}_{pumps,l}\) are the curve coefficients of pump \(l\); \({n}_{p,l}^{t}\) is the pump speed of pump \(l\); \({n}_{max}\) is the maximum pump speed [rpm]; \({H}_{min}\) and \({H}_{max}\) are vectors of the upper and lower head bounds at the nodes [m]; and \({n}_{min}\) and \({n}_{max}\) are the minimum and maximum pump rotation speeds, respectively.
This formulation is novel in the sense that it allows formulating the problem as a non-linear program (NLP), instead of a mixed-integer non-linear program (MINLP). The optimal operation problem of WDS is categorized as an MINLP, as it required the use of integer variables to correctly model the operation of pumps. When a pump is off, the result is a disconnection of a source from the network. In that case, (5) will require the inclusion of integer variables. When using variable speed pumps, modelling such a disconnection could be achieved without the use of integer variables. A pump will be considered off when pump flow is equal to zero. The pump rotation speed variable will then act as a free variable to adjust the head at the pump’s downstream node, virtually allowing the pump to function as a valve and regulate a virtual head gain along the pump.
On the PG side, the problem includes the following constraints, which constitute the relationship between power flow and voltage drop along transmission lines, bus power balance, generation limits, and voltage magnitude bounds:
(10)\({Y}_{m}=\frac{1}{2}\left({e}_{m}{e}_{m}^{T}Y+{Y}^{H}{e}_{m}{e}_{m}^{T}\right) \forall m\in M\)
(11)\({\stackrel{-}{Y}}_{m}=\frac{j}{2}\left({e}_{m}{e}_{m}^{T}Y-{Y}^{H}{e}_{m}{e}_{m}^{T}\right) \forall m\in M\)
(12)\({p}_{m}^{t}={\left({v}^{t}\right)}^{H}\bullet {Y}_{m}\bullet {v}^{t} \forall m\in M \forall t\in \mathcal{T}\)
(13)\({q}_{m}^{t}={\left({v}^{t}\right)}^{H}\bullet {\stackrel{-}{Y}}_{m}\bullet {v}^{t} \forall m\in M \forall t\in \mathcal{T}\)
(14)\({p}_{m}^{t}={p}_{in,m}^{t}-{p}_{L,m}^{t}-\frac{1}{FOC}\sum _{l\in L}\left({I}_{m,l}\bullet \frac{\gamma {H}_{p,l}^{t}{Q}_{p,l}^{t}}{{\eta }_{p}}\right)\)
(15)\({q}_{m}^{t}={q}_{in,m}^{t}-{q}_{L,m}^{t}-\frac{1}{FOC}\sum _{l\in L}\left({I}_{m,l}\bullet {\alpha }_{qp}\frac{\gamma {H}_{p,l}^{t}{Q}_{p,l}^{t}}{{\eta }_{p}}\right)\)
(16)\({\underset{\_}{p}}_{in,k}{\le p}_{in,k}^{t}\le {\overline{p}}_{in,k}\forall t\in \mathcal{T}\)
(17)\({\overline{p}}_{in,k}=S{F}_{k}\bullet SGC \forall k\in {m}^{*}\)
(18)\({\left(\underset{\_}{v}\right)}^{2}\le {\left({v}^{t}\right)}^{H}\bullet {M}_{m}\bullet {v}^{t}\le {\left(\overline{v}\right)}^{2} \forall m\in M \forall t\in \mathcal{T}\)
(19)\({v}_{1}^{t}={v}_{ref} \forall t\in \mathcal{T}\)
where: \(Y\) is the power grid’s admittance matrix; \({e}_{m}\) is a standard basis vector containing \(1\) in the mth element; \(m\) is the bus index; \(M\) is set of buses in the grid; \(t\) is the timestep; \({\left(\bullet \right)}^{H}\) is the complex-conjugate transpose; \({p}_{in,m}^{t}\) and \({q}_{in,m}^{t}\) are the active and reactive power generated at bus \(m\), respectively; \({p}_{L,m}^{t}\) and \({q}_{L,m}^{t}\) are the active and reactive loads, respectively; \({I}_{m,l}\) is an indicator parameter indicating which buses are connected to pumps in the grid; \(\gamma\) is water specific weight [N/m3]; \({H}_{p,l}^{t}\) is the pump head gain [m]; \({Q}_{p,l}^{t}\) is the pump flow [m3/sec];\({\alpha }_{qp}\) is the pump's active and reactive ratio; \({\eta }_{p}\) is the pump efficiency; \(FOC\) is the factor of conversion, converting pump power from metric units into the p.u. unit system, set here as 1 [MW/p.u.]; \({m}^{*}\) is the index corresponding to the bus where solar power generators are installed; \({\underset{\_}{p}}_{in,k}\) and \({\overline{p}}_{in,k}\) are the lower and upper generation bounds for generator k, respectively; \(SGC\) is the solar generation curve, presented in Fig. 1; \(\underset{\_}{\text{v}}\) and \(\overline{\text{v}}\) are the lower and upper voltage bounds, respectively; \({M}_{m}\) is a matrix defined by the standard basis vectors, such that \({M}_{m}={e}_{m}\bullet {e}_{m}^{T}\); and \({v}_{ref}\) is the voltage value defined for the reference bus.
2.3. Objective function
Usually, the objective function of optimal WDS design problems is defined as the minimal cost of both operation and construction derived from the design. In this paper, the results of two different problems will be presented and compared: the first being an optimal design problem of an independent WDS, with its results imposed on the power grid as a given load; and the second being the conjunctive optimal design problem, when both systems are designed simultaneously. Here, both the objective function of the independent WDS design problem and the objective function of the conjunctive design problem will be presented. When formulating the independent design problem, energy tariffs were not considered for the operation of pumps. Those tariffs are usually the result of an optimization done by the PG operators. It was the authors’ goal to present a reasonable comparison between the solutions to the two design problems, and therefore the decision was made not to consider energy tariffs. This matter is addressed in more detail in the conclusion section. The objective function of the independent problem comprises of the cost of constructing all pumping stations and tanks in the system. Those costs can be calculated by the following expressions:
(20)\(PUCC=\frac{CPUMP}{270\bullet {\eta }_{p}}\sum _{l=1}^{npumps}\text{m}\text{a}\text{x}({H}_{p,l}^{1}\bullet {Q}_{p,l}^{1},\dots ,{H}_{p,l}^{T}\bullet {Q}_{p,l}^{T})\)
(21)\(TCC=\sum _{s=1}^{ntanks}UT{C}_{s}\bullet \text{m}\text{a}\text{x}({H}_{tanks,s}^{1},\dots ,{H}_{tanks,s}^{T})\)
where \(PUCC\) is pump construction cost [MU]; \(CPUMP\) is unit pump cost of pump construction [MU/hp]; \(TCC\) is tank construction cost [MU]; \(UT{C}_{s}\) is unit water level cost of tank \(s\)[MU/m].
And the objective function for the independent design problem is given in (22):
(22)\({Z}_{1}=\text{min}TCC+PUCC\)
For the conjunctive design problem, two cost components are added to the objective – the cost of generating power, and the cost of constructing a solar field. Those are given by the following expressions:
(23)\(PGC=\left(\sum _{t\in \mathcal{T}}\sum _{k\in K}{a}_{k}\left({{p}_{in,k}^{t}}^{2}+{{q}_{in,k}^{t}}^{2}\right)+{b}_{k}({p}_{in,k}^{t}+{q}_{in,k}^{t})+{c}_{k}\right)\bullet AD\bullet APGPV\)
(24)\(SFC=w\bullet \left({\sum }_{k=1}^{nsolargenerators}S{F}_{k}\right)\)
where \(PGC\) is power generation cost [MU]; \({a}_{k}\)[MU/p.u2] \({b}_{k}\)[MU/p.u] and \({c}_{k}\)[MU] are generation cost coefficients of generator \(k\); \(AD\) is annual duration [days]; \(APGPV\) is annual power generation present value coefficient; \(SFCC\) is solar field construction cost [MU]; and \(w\) is the solar field construction cost coefficient [MU/MW].
And the objective function for the conjunctive design problem is given in (25):
(25)\({Z}_{2}=\text{min}TCC+PUCC+PGC+SFCC\)
Table 1 details the cost parameters that were used. Where WDS cost parameters are based on Ostfeld (2005).
Table 1