# Pressure Retarded Osmosis Power Units Modelling for Power Flow Analysis of Electric Distribution Networks

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## Abstract

**:**

## 1. Introduction

^{2}were reached [8]. The initial objective of the Mega-ton project was to produce fresh water from seawater desalination; the outcoming high salinity concentration brine water is currently used by the PRO plant, mainly to recover energy and reduce fresh water production costs. Additionally, in 2013, the Statkraft company explored the construction of a 2 MW power plant [5]. It is important to point out that the semipermeable membrane is a critical core feature of harnessing electric power from a salinity gradient [5]. To make feasible electric power production at a commercial level, PRO membranes must be able to provide a power density higher than 6.5 W/m

^{2}[9]. The Mega-ton project revealed that it was possible to surpass this technical barrier. In addition, this project proved that electric power can be harvested from hybrid processes, where the salinity gradient is not directly given by a natural source [6,8,9]. Arguably, PRO power is a promising technology to produce electric energy from salinity gradients in a continuous shape, with high expectations for its integration into distribution networks located at coastlines over the next few years [6,9].

## 2. Models of Distribution Network Conventional Components

#### 2.1. Modelling of Conventional Series Components

**a**,

**b**,

**c**,

**d**,

**A**and

**B**, are constant matrices computed based on the series impedance

**Z**

_{abc}and shunt admittance

**Y**

_{abc}of the line segment. These two latter matrices are obtained from the parameters of the conductors composing the line segment, as detailed in [21]. Thus, on the one hand, the linear systems (1) and (2) are used in the F-th forward sweep step to assess ${\mathit{V}}_{abc}^{n}$ and ${\mathit{I}}_{abc}^{n}$ at node n, respectively, from the phasors ${\mathit{V}}_{abc}^{m}$ and ${\mathit{I}}_{abc}^{m}$ at node m. On the other hand, the linear system (3) is used in the B-th backward sweep step to update ${\mathit{V}}_{abc}^{m}$ at node m from phasors ${\mathit{V}}_{abc}^{n}$ and ${\mathit{I}}_{abc}^{n}$ at node n.

**a**

_{t},

**b**

_{t},

**c**

_{t},

**d**

_{t},

**A**

_{t}and

**B**

_{t}are obtained from the nominal ratings, parameters, and type of connection of the transformer, as detailed in [21]. The linear system of Equations (4) and (5) are used in the F-th forward sweep step to assess ${\mathit{V}}_{abc}^{n}$ and ${\mathit{I}}_{abc}^{n}$, respectively, at the high voltage terminal (node n) from the known phasors ${\mathit{V}}_{abc}^{m}$ and ${\mathit{I}}_{abc}^{m}$ at the low voltage terminal (node m). Note that unlike (3), the set of linear Equations (6) considers the phasors ${\mathit{V}}_{abc}^{n}$ and ${\mathit{I}}_{abc}^{m}$ at node n and m, respectively, to update the phasor voltages ${\mathit{V}}_{abc}^{m}$ at node m in the B-th backward sweep step.

#### 2.2. Modelling of Conventional Shunt Components

**D**

_{I}defined by (26).

## 3. Modelling of PRO Power Units

#### 3.1. Model of the Harvested Hydraulic Power

_{Qr}and A

_{Qm}, respectively. These bins are separated by a semipermeable membrane of effective permeation area A

_{mem}, length L, and thickness δ

_{mem}. The river water and seawater are pumped into their corresponding bins. The river water bin is at atmospheric pressure, whereas the seawater bin is mechanically pressurized at the hydraulic pressure ΔP

_{h}by means of the external pressure exchanger and the booster pump (see Figure 3). At the inlet of these bins of the membrane module, the pumped river water (resp. seawater) has the salinity concentration and flow rate denoted by C

_{r}(resp. C

_{m}) and Q

_{r}(resp. Q

_{m}), respectively.

_{m}denote the universal gas constant, the solution temperature, the molar mass of the salt, and the Van’t Hoff factor, respectively. The osmotic pressure difference Δπ is higher than the opposing hydraulic pressure ΔP

_{h}imposed inside the seawater bin, hence it drives the permeation of river water to the seawater bin through the membrane area A

_{mem}. The permeate flux J

_{w}and the flow rate Q

_{mem}of this river water transference are given by (29) and (30), respectively. In (29), A is the water permeability coefficient. The permeation of the river water at the opposing hydraulic pressure ΔP

_{h}is known as the pressure retarded osmosis mechanism [2,6].

_{salt}and permeate flux J

_{salt}equal to zero. Hence, for this ideal condition, the hydraulic power harvested in the membrane module through the pressure retarded osmosis mechanism can be assessed by the product of the flow rate Q

_{mem}and the opposing hydraulic pressure ΔP

_{h}, as given by (31).

_{salt}and permeate flux J

_{salt}different from zero, whereas the river water permeates to the seawater bin at flow rate Q

_{mem}and permeate flux J

_{w}. On the one hand, the salt mass transference produces a zone inside the river water bin where the river water salinity concentration increases from the value C

_{r}to a value C

_{rp}at the membrane surface. On the other hand, the river water mass transference produces a zone inside the seawater bin where the seawater salinity concentration decreases from the value C

_{m}to a value C

_{mp}at the membrane surface, as illustrated in the cross-section of the membrane module shown in Figure 5. These zones of different salinity concentrations are referred to as the salinity concentration polarization phenomenon [23]. Accordingly, this phenomenon produces the effective salinity gradient ΔC

_{pol}given by (32). Since C

_{m}> C

_{mp}and C

_{r}< C

_{rp}, the effective salinity gradient ΔC

_{pol}is considerably lower than the ideal one ΔC estimated from (27), as illustrated in Figure 5.

_{mp}and C

_{rp}, however, are not known in advance; hence, the effective salinity gradient ΔC

_{pol}must be estimated from the solution of the non-linear Equation (33), where $g\left(\mathsf{\Delta}{C}_{pol}\right)$ and $f\left(\mathsf{\Delta}{C}_{pol}\right)$ are the functions given by (34) and (35). According to (28) and (29), the water permeate flux ${J}_{w}$ depends on the salinity gradient through the membrane surfaces. Arguably, due to the polarization phenomenon, that salinity gradient must be considered as ΔC

_{pol}instead of the ideal one ΔC. Thus, in (34) and (35) ${J}_{w}$ is expressed as the function of ΔC

_{pol}given by (36), where $\mathsf{\Delta}{\pi}_{pol}$ and ΔP

_{h}are given by (37) and (38), respectively. In these equations, S, δ

_{j}, and D

_{j}represent the membrane structure parameter, the thickness of the salinity increasing and decreasing zones (see Figure 5), and the salt diffusion coefficient (for j = r,m), which are detailed in [23]. In addition, B is the salt permeability coefficient, whereas C

_{r}, C

_{m}, R, T, i

_{m}, and M are those previously denoted for (27) and (28). Once ΔC

_{pol}is obtained from the solution of (33), the river water permeate flux ${J}_{w}$, the osmotic pressure $\mathsf{\Delta}{\pi}_{pol}$, the hydraulic pressure ΔP

_{h}, and the salt permeate flux J

_{salt}can be evaluated from (36), (37), (38), and (39), respectively.

_{pol}is obtained from the solution of (33) by assuming that the river water and seawater salinity concentrations C

_{r}and C

_{m}, respectively, remain constant along the membrane length. However, the mass transference of river water and salt occurs gradually through the membrane. Accordingly, the water (resp. salt) total mass inside the river water bin gradually decays (resp. increases), whereas progressively increases (resp. decays) in the seawater bin. This mainly produces gradual spatial variations of the flow rates Q

_{r}and Q

_{m}, the permeate fluxes J

_{w}and J

_{salt}, the salinity concentrations C

_{r}and C

_{m}, and the effective salinity gradient ΔC

_{pol}along the membrane length L. The gradual variation of these variables owing to the progressive mass transference of river water and salt along the membrane is known as the spatial variation phenomenon [23].

_{s}= L/n and the values of the variables for each segment are then computed, as detailed below. Figure 6 depicts two consecutive segments. Consider that the values of the flow rates and salinity concentrations Q

_{j}(k) and C

_{j}(k) (for j = r,m), respectively, are known at the input of the k-th segment. Take these values to solve (40)–(45) for ΔC

_{pol}(k). Once ΔC

_{pol}(k) is obtained, the values of the osmotic pressure Δπ(k) and the permeate flux J

_{w}(k) and J

_{salt}(k) at the k-th segment can be directly computed from (44), (43) and (46), respectively. Using Q

_{j}(k) and C

_{j}(k), as well as the resulting values of J

_{w}(k) and J

_{salt}(k), the flow rates and salinity concentrations Q

_{j}(k + 1) and C

_{j}(k + 1) (for j = r,m), respectively, at the input of the (k + 1)-th segment must be evaluated from the model (47)–(50), as illustrated in Figure 6. To compute the spatial variations along the membrane length, the (k + 1)-th segment is set as the k-th segment and this process is repeated until processing the last (n-th) segment. Clearly, this procedure starts at the first segment, i.e., for k = 1, where the values of Q

_{j}(k = 1) and C

_{j}(k = 1) are known. Note that the model (40)–(45) corresponds to the discretized form of (33)–(38). Furthermore, in (45) the value of the effective salinity gradient is fixed at the value $\mathsf{\Delta}{C}_{pol}\left(k=1\right)$ computed at the very first segment, in order to neglect the pressure losses along the membrane module.

_{h}(k) computed during the evaluation of the spatial variation phenomenon are used to perform an accurate evaluation of the hydraulic power from (51).

#### 3.2. Model of the Mechanical Power Produced by the Turbine

_{Tur}. However, the river and seawater pumping systems and the hydraulic pressurization system of the PRO power unit have power consumptions that must be considered in the transformation process. These power consumptions, denoted by P

_{rps}, P

_{mps}, and P

_{hps}, respectively, are evaluated from (52), (53), and (54), respectively, and should be subtracted from the hydraulic power ${P}_{hyd}$ to obtain the net hydraulic power ${P}_{hyd\text{\_}net}$ harvested from the PRO mechanism, as given in (55).

_{md}represents the pressure drop along the pipeline connecting the output of the pressure exchanger to the inlet of the seawater bin, whereas the term $\mathsf{\Delta}{P}_{h}\left(1-{\eta}_{px}\right)$ represents the pressure that is not recovered from the mixed solution by the pressure exchanger of efficiency ${\eta}_{px}$. Note that these two pressure drops are compensated by the booster pump of efficiency ${\eta}_{bp}$ located near the inlet of the seawater bin (see Figure 3) in order to ensure that the seawater enters its bin at the hydraulic pressure ΔP

_{h}.

_{mod}of identical membrane modules supplies a total hydraulic power P

_{hyd_total}to the single turbine of the PRO power unit, which is given as P

_{hyd_total}= P

_{hyd_total}n

_{mod}. In this context, an accurate evaluation of the net mechanical power ${P}_{mech}$ produced by the turbine from the pressure retarded osmosis mechanism can be directly obtained from (56).

#### 3.3. Model of the Electric Generator

_{mech}produced by the turbine from the pressure retarded osmosis mechanism. Furthermore, for the sake of simplicity but without loss of generality, it is considered the generator is directly connected to the distribution network through a coupling distribution transformer. Hence, the reactive power, voltage, and power factor controls can be neglected.

_{mech}is formulated below, based on the asynchronous generator model described in [21].

**V**

_{LNi},

**I**

_{si}, R

_{s}, and X

_{s}represent the line-to-neutral phasor voltages, the line phasor currents, resistance, and reactance of the stator, respectively. Furthermore, X

_{m}is the magnetization reactance, whereas

**I**

_{ri}, R

_{r}, and X

_{r}are the line phasor currents, resistance, and reactance of the rotor, respectively. The load resistance R

_{Li}is used to represent the transformation of the mechanical power P

_{mech}into active electric power ${P}_{R}$ in the rotor of the generator. The positive (resp. negative) sequence model is obtained by setting the value of the subscript i as 1 (resp. as 2) in these quantities.

_{mech}into the active electric power ${P}_{R}$ is formulated by the power balance Equation (57), where the sequence load resistances R

_{Li}and the sequence rotor currents

**I**

_{ri}(for i = 1, 2) are given by (58) and (59), respectively. Note that these rotor currents depend on the stator currents

**I**

_{si}(for i = 1, 2), which are given by (60). In addition, the positive and negative sequence slips of the machine are represented by s

_{i}(for i = 1, 2) in (58)–(60). The negative sequence slip s

_{2}can be expressed in terms of the positive sequence slip s

_{1}, as denoted by (61). Lastly, the line-to-neutral phasor voltages

**V**

_{LNi}in (60) can be readily obtained from the known line-to-ground voltages ${\mathit{V}}_{abc}^{n}$ at the connecting node n according to (62), where ${\mathit{A}}_{s}{}^{-1}$ represents the inverse of the well-known symmetrical component transformation matrix. Thus, according to (58)–(62), in the non-linear power balance Equation (57) the single variable to be known is the positive sequence slip s

_{1}. To obtain the value of s

_{1}from (57), the fsolve solver of MatLab

^{®}for the solution of non-linear equations is used. For this purpose, an initial condition ${s}_{1}^{0}$ for s

_{1}must be assessed, as explained at the end of this section.

_{1}is obtained from the solution of (57), the negative sequence slip s

_{2}must be obtained from (61). Based on the value of these slips, the positive and negative stator currents

**I**

_{si}(for i = 1, 2) can be directly evaluated from (60). Lastly, using the resulting sequence currents

**I**

_{si}, the line currents ${\mathit{I}}_{abc}^{{n}_{PRO}}$ injected by the asynchronous generator at the k-th forward sweep step can be expressed as in (63), where the value of the zero-sequence current

**I**

_{s0}must be set as zero (i.e.,

**I**

_{s0}= 0). This completes the PRO power unit current injection model sought.

_{3ϕ}generated by the PRO power unit is obtained from (64), where the superscript T denotes transposed vector.

_{L1}must be first carried out to obtain the initial condition ${s}_{1}^{0}$ of the positive sequence slip s

_{1}required for solving (57). The coefficients of (65) are given by (66). The general guidelines to Formulates (65) and (66) are given in [19]. The solution of (65) for R

_{L}

_{1}is directly obtained from (67), which yields the values R

_{L}

_{1a}and R

_{L}

_{1b}. Evaluating (68) with the resulting values of R

_{L}

_{1a}and R

_{L}

_{1b}yields two possible initial conditions ${s}_{1a}^{0}$ and ${s}_{1b}^{0}$ for the positive sequence slip s

_{1}. Since the induction machine is operated as generator, ${s}_{1a}^{0}$ and ${s}_{1b}^{0}$ must lower than zero. Thus, the largest of these values is considered as the initial condition ${s}_{1}^{0}$, as denoted by (69). This value can be readily used to solve the non-linear Equation (57).

## 4. Forward-Backward Sweep Method for Power Flow Analysis

#### 4.1. Preliminaries of the FBS Method

_{b}nodes, n

_{s}serial components, and n

_{sh}shunt components. The system has a distribution substation, with its terminal defined as the source node and denoted by n

_{f}. Furthermore, the actual line-to-ground voltages are known from measurements and are denoted by ${\mathit{V}}_{abc}^{nf}$.

_{b}= 5, n

_{s}= 4, n

_{sh}= 3, and n

_{f}= n5.

_{n}of a given node n is defined as the total number of series elements composing the radial path from that node to the source node n

_{f}. Thus, the depth level of the given node n is straightforwardly evaluated by counting the serial components composing that radial path. For example, the system’s node n1 has a depth level Dp

_{n}

_{1}= 2 because its radial path is composed of two series elements, as shown in Figure 8. Note that the source node n

_{f}is the only one with a depth level Dp

_{nf}equal to 0. Once the depth levels Dp

_{n}of all the system nodes are evaluated, the sorting is performed as follows. Let

**S**or be a set having n

_{b}empty elements, thus having a cardinality of n

_{b}. Then, assign to the first empty element of this set the name of the source node having the depth level equal to 0. In the following adjacent empty elements of

**S**or, assign the names of all the nodes having a depth level equal to 1, and so on until assigning in adjacent empty elements the names of all the remaining nodes according to their increasing depth levels. It is noted that the farthest node of the system can be defined as that node whose name is in the last element of

**S**or (i.e.,

**S**or(n

_{b})). In this way, the nodes of the system have been sorted. For instance, in the sample system given in Figure 8, the nodes n1, n2, n3, n4, and n5 have depth levels of 2, 1, 2, 3, and 0, respectively. In this case, the

**S**or set is given by

**S**or = {n5, n2, n1, n3, n4}, where the name of the farthest node is n4 =

**S**or(5). It is pointed out that a computational routine can be implemented to readily assess the

**S**or set.

**S**or(n

_{FS}) for n

_{FS}= n

_{b}, (n

_{b}−1),…,1. On the contrary, a complete backward sweep step goes over the nodes of the system in the order defined by

**S**or(n

_{BS}) for n

_{BS}= 1, 2,…,n

_{b}. In addition, when performing both the forward and backward sweep steps, the series and shunt elements connected to a given node n (n∈

**S**or) of depth level Dp

_{n}and without any terminal connected to nodes of depth levels lower than Dp

_{n}are called in this work as the downward elements connected to node n. It must be noted that the series downward elements connected to node n have their second terminal connected to different nodes of depth levels higher than Dp

_{n}, such that these nodes are termed as child nodes of node n. The sum of the line currents transported by the downward elements connected to a given node n corresponds to the total line currents demanded at node n, and are denoted by ${\mathit{I}}_{abc}^{n}$. For instance, in the illustrative system shown in Figure 8, the downward elements connected to node n2 are given by the shunt element 3, the series element 23, and the series element 21. Hence the total line current demanded at node n2 is given by ${\mathit{I}}_{abc}^{n2}={\mathit{I}}_{abc}^{23}+{\mathit{I}}_{abc}^{21}+{\mathit{I}}_{abc}^{2sh3}$. In addition, the child nodes of node n2 are n1 and n3.

**S**or(n

_{FS}) and ∀n

_{FS}= n

_{b}, (n

_{b}− 1), …, 1) are initialized in a balanced way according to the different voltage rating levels of the distribution system. In addition, the total line currents ${\mathit{I}}_{abc}^{n}$ for all nodes are initialized at zero. It is assumed that all the parameters of the line segments, transformer, and loads are already known.

_{mech}produced by the turbine must be evaluated from (56), which will be useful to compute from (63) the currents ${\mathit{I}}_{abc}^{{n}_{PRO}}$ injected by the PRO power unit along the entire iterative procedure. For this purpose, however, the known values of the salinity concentration C

_{r}(resp. C

_{m}) and the flow rate Q

_{r}(resp. Q

_{m}) of the river (resp. sea) water at the inlet of the membrane module must first be used to assess the spatial variations of the osmotic $\mathsf{\Delta}{\pi}_{pol}$ and the hydraulic $\mathsf{\Delta}{P}_{h}$ pressures (as explained in Section 3.1). The resulting values are used to evaluate the hydraulic power P

_{hyd}from (51). Once P

_{hyd}is computed, the turbine mechanical power P

_{mech}is readily obtained from (56). It must be pointed out that the salinity gradient does not depend on the distribution system variables, hence the mechanical power P

_{mech}remains constant to compute the currents injected by the PRO power unit along the entire procedure of the FBS method. However, note from (60) and (63) that the currents ${\mathit{I}}_{abc}^{{n}_{PRO}}$ injected by the PRO power unit do not remain constant, since they depend on the line-to-ground voltages at the node of connection. Lastly, it is assumed that all parameters involved in (27)–(60) are known.

#### 4.2. The FBS Method

**S**or set is known. In addition, consider that the line-to-ground voltages ${\mathit{V}}_{abc}^{n}$ and the total line currents ${\mathit{I}}_{abc}^{n}$ demanded for all the system nodes (i.e., for n =

**S**or(n

_{FS}) and ∀n

_{FS}= n

_{b}, (n

_{b}− 1),…,1) have been initialized, as previously explained in Section 4.1. Based on this information, the FBS method illustrated in Figure 9 is executed, as explained below. The method starts by performing a complete forward sweep step to update the values of ${\mathit{V}}_{abc}^{n}$ and ${\mathit{I}}_{abc}^{n}$ for all system nodes. Once this forward sweep step is completed, a given convergence criterion is verified. If this criterion is satisfied, the FBS method ends. Otherwise, a complete backward sweep step is executed to update only the values ${\mathit{V}}_{abc}^{n}$ for all nodes computed in the previous forward sweep step. Finally, a new complete forward sweep step is newly executed. This iterative process is repeated until the given convergence criterion is satisfied. At this end, the steady-state operating point of the system is known in terms of the values of ${\mathit{V}}_{abc}^{n}$ and ${\mathit{I}}_{abc}^{n}$. The iterative process shown in Figure 9 is detailed below.

**S**or (n

_{b}), and goes over the system nodes in the order n =

**S**or(n

_{FS}) (for n

_{FS}= n

_{b}, (n

_{b}− 1), …, 1) until reaching the source node n =

**S**or (1). When this F-th forward sweep step is going over a given node n =

**S**or(k∈n

_{FS}), the forward sweep sub-step explained in Section 4.3 must be performed to update the values of the line-to-ground voltages ${\mathit{V}}_{abc}^{n}$ and the total line currents ${\mathit{I}}_{abc}^{n}$ demanded at that node. For this purpose, the forward sweep sub-step uses the models previously reported in Section 2 and the PRO power unit model included in Section 3. Thus, when the F-th forward sweep step is completed (i.e., the source node n =

**S**or (1) has been reached), the line-to-ground voltages ${\mathit{V}}_{abc}^{n}$ and the total line currents ${\mathit{I}}_{abc}^{n}$ demanded for all the system nodes have been updated.

**S**or (1) are compared to the magnitudes of the voltages $\left|{\mathit{V}}_{abc}^{nf}\right|$ known at that node. If the resulting maximum absolute error Emax is less than a specified tolerance (says 1 × 10

^{−3}), as denoted by (70), the FBS method ends. Otherwise, the line-to-ground voltages ${\mathit{V}}_{abc}^{n}$ estimated for the source node n =

**S**or (1) in the previous F-th forward sweep step are set as the known voltages ${\mathit{V}}_{abc}^{nf}$, as denoted by (71). The backward sweep step described below is then executed.

**S**or (1) and goes over the system nodes in the order n =

**S**or(n

_{BS}) (for n

_{BS}= 1, 2,…,n

_{b}) until reaching the farthest node n =

**S**or (n

_{b}). When this B-th forward sweep step is going over a given node n =

**S**or(k∈n

_{BS}), the backward sweep sub-step explained in Section 4.4 must be performed in order to update the values of the line-to-ground voltages of the child nodes of node n. For this purpose, the backward sweep sub-step uses the models of the series components previously reported in Section 2. The PRO power unit and loads are classified as shunt elements; thus, they are not considered to perform the backward sweep sub-step. When the B-th backward sweep step is completed, the line-to-ground voltages ${\mathit{V}}_{abc}^{n}$ for all nodes have been updated. Lastly, the total line currents ${\mathit{I}}_{abc}^{n}$ computed for all nodes in the previous F-th forward sweep step are reset to a value of zero. The FBS method then returns to perform a new forward sweep step, by considering as the new initial condition the values of the line-to-ground voltages ${\mathit{V}}_{abc}^{n}$ updated in this latter B-th backward sweep step and the recently reset currents ${\mathit{I}}_{abc}^{n}$.

#### 4.3. Forward Sweep Sub-Step

**S**or(k∈n

_{FS}). Therefore, the actual values of the line-to-ground voltages ${\mathit{V}}_{abc}^{n}$ and the total line currents ${\mathit{I}}_{abc}^{n}$ demanded at node n are provided as initial conditions to perform the forward sweep sub-step formulated below.

**S**or set (or from the system diagram) the child node m at which the second terminal is connected. The present values of the line-to-ground voltages ${\mathit{V}}_{abc}^{m}$ and the total line currents ${\mathit{I}}_{abc}^{m}$ demanded at node m are already known. Then, these values together with the

**a**,

**b**,

**c,**and

**d**matrices for the j-th line segment connecting nodes n and m are used to evaluate the model given by (1) and (2). This evaluation yields a new value for the line-to-ground voltages ${\mathit{V}}_{abc}^{n}$ at node n, as well as the line currents ${\mathit{I}}_{abc}^{nm}$ flowing from node n to m through the j-th line segment. The line currents ${\mathit{I}}_{abc}^{nm}$ are used to update the present values of the total line currents ${\mathit{I}}_{abc}^{n}$ demanded at node n according to (70).

**a**

_{t},

**b**

_{t},

**c**

_{t}, and

**d**

_{t}matrices for the j-th transformer to evaluate (4) and (5), instead of (1) and (2).

_{mech}, are already known. Based on these values, compute the positive sequence slip s

_{1}from (57) and then compute the sequence currents ${\mathit{I}}_{si}$ from (60). Lastly, evaluate the line currents ${\mathit{I}}_{abc}^{{n}_{PRO}}$ injected at node n by the j-th PRO power unit from (63), which are used to update the present values of the total line currents ${\mathit{I}}_{abc}^{n}$ demanded at node n according to (74).

#### 4.4. Backward Sweep Sub-Step

**S**or(k∈n

_{BS}). Therefore, the actual line-to-ground voltages ${\mathit{V}}_{abc}^{n}$ at node n are provided to perform the backward sweep sub-step described below. It must be pointed out that if n =

**S**or(1), the values of ${\mathit{V}}_{abc}^{n}$ are those recently set according to (71). Otherwise, the values of ${\mathit{V}}_{abc}^{n}$ are given by those recently updated in the previous backward sweep sub-step executed in the present B-th forward sweep step.

**S**or set (or from the system diagram) the child node m at which the second terminal is connected. The line currents ${\mathit{I}}_{abc}^{nm}$ flowing through the j-th line segment were computed at the previous F-th forward sweep step from (2) (see paragraph above (72)), and thus are known. The values of the line-to-ground voltages ${\mathit{V}}_{abc}^{n}$ and the line currents ${\mathit{I}}_{abc}^{nm}$ are used to evaluate the model (3) yielding the new value of the line-to-ground voltage ${\mathit{V}}_{abc}^{m}$ at the child node m.

**S**or set (or from the system diagram) the child node m at which the second terminal is connected. The values of the line currents ${\mathit{I}}_{abc}^{m}$ at the secondary side of the transformer correspond to the total currents demanded at the child node m, which are obtained at the previous F-th forward sweep step and thus are known. Values of the line-to-ground voltages ${\mathit{V}}_{abc}^{n}$ and the line currents ${\mathit{I}}_{abc}^{m}$ are used to evaluate the model (6) and to obtain the new value of the line-to-ground voltage ${\mathit{V}}_{abc}^{m}$ at the child node m.

## 5. Numerical Results

_{f}= n1, where the known line-to-ground voltages ${\mathit{V}}_{abc}^{nf}$ are given by (78). According to the depth levels of the system nodes, the

**S**or set is as given in (79). Lastly, the numerical results were obtained by using a computational program written in MatLab

^{®}language, where the convergence tolerance Tol for the FBS method was set at a value of 1 × 10

^{−3}.

#### 5.1. Steady-State Operating Condition for the Base Case

^{−3}was satisfied at the 4th iteration. The results obtained are discussed below.

_{f}= n1) to those obtained for the node n4, as shown in rows 2 to 5 of Table 1. The voltage profile of the nodes in the main feeder, however, is acceptable, since the voltage drops do not exceed 5% of the nominal value of 2.402 kV. It must be noted, however, that the line-to-ground voltages levels of nodes n6, n7, n12, and n13 in the load supply areas present drops higher than 5% of the nominal value of 0.277 kV (i.e., $0.48/\sqrt{3}$ kV). Lastly, due to the balanced load conditions and the short length of the distribution line segments, the line-to-ground voltages of all the system nodes are almost balanced, as observed in Table 1.

#### 5.2. Steady-State Operating Conditions Considering the Integration of PRO Power Plants

**S**or set are the same as those used in the base case. In addition, according to the preliminaries of the FBS method explained in Section 4.1, the mechanical power P

_{mech}of each of the 50 PRO power units composing the two PRO power plants must be also assessed, which is performed as follows. Since the PRO power units are considered to have identical parameters and operating characteristics, it is only necessary to evaluate the mechanical power P

_{mech}of a single PRO power unit from (56). To obtain P

_{mech}, however, the hydraulic power P

_{hyd}is firstly evaluated from (51), implying the solution of the polarization model (40)–(46) along with the evaluation of the spatial variation model (47)–(50) for the single PRO power unit. For this purpose, at the inlet of the membrane module of the single PRO power unit, the considered values of the salinity concentration C

_{r}and C

_{m}of the river and seawater were 0 g/L and 35 g/L, respectively, whereas the values of the flow rate Q

_{r}and Q

_{m}were set as 0.0012 m

^{3}/s and 0.0011 m

^{3}/s, respectively. Lastly, the parameters of the membrane module, the hydraulic system involved in (40)–(56), and the induction generator are given in Appendix A. According to these conditions, the mechanical power P

_{mech}evaluated from (56) for a single PRO power unit is 7.49 kW.

^{−3}was satisfied at the 4th iteration. It is important to point out that the number of iterations required to satisfy the convergence tolerance has not increased with respect to the base case. The resulting steady-state operating condition of the system under the integration of the PRO power plants is discussed below.

_{f}= n1), as observed in rows 2 to 5 of this table. It must be pointed out, however, that the voltage profile along the main feeder is still acceptable because voltage raises do not exceed 5% of the nominal value of 2.402 kV. In addition, in the load supply areas, none of the line-to-ground nodal voltages exhibits a voltage drop higher than 5% of the nominal value of 0.277 kV, as shown in rows 6 to 15 of Table 3. Lastly, these results clearly show that, for the conditions considered in this case study, the nodal voltages remain almost balanced under the integration of the PRO power plants.

#### 5.3. Comparison of the Steady-State Operating Conditions

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Distribution Line Segments

**Z**

_{abc}and the shunt admittance

**Y**

_{abc}matrices as well as the constant matrices involved in (1)–(3) of the distribution line segments can be readily performed [21].

Line Segment | Node n | Node m | Length (miles) |
---|---|---|---|

Line12 | n1 | n2 | 0.8 |

Line23 | n2 | n3 | 0.8 |

Line34 | n3 | n4 | 0.8 |

Line56 | n5 | n6 | 0.1 |

Line57 | n5 | n7 | 0.1 |

Line58 | n5 | n8 | 0.1 |

Line1012 | n10 | n12 | 0.05 |

Line1013 | n10 | n13 | 0.05 |

Line1014 | n10 | n14 | 0.05 |

#### Appendix A.2. PRO Coupling and Supply Transformers

Transformer | Node n | Node m | Rated Power (kVA) | Rated High Side Voltage (kV) | Rated Low Side Voltage (kV) | Resistance (%) | Reactance (%) |
---|---|---|---|---|---|---|---|

Trans39 | n3 | n9 | 250 | 4.16 | 0.48 | 1 | 6 |

Trans35 | n3 | n5 | 100 | 4.16 | 0.48 | 1 | 6 |

Trans411 | n4 | n11 | 250 | 4.16 | 0.48 | 1 | 6 |

Trans410 | n4 | n10 | 100 | 4.16 | 0.48 | 1 | 6 |

#### Appendix A.3. System Power Loads

Load | Node n | Rated Voltage (kV) | Phase a Power | Phase b Power | Phase c Power | |||
---|---|---|---|---|---|---|---|---|

Active (kW) | Reactive (kVAR) | Active (kW) | Reactive (kVAR) | Active (kW) | Reactive (kVAR) | |||

Load6 | n6 | 0.48 | 10 | 2 | 10 | 2 | 10 | 2 |

Load7 | n7 | 0.48 | 11 | 1 | 11 | 1 | 11 | 1 |

Load8 | n8 | 0.48 | 0 | 4 | 0 | 4 | 0 | 4 |

Load12 | n12 | 0.48 | 15 | 3 | 15 | 3 | 15 | 3 |

Load13 | n13 | 0.48 | 10 | 2 | 10 | 2 | 10 | 2 |

Load14 | n14 | 0.48 | 0 | 3 | 0 | 3 | 0 | 3 |

#### Appendix A.4. PRO Power Unit

_{mod}of identical membrane modules with equal operating conditions may feed and provide a total hydraulic power to the turbine of a PRO power unit, as denoted by (56). To obtain the results discussed in Section 5.2, n

_{mod}was set to 20. The parameters of the membrane modules composing the stack are given in Table A4. The properties of the seawater and the river water at the inlet of any of these membrane modules are given in Table A5. Furthermore, the efficiencies of the hydraulic system pumping elements, the pressure exchanger, and the turbine of the PRO unit were set at the values given in Table A6. The pipeline’s pressure losses were neglected, as indicated in this table. Lastly, the parameters of the asynchronous generator of the PRO power unit are given in Table A7.

Name | Symbol | Value |
---|---|---|

Water permeability | A | 1.87 × 10^{−9} m/(s kPa) |

Salt permeability | B | 1.11 × 10^{−7} m/s |

Structural parameter | S | 6.78 × 10^{−4} m |

Surface area | A_{mem} | 222 m^{2} |

Membrane length | L | 1.52 m |

Salt diffusion coefficient for the river water side | D_{r} | 1.4285 × 10^{−9} m^{2}/s |

Salt diffusion coefficient for the sea water side | D_{m} | 1.4350 × 10^{−9} m^{2}/s |

River water side boundary layer thicknesses | δ_{r} | 3.0710 × 10^{−5} m |

Seawater side boundary layer thicknesses | δ_{m} | 3.1014 × 10^{−5} m |

Membrane segmentation for spatial variation | n | 165 segments |

Name | Symbol | Value |
---|---|---|

Temperature | T | 297.15 K |

River water concentration | C_{r} | 0 g/L |

Seawater concentration | C_{m} | 35 g/L |

River water flow rate | Q_{r} | 0.0012 m^{3}/s |

Seawater flow rate | Q_{m} | 0.0011 m^{3}/s |

River water density | ρ_{r} | 1000 kg/m^{3} |

Seawater density | ρ_{m} | 1027 kg/m^{3} |

Name | Symbol | Value |
---|---|---|

Pumping element efficiencies | η_{rps}, η_{mps}, η_{bp} | 100% |

Exchanger efficiency | η_{px} | 100% |

Pipeline pressure losses | P_{rpu}, P_{mpu}, P_{md} | 0 kPa |

Turbine efficiency | η_{Turb} | 85% |

Name | Symbol | Value |
---|---|---|

Rated power | P_{rated} | 10 HP |

Rated voltage | V_{rated} | 480 V |

Stator resistance | R_{s} | 0.740 (Ω) |

Stator reactance | X_{s} | 1.33 (Ω) |

Rotor resistance | R_{r} | 0.647 (Ω) |

Rotor reactance | X_{r} | 2.01 (Ω) |

Magnetization reactance | X_{m} | 77.6 (Ω) |

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**Figure 7.**Sequence line-to-neutral circuit model of the induction generator [21].

**Figure 13.**Line-to-ground voltage profiles for (

**a**) the main feeder and (

**b**) the load areas and transformers.

Node | Voltages (kV) | ||
---|---|---|---|

${\mathit{V}}_{\mathit{a}\mathit{n}}$ | ${\mathit{V}}_{\mathit{b}\mathit{n}}$ | ${\mathit{V}}_{\mathit{c}\mathit{n}}$ | |

n1 | 2.403 | 2.401 | 2.402 |

n2 | 2.379 | 2.380 | 2.378 |

n3 | 2.356 | 2.359 | 2.354 |

n4 | 2.343 | 2.347 | 2.342 |

n5 | 0.266 | 0.266 | 0.266 |

n6 | 0.261 | 0.262 | 0.261 |

n7 | 0.260 | 0.262 | 0.261 |

n8 | 0.265 | 0.265 | 0.265 |

n9 | 0.272 | 0.272 | 0.272 |

n10 | 0.263 | 0.264 | 0.264 |

n11 | 0.270 | 0.271 | 0.271 |

n12 | 0.259 | 0.261 | 0.260 |

n13 | 0.261 | 0.262 | 0.261 |

n14 | 0.263 | 0.263 | 0.263 |

Component | Node n | Node m | Current at Node n (A) | Current at Node m (A) | ||||
---|---|---|---|---|---|---|---|---|

${\mathit{I}}_{\mathit{a}}$ | ${\mathit{I}}_{\mathit{b}}$ | ${\mathit{I}}_{\mathit{c}}$ | ${\mathit{I}}_{\mathit{a}}$ | ${\mathit{I}}_{\mathit{b}}$ | ${\mathit{I}}_{\mathit{c}}$ | |||

Line12 | n1 | n2 | 21.4 | 21.4 | 21.4 | 21.4 | 21.4 | 21.4 |

Line23 | n2 | n3 | 21.4 | 21.4 | 21.4 | 21.4 | 21.4 | 21.4 |

Line34 | n3 | n4 | 11.6 | 11.6 | 11.6 | 11.6 | 11.6 | 11.6 |

Line56 | n5 | n6 | 39.1 | 39.0 | 39.1 | 39.1 | 39.0 | 39.1 |

Line57 | n5 | n7 | 42.4 | 42.2 | 42.3 | 42.4 | 42.2 | 42.3 |

Line58 | n5 | n8 | 15.1 | 15.1 | 15.1 | 15.1 | 15.1 | 15.1 |

Line1012 | n10 | n12 | 59.0 | 58.7 | 58.9 | 59.0 | 58.7 | 58.9 |

Line1013 | n10 | n13 | 39.1 | 39.0 | 39.0 | 39.1 | 39.0 | 39.0 |

Line1014 | n10 | n14 | 11.4 | 11.4 | 11.4 | 11.4 | 11.4 | 11.4 |

Trans39 | n3 | n9 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Trans35 | n3 | n5 | 9.8 | 9.8 | 9.8 | 85.1 | 84.8 | 85.0 |

Trans411 | n4 | n11 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Trans410 | n4 | n10 | 11.6 | 11.6 | 11.6 | 101.0 | 100.6 | 100.9 |

Load6 | n6 | - | 39.1 | 39.0 | 39.1 | - | - | - |

Load7 | n7 | - | 42.4 | 42.2 | 42.3 | - | - | - |

Load8 | n8 | - | 15.1 | 15.1 | 15.1 | - | - | - |

Load12 | n12 | - | 59.0 | 58.7 | 58.9 | - | - | - |

Load13 | n13 | - | 39.1 | 39.0 | 39.0 | - | - | - |

Load14 | n14 | - | 11.4 | 11.4 | 11.4 | - | - | - |

Node | Voltages (kV) | ||
---|---|---|---|

${\mathit{V}}_{\mathit{a}\mathit{n}}$ | ${\mathit{V}}_{\mathit{b}\mathit{n}}$ | ${\mathit{V}}_{\mathit{c}\mathit{n}}$ | |

n1 | 2.404 | 2.402 | 2.398 |

n2 | 2.407 | 2.402 | 2.400 |

n3 | 2.412 | 2.404 | 2.403 |

n4 | 2.414 | 2.404 | 2.403 |

n5 | 0.272 | 0.272 | 0.271 |

n6 | 0.267 | 0.268 | 0.266 |

n7 | 0.267 | 0.268 | 0.266 |

n8 | 0.271 | 0.271 | 0.270 |

n9 | 0.273 | 0.273 | 0.272 |

n10 | 0.272 | 0.272 | 0.270 |

n11 | 0.273 | 0.273 | 0.272 |

n12 | 0.268 | 0.268 | 0.267 |

n13 | 0.269 | 0.269 | 0.268 |

n14 | 0.271 | 0.271 | 0.270 |

Component | Node n | Node m | Current at Node n (A) | Current at Node m (A) | ||||
---|---|---|---|---|---|---|---|---|

${\mathit{I}}_{\mathit{a}}$ | ${\mathit{I}}_{\mathit{b}}$ | ${\mathit{I}}_{\mathit{c}}$ | ${\mathit{I}}_{\mathit{a}}$ | ${\mathit{I}}_{\mathit{b}}$ | ${\mathit{I}}_{\mathit{c}}$ | |||

Line12 | n1 | n2 | 49.4 | 49.5 | 47.9 | 49.4 | 49.5 | 47.9 |

Line23 | n2 | n3 | 49.4 | 49.5 | 47.9 | 49.4 | 49.5 | 47.9 |

Line34 | n3 | n4 | 24.4 | 24.5 | 23.6 | 24.4 | 24.5 | 23.6 |

Line56 | n5 | n6 | 38.2 | 38.1 | 38.3 | 38.2 | 38.1 | 38.3 |

Line57 | n5 | n7 | 41.4 | 41.2 | 41.5 | 41.4 | 41.2 | 41.5 |

Line58 | n5 | n8 | 14.8 | 14.8 | 14.8 | 14.8 | 14.8 | 14.8 |

Line1012 | n10 | n12 | 57.1 | 57.0 | 57.4 | 57.1 | 57.0 | 57.4 |

Line1013 | n10 | n13 | 37.9 | 37.9 | 38.1 | 37.9 | 37.9 | 38.1 |

Line1014 | n10 | n14 | 11.1 | 11.1 | 11.1 | 11.1 | 11.1 | 11.1 |

Trans39 | n3 | n9 | 29.8 | 30.1 | 29.3 | 254.3 | 261.3 | 257.1 |

Trans35 | n3 | n5 | 9.6 | 9.6 | 9.6 | 83.0 | 82.8 | 83.3 |

Trans411 | n4 | n11 | 29.8 | 30.2 | 29.2 | 253.4 | 262.2 | 257.0 |

Trans410 | n4 | n10 | 11.3 | 11.3 | 11.3 | 97.9 | 97.7 | 98.3 |

Load6 | n6 | - | 38.2 | 38.1 | 38.3 | - | - | - |

Load7 | n7 | - | 41.4 | 41.2 | 41.5 | - | - | - |

Load8 | n8 | - | 14.8 | 14.8 | 14.8 | - | - | - |

Load12 | n12 | - | 57.1 | 57.0 | 57.4 | - | - | - |

Load13 | n13 | - | 37.9 | 37.9 | 38.1 | - | - | - |

Load14 | n14 | - | 11.1 | 11.1 | 11.1 | - | - | - |

PRO plant 1 | n9 | - | 254.3 | 261.3 | 257.1 | - | - | - |

PRO plant 2 | n11 | - | 253.4 | 262.2 | 257.0 | - | - | - |

**Table 5.**Effect of the integration of the PRO plants on the three-phase active and reactive power flow and system losses.

Description | Percentual Increment |
---|---|

Three-phase active power imported from the source node n1 | −250% |

Three-phase reactive power imported from the source node n1 | 400% |

Three-phase active power total power losses | 244% |

Three-phase reactive power total power losses | 339% |

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**MDPI and ACS Style**

Llamas-Rivas, M.; Pizano-Martínez, A.; Fuerte-Esquivel, C.R.; Merchan-Villalba, L.R.; Lozano-García, J.M.; Zamora-Cárdenas, E.A.; Gutiérrez-Martínez, V.J. Pressure Retarded Osmosis Power Units Modelling for Power Flow Analysis of Electric Distribution Networks. *Energies* **2021**, *14*, 6649.
https://doi.org/10.3390/en14206649

**AMA Style**

Llamas-Rivas M, Pizano-Martínez A, Fuerte-Esquivel CR, Merchan-Villalba LR, Lozano-García JM, Zamora-Cárdenas EA, Gutiérrez-Martínez VJ. Pressure Retarded Osmosis Power Units Modelling for Power Flow Analysis of Electric Distribution Networks. *Energies*. 2021; 14(20):6649.
https://doi.org/10.3390/en14206649

**Chicago/Turabian Style**

Llamas-Rivas, Mario, Alejandro Pizano-Martínez, Claudio R. Fuerte-Esquivel, Luis R. Merchan-Villalba, José M. Lozano-García, Enrique A. Zamora-Cárdenas, and Víctor J. Gutiérrez-Martínez. 2021. "Pressure Retarded Osmosis Power Units Modelling for Power Flow Analysis of Electric Distribution Networks" *Energies* 14, no. 20: 6649.
https://doi.org/10.3390/en14206649