Asymptotic Assessment of Distribution Voltage Profile Using a Nonlinear ODE Model
AAsymptotic assessment of distribution voltage profileusing a nonlinear ODE model ∗ Haruki Tadano, Yoshihiko Susuki † , and Atsushi Ishigame ‡ Abstract
The promising increase of Electric Vehicles (EVs) in our society poses a challengingproblem on the impact assessment of their charging/discharging to power distributiongrids. This paper addresses the assessment problem in a framework of nonlinear dif-ferential equations. Specifically, we address the nonlinear ODE (Ordinary DifferentialEquation) model for representing the spatial profile of voltage phasor along a distribu-tion feeder, which has been recently introduced in literature. The assessment problemis then formulated as a two-point boundary value problem of the nonlinear ODE model.In this paper we then derive an asymptotic charcterisation of solutions of the problemthrough the standard regular perturbation method. This provides a mathematically-rigorand quantitative method for assessing how the charging/discharging of EVs affects thespatial profile of distribution voltage. Effectiveness of the asymptotic charcterisation isestablished with simulations of both simple and practical configurations of the powerdistribution grid.
Technological motivation.—
The high penetration of Electric Vehicles (EVs) is a promisingfuture in our society [1]. It is reported in [2] that the power demand from global EV fleetreached the total electricity consumption in Germany and the Netherlands in 2017. In this,concerns with their substantial impacts to power distribution grids have raised such as con-gestion management and voltage amplitude regulation: see, e.g., [3, 4]. In the Nordic regionwhere EVs are penetrating in advance, it is pointed out in [5] that there is a possibility ofoverloading distribution transformers in urban areas due to EV charging in the future.The concerns pose a problem on the impact assessment of EV charging to power distri-bution grids. There are multiple reasons why it is now important and challenging. A newtechnology called fast charging has been developed with the high increase of the amount ofcharged power with grid-faced inverters [2]. Also, the so-called ancillary service using a coop-erative use of a large population of EVs has been developed to provide the fast responsivenessof frequency control in power transmission grids: see, e.g., [6, 7, 8]. This kind of technologyrelated to vehicles is generally referred to as V2X [9]. An EV is regarded as an autonomously ∗ Y.S. acknowledges supports from Japan Science and Technology Agency, Core Research for EvolutionalScience and Technology (JST-CREST) Program † Corresponding author, [email protected] ‡ They are with Department of Electrical and Information Systems, Osaka Prefecture University, 1-1Gakuen-cho, Naka-ku, Sakai, 599-8631 Japan. a r X i v : . [ ee ss . S Y ] J a n oving battery in the spatial domain and can conduct the charging and discharging (inprinciple) anywhere in a distribution grid, which has a large number of spatially distributedconnecting points such as households, charging points in shopping malls, and charging sta-tions. This is completely different from other Distributed Energy Resources (DERs) such asPhoto-Voltaic (PV) generation units, which do not move spatially. The impact assessment isthus important to solve from the new power technologies and challenging as a new problemarising in a mixed domain of power and transportation systems.The impact, in particular, to the voltage amplitude of power distribution feeders, ishistorically assessed with the so-called power-flow equation that is an algebraic mathematicalmodel for the discretized evaluation of distribution voltage: see, e.g., [10]. Although themodel is nonlinear, it has been widely used for the impact assessment due to EV charging:see [11, 12, 13, 4, 14] and references therein. However, the assessment is numerics-based, stillcomputationally costly, and does not provide information of the impact with a clear referenceto its physical origin. In particular, it is hard for us to gain quantitative measures on thespatial impact on distribution voltage, such as how far EV charging at a particular locationaffects the distribution voltage, which is crucial to the current assessment problem regardingthe autonomously moving battery. A new modeling of the spatial profile of distributionvoltage—distribution voltage profile—and associated assessment methodology are thereforerequired. Purpose and contributions.—
The purpose of this paper is to solve the assessment prob-lem in a framework of nonlinear differential equations. Specifically, we address the nonlinearOrdinary Differential Equation (ODE) model for representing the distribution voltage profilederived by Chertkov et al. [15]. Unlike the power-flow equation, the nonlinear ODE model iscapable of representing the intrinsic spatial (continuous in space) characteristics of distribu-tion voltage profile. The nonlinear ODE model therefore explicitly keeps spatial informationof (balanced) distribution grids and hence enables us to quantify the spatial impact of EVson the distribution voltage profile. The nonlinear ODE is used for evaluating and mitigat-ing the impact of DERs including PV units and EVs [16, 17, 18, 19]. In [15] the authorsformulate the assessment problem as a
Two-Point Boundary Value (TPBV) problem of thenonlinear ODE, and in [16, 17] the authors provide a numerical scheme for approximatelyderiving its solution via discretisation. The boundary value problem of nonlinear ODEs hasbeen historically studied in applied and computational mathematics [20, 21, 22]. In this pa-per, as the theoretical foundation of the preceding work [16, 17, 18, 19], we characterize thesolution of the nonlinear TPBV problem using the regular perturbation technique [23] andderive a sequence of Initial Value (IV) problems of linear ODEs whose solutions asymptoti-cally approximate the original solution. This has benefits from the technological viewpoint:for instance, as shown in this paper, it enables us to quantify the impact of EV chargingin a separation manner from the those of loads and others DERs. This is never archivedwith the power-flow equations and one of the novelty of our ODE approach. Effectiveness ofthe asymptotic charcterisation is established with simulations of both simple and practicalconfigurations of the distribution grid.The contributions of the paper are three-fold. First, we newly derive an asymptotic rep-resentation of the distribution voltage profile by applying the regular perturbation techniqueto the nonlinear ODE model. It enables us to approximately evaluate the profile, which is thesolution of the nonlinear TPBV problem, by solving solutions of the associated IV problemsof linear ODEs. Needless to say, the linear ODEs are simple to solve both analytically andnumerically, and are thus direct to the assessment in complex power grids. Second, to provide2igure 1: Balanced, straight-line distribution feeder that starts a substation transformer(bank) and ends at a non-loading point.the mathematical background of the representation, we collect a series of proofs for existenceof solutions for the nonlinear TPBV problem and associated IV problems of linear ODEs.A regularity result for the solutions is also proved. Third, we demonstrate effectiveness ofthe representation with numerical simulations of simple and practical configurations of thedistribution grid. Preliminary work of this paper is presented in [24] as a non-reviewed reportin a domestic conference. This paper is a substantially enhanced version of [24] by newlyadding a series of proofs for the existence and regularity of solutions and presenting a newset of simulation results for the practical power grid model.
Organisation of this paper.—
Section 2 introduces the nonlinear ODE model for distribu-tion voltage profile and states the TPBV problem of the nonlinear ODE. In Section 3, wederive the asymptotic representation of solutions of the nonlinear problem using the standardperturbation technique. A series of theoretical results on solutions of the original nonlinearTPBV and associated linear IV problems are also derived. Section 4 presents numerical sim-ulations to validate the asymptotic expansion result in Section 3. The conclusion is made inSection 5 with a brief summery and future directions.
At the beginning, we introduce a mathematical approach to model the voltage profile ofdistribution systems in [15]. In this paper, our concerns pose a problem on the impact as-sessment of the spatially distributed EV charging at each location to distribution systems.Thus, it becomes relevant to consider the voltage profile starting at a distribution substation(bank) that is continuous in space (length). Now consider a single distribution feeder shownin Figure 1, starting at a transformer where the origin of the one-dimensional displacement(location) is introroduced. In AC electrical networks, phasor representations of voltage am-plitide and phase are used as physical quantities. The voltage phasor at the location x isrepresented with v ( x )e i θ ( x ) , where i stands for the imaginary unit, v ( x ) the voltage amplitude[V], and θ ( x ) the voltage phase [rad]. At the starting point x = 0 in Figure 1, due to voltageregulation at the substation, we naturally set v (0) to be constant. Throughout this paper, v (0) coincides with unity in per-unit system and θ (0) with zero as a reference. Then, the twofunctions θ and v are described as the following nonlinear ODE [15]: − dd x (cid:18) v d θ d x (cid:19) = b ( x ) p ( x ) − g ( x ) q ( x ) g ( x ) + b ( x ) , (1a)d v d x = v (cid:18) d θ d x (cid:19) − g ( x ) p ( x ) + b ( x ) q ( x ) v ( g ( x ) + b ( x ) ) . (1b)3he functions g ( x ) and b ( x ) in (2) are the position-dependent conductance and susceptanceper unit-length [S/km] and assumed to be continuous in x . Also, the functions p ( x ) (or q ( x ))is the active (or reactive) power flowing into the feeder (note that p ( x ) > x ). In this paper, we refer to p ( x ) and q ( x ) asthe power density functions in [W/km] and [Var/km]. Here, because Eq. (1) is complicated,the two ancillary functions s ( x ) and w ( x ) are introduced as s ( x ) := − v ( x ) dd x θ ( x ) , (2a) w ( x ) := dd x v ( x ) . (2b)The function w ( x ) is called the voltage gradient [V/km]. At the end point x = L in Figure 1(namely, no feeder and load exist for x > L ), by supposing that the end is not loaded, wehave the conditions s ( L ) = 0 and w ( L ) = 0. With the above, the nonlinear ODE model ofdistribution voltage profile is derived asd θ d x = − sv ( v (cid:54) = 0) , (3a)d v d x = w, (3b)d s d x = b ( x ) p ( x ) − g ( x ) q ( x ) g ( x ) + b ( x ) , (3c)d w d x = s v − g ( x ) p ( x ) + b ( x ) q ( x ) v ( g ( x ) + b ( x ) ) , (3d)with boundary condition as θ (0) = 0 , v (0) = 1 , s ( L ) = 0 , w ( L ) = 0 . (4)Therefore, as the assessment problem in this paper, we have the TPBV problem of thenonlinear ODE (3) with (4).In this paper, we compute numerical solutions of the TPBV problem of the nonlinear ODE(3)-(4) with the iterative method based on [17]. As far as we have used in this and [16, 17, 18,19, 24], the iterative method based on discretisation works correctly (i.e., provides physicallyrelevant outputs of voltage profile). Note that the authors of [18] report a comparison of thecomputation of distribution voltage profile with the nonlinear ODE model and the standardpower-flow equation. It is shown that the distribution voltage profiles computed with thetwo different mathematical models are consistent for a rudimental feeder configuration.For simplicity of the above introduction, we have assumed that except for the substation,no voltage regulator device such as load ratio control transformer and step voltage regulatoris operated. Note that it is possible to include the effect of such voltage regulation devices inthe ODE model: see Appendix C. This section is devoted to theoretical studies of the TPBV problem of the nonlinear ODE (3).For this, we apply the regular perturbation technique to the nonlinear ODE (3) and derive aseries of IV problems of linear ODEs. Also, we provide the theorem of existence of solutions4f the nonlinear TPBV problem, which will be the basis of our asymptotic charcterisation.In the rest of this section, we suppose that the independent variable x belongs in a closedinterval [0 , L ] where 0 < L < ∞ , and that the power density functions p ( x ) and q ( x ) takethe following forms with a small positive parameter ε : p ( x ) = ε ˜ p ( x ) , q ( x ) = ε ˜ q ( x ) . (5)The parameter ε controls the magnitude of the impacts of loads and EVs on the distributionfeeder. Also, the position-dependent function ˜ p ( x ) (or ˜ q ( x )) determines the spatial shape ofdemand of active power (or reactive power) along the feeder and can be taken from a suitablespace of functions for the current theoretical study and numerical one in Section 4. For thesimplicity of the theoretical analysis, it is supposed that the conductance and susceptance ofthe feeder are constant in x : g ( x ) = G, b ( x ) = B, (6)where G and B are constants and can be determined from practice. First of all, we apply the brute-force application of the regular perturbation technique [23]to the nonlinear ODE (3). It is supposed that the solutions θ ( x ), v ( x ), s ( x ), and w ( x ) ofthe TPBV problem of the nonlinear ODE (3) with the boundary condition (4) are expandedwith polynomials of ε , given by θ ( x ) ∼ εθ ( x ) + ε θ ( x ) + ε θ ( x ) + ε θ ( x ) + · · · v ( x ) ∼ εv ( x ) + ε v ( x ) + ε v ( x ) + ε v ( x ) + · · · s ( x ) ∼ εs ( x ) + ε s ( x ) + ε s ( x ) + ε s ( x ) + · · · w ( x ) ∼ εw ( x ) + ε w ( x ) + ε w ( x ) + ε w ( x ) + · · · , (7)where the perturbation terms θ i ( x ), v i ( x ), s i ( x ), and w i ( x ) ( i = 1 , , . . . ) are functions definedon the closed interval [0 , L ] with the following conditions at x = 0 and x = L : θ (0) = θ (0) = · · · = 0 v (0) = v (0) = · · · = 0 s ( L ) = s ( L ) = · · · = 0 w ( L ) = w ( L ) = · · · = 0 . (8)Note that the degree of regularity of solutions of the TPBV problem with respect to ε ishard to know from the generality of the choice of ˜ p ( x ) and ˜ q ( x ). The above expansion is aformalism, and its justification remains to be solved.Thus, the associated linear ODEs for each order of the perturbation terms are derived.By substituting (7) into (3) with (5) and (6), and picking up the coefficients of the smallparameter ε , the following linear ODEs for the 1st-order perturbation terms θ ( x ), v ( x ), s ( x ), and w ( x ) are derived: 5 θ d x = − s d v d x = w d s d x = B ˜ p ( x ) − G ˜ q ( x ) G + B d w d x = − G ˜ p ( x ) + B ˜ q ( x ) G + B . (9)That is, we have the IV problem of the linear ODE (9) with the initial values (8). Thisproblem is self-consistent in the sense that except for the unknown functions θ ( x ), v ( x ), s ( x ), and w ( x ), all the parameters are given. In the similar manner as above, the linearODEs for the 2nd-, 3rd-, and 4th-order perturbation terms are derived as follows:d θ d x = 2 v ( x ) s ( x )d v d x = w d s d x = 0d w d x = s ( x ) + G ˜ p ( x ) + B ˜ q ( x ) G + B v ( x ) , (10)d θ d x = 4 v ( x ) s ( x ) + { v ( x ) + v ( x ) } s ( x )d v d x = w d s d x = 0d w d x = − s ( x ) v ( x ) + G ˜ p ( x ) + B ˜ q ( x ) G + B v ( x ) , (11)and d θ d x = − v ( x ) (cid:2) v ( x ) s ( x ) + { v ( x ) + v ( x ) } s ( x ) (cid:3) −{ v ( x ) + v ( x ) } · v ( x ) s ( x ) − { v ( x ) + v ( x ) v ( x ) } · ( − s ( x ))d v d x = w d s d x = 0d w d x = − s ( x ) v ( x ) + G ˜ p ( x ) + B ˜ q ( x ) G + B v ( x ) . (12)Namely, we have the IV problems of the linear ODEs (10), (11), and (12) with the initialvalues (8). The i -th order IV problem becomes self-consistent if all the problems with orderlower than the i -th have unique solutions with appropriate regularity. This implies that itis possible to determine the perturbation terms in a recursive manner from the 1st orderproblem. Also, since all the ODEs are linear, it is possible to derive analytical forms of6he solutions that are direct to assessing the impacts of ˜ p ( x ) and ˜ q ( x ) on the variables suchas the voltage amplitude v ( x ), which is one of the benefits of our ODE approach from thetechnological viewpoint. Here we collect a series of theoretical results on existence and regularity of solutions for thenonlinear TPBV problem and on existence, uniqueness, and regularity of solutions for thederived linear IV problems. The notation C r [0 , L ] used below represents the space of r -timesdifferentiable functions defined on the finite, closed-interval [0 , L ]. The case r = 0 implies thespace of continuous functions on [0 , L ]Before applying the perturbation technique, it is the first study to prove that the originalnonlinear problem, namely, the nonlinear TPBV problem, has a solution under presence ofthe perturbation terms. Theorem 3.1.
Consider the TPBV problem of the nonlinear ODE (3) with the boundarycondition (4), and assume ˜ p ( x ) , ˜ q ( x ) ∈ C [0 , L ]. Then, there exists a constant ε > C regularity. Proof.
See Appendix A.
Remark 1.
The C regularity of solutions validates the application of finite-difference scheme[20] for locating them numerically. We use the so-called central finite-difference scheme [20]in this paper.Next, we consider the derived IV problems of the linear ODEs (9) to (12). The existenceand uniqueness of solutions of the problems are a simple outcome of applying the standardtheory of ODEs [26]. For this, let us denote the n -th order problems described by (9) to (12)as follows: for u n = ( θ n , v n , s n , w n ) (cid:62) ( n = 1 , . . . , u n d x = F n ( x, u , u , u , u ) , (13)with the boundary conditions (8). The F n describe the right-hand sides of (9) to (12). Thefollowing theorem with C regularity now holds. Theorem 3.2.
Consider the IV problem of the n -th linear ODE for n = 1 , . . . ,
4, and assume˜ p ( x ) , ˜ q ( x ) ∈ C [0 , L ]. Then, the problem has a unique solution u n ( x ) ∈ C ([0 , L ] ). Proof.
First, consider the 1st order problem. Since ˜ p ( x ) , ˜ q ( x ) ∈ C [0 , L ] is assumed, by directintegration of the right-hand sides of (9) for d s / d x and d w / d x , the solutions s ( x ) and w ( x )are unique and in C [0 , L ]. Then, from the ODEs for θ and v in (9), the solutions θ ( x )and v ( x ) are in C [0 , L ]. In the same manner, the higher order problems are also solvablerecursively and have unique solutions in C [0 , L ].In addition to this, regarding the voltage amplitude v n (and voltage gradient w n ), it ispossible to state a stronger result because the following equations of the derivatives w n and w n are derived for all n = 3 , , . . . :d v n d x = w n , d w n d x = − s ( x ) v n − ( x ) + G ˜ p ( x ) + B ˜ q ( x ) G + B v n − ( x ) . (14)Thus, we have the following theorem that is fundamental for the asymptotic charcterisationof distribution voltage profile: 7 heorem 3.3. Consider the IV problem of the n -th linear ODE (14) with v n (0) = 0 and w n ( L ) = 0 for n = 3 , , . . . , and assume ˜ p ( x ) , ˜ q ( x ) ∈ C [0 , L ]. Then, the problem has a uniquesolution ( v n ( x ) , w n ( x )) (cid:62) ∈ C ([0 , L ] ). Proof.
The proof is almost the same as Theorem 3.2. Since p ( x ) , q ( x ) ∈ C [0 , L ] and s ( x ) , v ( x ) , v ( x ) ∈ C [0 , L ] hold, by integration of the right-hand sides of (14), we im-mediately see w n ( x ) ∈ C [0 , L ] and also v n ( x ) ∈ C [0 , L ] at least. Remark 2.
For the rest of the independent variables, namely θ n ( x ) and s n ( x ), we do nothave the result parallel to Theorem 3.3 unfortunately because no equation parallel to (14) isderived for θ n ( x ) and s n ( x ). In Section 3.1, we introduced the small positive parameter ε and associated asymptoticexpansion. Here, let us introduce a physical interpretation of ε in order to make it clearto show its utility. The parameter controls the magnitude of the impacts of loads and EVson the distribution feeder. Technically and interestingly, it can be interpreted as the ratioof demand’s utilization (in kW) with respect to the capacity (also in kW) of a distributiontransformer. This ratio is related to the congestion management of distribution grids [4] inwhich overloading distribution transformers is taken into consideration as in [5]. Now, wedecompose ε into the two parameters: the contribution by EV charging, denoted by ε ev , andthe ontribution by the other loads and DERs by ε load : ε = ε ev + ε load , (15)where we will see that the two parameters are bounded above by 1. Accordingly, the powerdensity functions p ( x ) and q ( x ) are rewritten as follows: p ( x ) = ε ev ˜ p ( x ) + ε load ˜ p ( x ) , q ( x ) = ε ev ˜ q ( x ) + ε load ˜ p ( x ) . (16)Then, from (7), the voltage amplitude v ( x ) is expanded in terms of ε ev and ε load as v ( x ) = 1 + ε load v ( x ) + ε v ( x ) + ε v ( x ) + · · · + ε ev v ( x ) + ε v ( x ) + ε v ( x ) + · · · + 2 ε ev ε load v ( x ) + 3 ε ev ε load ( ε ev + ε load ) v ( x ) + · · · . (17)To explain each of the lines on the right-hand side, let us consider a loading condition ofthe distribution feeder where the loads except for EVs are originally connected and knowna prior. If no EV is connected, that is ε ev = 0, then the voltage amplitude is completelyevaluated with the first line. Thus, the impact of EV charging conditioned by the loads isquantified with the second and third lines, defined as∆ v ev | load ( x ) := ε ev v ( x ) + ε v ( x ) + ε v ( x ) + · · · + 2 ε ev ε load v ( x ) + 3 ε ev ε load ( ε ev + ε load ) v ( x ) + · · · . (18)Here, it should be noted that the perturbation terms v n ( x ) are determined solely by the scaled power density functions ˜ p ( x ) and ˜ q ( x ) not the ratio ε ev , and that ˜ p ( x ) and ˜ q ( x ) represent the8igure 2: Basic configuration of three feeders with one bifurcation point. Three branch linesA, B, and C are connected to the bifurcation point T. The arrows respect the referencedirections of current flows.spatial shape of demand of power that is determined mainly by locations of loading centersand charging points. This implies that (18) provides a simple method for quantifying howthe EV penetration affects the voltage amplitude, which will be demonstrated in Figure 5and Table 2. It becomes realized for the first time using the ODE approach and thus showsits technological benefit in comparison with the conventional method based on power-flowequations, which requires inevitable iterative computations of nonlinear programming. In assessment of realistic configurations, it is inevitable to consider the bifurcation of distri-bution feeder in our framework of asymptotic expansion. To take it into account, we considerthe three feeders with one bifurcation point shown in Figure 2. For this case, in order toformulate the nonlinear TPBV problem, we introduce as in [17, 18] an additional boundarycondition at the bifurcation point. It is shown in [17, 18] that at this point, the voltage phase θ and voltage amplitude v are continuous along any pair of the three feeders, labeled as A,B, and C, while the auxiliary variable s and voltage gradient w are not continuous. Theseare described as follows: θ A = θ B = θ C v A = v B = v C s A = s B + s C w A = w B + w C , (19)where θ A , v B , and so on represent the values of the dependent variables taken as limits tothe bifurcation point along feeders A, B, and C, respectively . It should be noted that theboundary conditions for s and w are dependent on the choice of reference directions of currentflows: (19) holds for the directions in Figure 2.Now, we show a set of boundary conditions associated with (19) in the asymptotic expan-sion. For this, using the three different independent variables x j ∈ R ( j ∈ { A , B , C } ) for thethree feeders, we suppose the following expansions of the functions θ j ( x j ), v j ( x j ), s j ( x j ), and w j ( x j ) around a neighborhood of the bifurcation point (represented in the x j -coordinates as For example, if feeder A corresponds to that in Figure 1 and is connected to be the bifurcation point at x = L , then θ A is defined as lim x → L θ ( x ). j = 0) in terms of a common small parameter ε : θ j ( x j ) ∼ ∞ (cid:88) i =1 ε i θ j,i ( x j ) , v j ( x j ) ∼ ∞ (cid:88) i =1 ε i v j,i ( x j ) s j ( x j ) ∼ ∞ (cid:88) i =1 ε i s j,i ( x j ) , w j ( x j ) ∼ ∞ (cid:88) i =1 ε i w j,i ( x j ) . (20)Thereby, the condition (19) at the bifurcation point is re-written in the asymptotic frameworkas follows: θ A1 = θ B1 = θ C1 , θ A2 = θ B2 = θ C2 , · · · v A1 = v B1 = v C1 , v A2 = v B2 = v C2 , · · · s A1 = s B1 + s C1 , s A2 = s B2 + s C2 , · · · w A1 = w B1 + w C1 , w A2 = w B2 + w C2 , · · · (21)where θ j,i is defined as lim x j → θ j,i ( x ) and so on. The derivation of (21) is presented in AppendixB. This shows that it is possible to perform a low-order approximation of the distributionvoltage profile in a self-consistent manner over the bifurcation point. This will be used inSection 43.2 for numerical simulations.Furthermore, it is practically inevitable to consider the case where a voltage regulationdevice (Step Voltage Regulator [10]) is installed in the feeder. In this case, an additionalboundary condition can be formulated in the similar manner as above and [18]. This issummarized in Appendix C for wider utility of the asymptotic assessment. This section is devoted to numerical demonstration of the asymptotic charcterisation of distri-bution voltage profile in Section 3. The demonstration is done with numerical solutions of thenonlinear TPBV problem for two distribution models. The main idea for the demonstrationis to compare the asymptotic expansions (7) up to the 1st, 2nd, 3rd, and 4th perturbationterms with direct numerical solutions for the simple feeder model (see Figure 3) and thepractical model (see Figure 6). The detailed setting of the numerical demonstrations in thissection is presented in Appendix D.Here, in practical situations, the power demand and generation along a feeder happen ina discrete manner. To precisely state this, we suppose that N number of load and stationsare located at x = ξ i ∈ (0 , L ) ( i = 1 , . . . , N ) satisfying ξ i +1 < ξ i . Then, by denoting as P i the active power consumed at x = ξ i , the power density functions p ( x ) is given as follows: p ( x ) = N (cid:88) i =1 P i δ ( x − ξ i ) , (22)where δ ( x − ξ i ) is the Dirac’s delta-function supported at x = ξ i . The formulation of p ( x ) isexcluded in the theoretical development of Section 33.2 because of the C assumption and,furthermore, makes it difficult to numerically approximate solutions of the nonlinear ODE.To avoid these, as in [18, 19], we use the following coarse-graining of p ( x ) with the Gaussianfunction: p ( x ) ∼ N (cid:88) i =1 P i √ πσ exp (cid:18) − ( x − ξ i ) σ (cid:19) , (23)10igure 3: Simple distribution feeder model. The feeder has the 3 loads and 2 EV stationslocated at a common interval (0.5 km).Table 1: Validation of asymptotic expansion (7) by differences from nonlinear ODE for simpledistribution feeder model in Figure 3 ∆ w (0 km) ∆ v ( L = 5 km) ∆ θ ( L = 5 km)asymptotic expansion up to 1st-order 0.02700 0.0542 0.0564asymptotic expansion up to 2nd-order 0.00951 0.0191 0.0250asymptotic expansion up to 3rd-order 0.00580 0.0102 0.0145asymptotic expansion up to 4th-order 0.00478 0.0078 0.0099where we regard x, ξ i as scalars, and σ is the variance. The coarse-grained p ( x ) is clearly C , can be treated as in Section 33.2, and hence lead to the existence of solutions for thenonlinear TPBV problem. The parameter σ ( >
0) is fixed at a constant distance, which issufficiently smaller than the interval between loads or EV charging stations.
First, we evaluate the asymptotic expansion (7) for the simple feeder model in Figure 3.Specifically, the three functions θ ( x ), v ( x ), and w ( x ) are addressed; s ( x ) is not addressedbecause it is equivalent to the 1st-order εs ( x ), and the higher-order terms s i ( x ) are identicallyzero. The feeder model in Figure 3 possesses the 3 loads and 2 EV charging stations locatedat a common interval (0.5 km). We assume that the rated capacity of the transformer is set at12 MVA, and no loads exists at the end point of the feeder for simplicity of the analysis. It isalso assumed that all the loads are connected via inverters like EVs and thus operated underunity power-factor mode. This implies that the reactive power q ( x ) is here negligible, namely q ( x ) = 0 for all x ∈ [0km , p ( x ) for simulations of thenonlinear and linear ODEs is shown in the top of Figure 4.Figure 4 and Table 1 show the proposed asymptotic expansions and direct numericalsolutions of the nonlinear ODE with the power density function in the top of Figure 4. Thechoice of the value of ε is an issue for numerical simulations of the asymptotic expansionbecause it is not guided by perturbation theory. Here, for a fixed p ( x ) we set ε at multiplesmall values for simulations , and the following description is consistent for ε = 1 × − , × − , . . . , × − . The second row of Figure 4 shows the computational results on voltage This implies in (23) with P i = ε ˜ P i that we tune both values of ε and ˜ P i while keeping P i . v ( L = 5 km) of results between (7) and the nonlinear ODE is computed at x = L = 5 km.The asymptotic expansion up to 4th is used here. ε ev /ε Error ∆ v ( L = 5km)30% 0.00428540% 0.00570450% 0.0179560% 0.03283gradient w ( x ): the proposed asymptotic expansions up to 1st- to 4th-order terms by pink,green, red, and orange solid lines, and the direct numerical solution of the nonlinear ODE by blue dashed line. The difference between each of the asymptotic expansions and the nonlinearODE increases from the end to the start of the feeder due to the effect of power consumptionby loads. Similarly, the associated voltage amplitude v ( x ) and voltage phase θ ( x ) are shownin the third row and bottom of Figure 4. For both, the difference between each of theasymptotic expansions and the nonlinear ODE increases from the start to the end of thefeeder. It is clearly shown in Figure 4 that the asymptotic expansions for w ( x ), v ( x ), and θ ( x ) approach to the direct numerical solution of the nonlinear ODE as the order increases.This is confirmed from the quantification of differences between the asymptotic expansionand the nonlinear ODE for the computed values, denoted as ∆ w (0km), ∆ v ( L = 5km), and∆ θ ( L = 5km), in Table 1.Here, the utility of the impact assessment (7) is demonstrated. Figure 5 and Table 2show numerical results on the impact assessment for the single feeder model. Specifically, inFigure 5 we consider the original (a prior) case where the loads (Load 1, 2, and 3) are originallyconnected; then we introduce the EV charging (EV stations 1 and 2). For this, by the orange line we plot (7) under the asymptotic expansion (up to 4th-order term) and ε ev = 0 . ε (hence ε load = 0 . ε ), and by the blue line we also plot the numerical simulation of the nonlinear ODE.It is clearly shown that these numerical results are similar. Table 2 shows the ε ev -dependenceof accuracy of the impact assessment using (7). The errors of results between (7) and thenonlinear ODE are small for different choices of ε ev . The accuracy slightly decreases as ε ev increases. The EV station 2 is placed close to the end of feeder and thus dominantly affectsthe voltage profile (if it extracts current flow from the bank through the feeder). In this,the increase of ε ev implies that the accuracy of asymptotic expansion up to 4th tends todeteriorate. This can be improved by adding higher-order terms to the assessment. Thus,the effectiveness of the asymptotic expansion for the impact assessment is confirmed.Consequently, the proposed asymptotic expansion is capable of evaluating the distributionvoltage profile for the simple distribution feeder model in Figure 3. Second, we evaluate the the asymptotic expansions (7) for the practical configuration withmultiple feeders and bifurcations shown in Figure 6. As in the previous sub-section, forcomparison of phase θ , voltage amplitude v , and voltage gradient w , we consider the theproposed asymptotic expansions (7) including the perturbation terms up to 1st to 4th orderand the numerical solution of the nonlinear ODE (3). The model in Figure 6 is based on a12able 3: L - and L ∞ -like norms for voltage amplitude differences for practical feeder modelin Figure 6 L -like norm L ∞ -like normasymptotic expansion up to 1st-order 1.9521 0.0154asymptotic expansion up to 2nd-order 0.4636 0.0037asymptotic expansion up to 3rd-order 0.2471 0.0019asymptotic expansion up to 4th-order 0.2113 0.0017practical distribution feeder of residential area in western Japan and provided by an utilitycompany. A similar distribution model is used in [19] and thus is summarized in AppendixD. The corresponding power density function is shown in the top of Figure 7. The functionis constructed in the same way as in the single-feeder model. In the figure, the positiveness implies the discharging operation by in-vehicle batteries, and the negativeness does their charging operation or the power consumption by loads. The blue part on the feeders rep-resents the locations connected to the residential loads through the pole transformers. Inaddition to the loads, we assume that each station has 9 EVs for simultaneous charging,where each EV has the rated charging power of 4 kVA based on [11].Figure 7 shows the difference the asymptotic expansion (7) up to 4th and the numericalsolution of the nonlinear ODE (3) incorporated with the power density function based on thetop of Figure 7. The simulations for the multiple feeders with bifurcations were performedwith the boundary condition (19). The voltage amplitude v ( x ) and voltage phase θ ( x ) atthe start of the feeders is set to unity. Similarly, the voltage gradient w ( x ) at each end ofthe feeders is set to zero. The difference of voltage gradient increases toward the start ofthe feeders due to the boundary condition at the end of the feeders. On the other hand, thedifferences of voltage amplitude and voltage phase increase toward each end of the feeders.Note that the observation is consistent for the choice of multiple values for ε : 1 × − , × − , . . . , × − .The practical feeder model is complicated, and hence the quantitative evaluation likeTable 1 is not straightforward. Here, we use the two norms of differences in voltage amplitudebetween the asymptotic expansions and the direct numerical solution, which are similar tothe standard L and L ∞ norms of functions. Although mathematically not rigor, the twonorms for the difference e ( x ) are described as | e ( x ) | := (cid:115)(cid:90) all feeders { e ( x ) } d x, | e ( x ) | ∞ := max x ∈ all feeders | e ( x ) | . (24)The L -like norm | e ( x ) | implies the RMS quantification of the difference, and the L ∞ -likenorm | e ( x ) | ∞ does the worst-case quantification. The computational results on the two normsare shown in Table 3. It is clearly shown in the table that the L - and L ∞ -like norms becomesmall as the order increases. Consequently, the proposed asymptotic expansions are capableof evaluating the practical feeder model in Figure 6.13 Concluding Remarks
Motivated by recent electrification of vehicle and its impact to power distribution grids, in thispaper we revisited the TPBV formulation of nonlinear ODE for the assessment problem ofdistribution voltage profiles. Its asymptotic assessment was newly proposed by applying theregular perturbation technique in ODEs, and its effectiveness was established with numericalsimulations of the simple and practical configurations of the power distribution grid. The keyderivation is the asymptotic expansion (7) of solutions of the nonlinear TPBV problem andprovides an analytical insight that can explain the impact of EV charging on the distributionvoltage profile.Several remarks on the work in this paper are presented. First, it is desirable to prove anyconvergence theorem for the asymptotic expansion. Our numerics suggest that the accuracyof asymptotic expansion could be improved by including higher-order perturbation terms.Second, it is of scientific interest and technological significance to consider the impact of thetime-dependent variation of EV charging on the distribution voltage profile. Third, it is alsointeresting to connect the asymptotic assessment with the regulation of distribution voltageprofile against such variation.
Acknowledgement
The authors would like to thank Mr. Shota Yumiki (Osaka Prefecture University) for valuablediscussions on the work presented in this paper.
Appendix A Proof of Theorem 3.1
In this proof, we consider the original nonlinear ODE (3) with (6), where p ( x ) , q ( x ) ∈ C [0 , L ],and θ, s, w ∈ R and v ∈ R > (set of all positive real numbers). The current proof is devotedto the existence of solutions of the nonlinear TPBV problem described by (3) and (4).First of all, we consider the differential equation (3c) for determining s ( x ), which is anIV problem. Because of p ( x ) , q ( x ) ∈ C [0 , L ], the right-hand side of (3c) can be explicitlyintegrated in x , and s ( x ) is expressed in a self-consistent manner as follows: s ( x ) = s ( L ) + (cid:90) xL Bp ( ξ ) − Gq ( ξ ) Y d ξ ∀ x ∈ [0 , L ] . Thus, s ( x ) is unique and C .Next, in order to determine v ( x ) and w ( x ), it is necessary consider the nonlinear TPBVproblem as d v d x = w d w d x = s ( x ) v − ε G ˜ p ( x ) + B ˜ q ( x ) v ( G + B ) , ∀ x ∈ [0 , L ] , (25)with v (0) = 1 , w ( L ) = 0 . (26)Since ˜ p ( x ), ˜ q ( x ) ∈ C [0 , L ] and s ( x ) ∈ C [0 , L ], the right-hand sides of (25) have continuousfirst derivatives with respect to v and w for ( x, v, w ) ∈ D (an open set in [0 , L ] × ( R > × R )).In the TPBV problem, we fix ε as non-negative.14o consider the existence of solutions of the nonlinear TPBV problem (25), let us definethe initial-value problem for (25) asd v d x = w d w d x = s ( x ) v − λ G ˜ p ( x ) + B ˜ q ( x ) v ( G + B ) , (27)with v (0) = 1 , w (0) = η, (28)where η is picked up from D . The parameter λ is picked up from an open interval in R including 0. According to the standard theorems of existence and uniqueness of solutionsfor IV problems [26], there exists an unique solution ( v ( x, , (1 , η ) , λ ) , w ( x, , (1 , η ) , λ )) of(27) passing through (0 , (1 , η )). This solution can be extend to x = L . Also, from thestandard theorem on the dependence of solutions on parameters and initial data [26], thesolution w ( x, , (1 , η ) , λ )) is continuously differentiable with respect to η and λ in its domainof definition. At λ = 0 (as ε → +0), because of s ( x ) = 0 , ∀ x ∈ [0 , L ] (see the ODE (3c) for s ), the solution w ( x, , (1 , η ) ,
0) is exactly η .Now, we are in a position to prove the existence of solutions of the nonlinear TPBVproblem (25). For this, we define φ ( η, λ ) := w ( L, , (1 , η ) , λ ) . (29)The proof is that we find a solution of φ ( η, λ ) = 0 , η ∈ D , λ > . (30)For this, we use the implicit function theorem [26]. First, we see φ ( η,
0) = η = 0. From above,the derivatives ∂φ/∂η and ∂φ/∂λ are continuous in an open set including ( η, λ ) = (0 , ∂φ/∂η estimated at ( η, λ ) = (0 ,
0) is exactly one (not zero). Therefore,from the implicit function theorem, there exists a map η ∗ from an open interval including 0 to R such that φ ( η ∗ ( λ ) , λ ) = 0, η ∗ (0) = 0, and (d η ∗ / d λ ) λ =0 (cid:54) = 0. Because of the open interval,this implies that there exists a solution of (30) for a positive λ , i.e. ε . This proves that thereexists ε > v ( x ) , w ( x ), is C from the above argument of IV problems on ODEs.Finally, since s ( x ) , v ( x ) are C and v ( x ) >
0, from the IV problem described by (3a) and θ (0) = 0, θ ( x ) exists for x ∈ [0 , L ] uniquely and is C . By collecting all the statements above,it follows that there exists ε > C regularity. Appendix B Derivation of the Boundary Conditions (21)
This appendix is devoted to the derivation of the boundary conditions (21) for the first order.The derivation for higher-order cases is the same as below. We refer to the bifurcation pointin Figure 8 as T. We also represent the value of θ j, at the point separated from T by δ ∈ R > along the feeder j ∈ { A , B , C } , as θ j, ( δ ) where δ = 0 implies x j = 0. The same notation is15sed for the other dependent variables. From the continuity of θ j, and v j, at T, the values θ T and v T at T are defined as follows: θ T , := lim δ → +0 θ A , ( δ ) = lim δ → +0 θ B , ( δ ) = lim δ → +0 θ C , ( δ ) , (31) v T , := lim δ → +0 v A , ( δ ) = lim δ → +0 v B , ( δ ) = lim δ → +0 v C , ( δ ) , (32)where we assume v T , > s j, and w j, at T are defined as follows: s A , (0) := − lim δ → +0 θ T , − θ A , ( δ ) δs B , (0) := − lim δ → +0 θ B , ( δ ) − θ T , δs C , (0) := − lim δ → +0 θ C , ( δ ) − θ T , δ , (33)and w A , (0) := lim δ → +0 v T , − v A , ( δ ) δw B , (0) := lim δ → +0 v B , ( δ ) − v T , δw C , (0) := lim δ → +0 v C , ( δ ) − v T , δ . (34)The voltage phasors at δ are also introduced with the independent variables θ and v : ˙ V j, ( δ ) = v j, ( δ )e i θ j, ( δ ) for j ∈ { A , B , C } . Then, the following equations are obtained from the first lawof Kirchhoff. ˙ V A , − ˙ V T , δ ˙ Z − ˙ V T , − ˙ V B , δ ˙ Z − ˙ V T , − ˙ V C , δ ˙ Z = 0 ,v A , ( δ )e i { θ A , ( δ ) − θ T , } − v T , δ + v B , ( δ )e i { θ B , ( δ ) − θ T , } − v T , δ + v C , ( δ )e i { θ C , ( δ ) − θ T , } − v T , δ = 0 , (35)where ˙ Z is the impedance per unit length. According tothe small amount of δ , the trigono-metric functions for θ A , ( δ ) and θ T , , it can be written as follows:cos( θ A , − θ T , ) (cid:39) , (36)sin( θ A , − θ T , ) (cid:39) θ A , ( δ ) − θ T , . (37)From (36) and (37), we expand the first term of the left-hand side of (35) as follows: v A , ( δ )e i { θ A , ( δ ) − θ T , } − v T , δ = v A , ( δ ) δ (cid:26) cos( θ A , ( δ ) − θ T , ) + i sin( θ A , ( δ ) − θ T , ) (cid:27) − v T , δ (cid:39) v A , ( δ ) − v T , δ + i v A , ( δ ) θ A , ( δ ) − θ T , δ . (38)By taking the limitation δ →
0, the above equation can be rewritten from (33) and (34) tothe following: lim δ → +0 v A , ( δ )e i { θ A , ( δ ) − θ T , } − v T , δ = − w A , (0) + i v T , s A , (0) . (39)16he same derivation holds for the feeders B and C, and from (35) we have − w A , (0) + i v T , s A , (0) + w B , (0) − i v T , s B , (0) + w C , (0) − i v T , s C , (0) = 0 , − w A , (0) + w B , (0) + w C , (0) + i v T , { s A , (0) − s B , (0) − s C , (0) } = 0 . (40)This clearly show (21), namely, w A , − w B , − w C , = 0 , s A , − s B , − s C , = 0 . (41) Appendix C Representation of Voltage Regulation Devices
In this section, we consider the new boundary conditions for the nonlinear TPBV problemin the case where a voltage regulation device (Step Voltage Regulator [10]; SVR) is installedin the feeder. In this case, the voltage amplitude changes discontinuously at the installedlocation of SVR. This poses a new boundary condition at the location. In Figure 1, weconsider that a single-phase tap-changing transformer with variable turn ration n ( >
0) isinstalled at the location x = ξ ∈ (0 , L ). Here, for the simple derivation, we assume no powersupply and demand at the location x = ξ , that is, p ( ξ ) = q ( ξ ) = 0. Thus, the followingboundary conditions are derived in [18]: θ ( ξ − ) = θ ( ξ +) v ( ξ − ) = 1 n v ( ξ +) s ( ξ − ) = s ( ξ +) w ( ξ − ) = n · w ( ξ +) (42)where the limit values for the location of the transformer such as θ ( ξ − ) and v ( ξ +) arerepresented by θ ( ξ − ) = lim δ →− θ ( ξ + δ ) or v ( ξ +) = lim δ → +0 v ( ξ + δ ). By a similar manner asabove, the condition (42) at the location of SVR is re-written in the asymptotic frameworkas follows: θ ( ξ − ) = θ ( ξ +) , θ ( ξ − ) = θ ( ξ +) , · · · v ( ξ − ) = 1 n v ( ξ +) , v ( ξ − ) = 1 n v ( ξ +) , · · · s ( ξ − ) = s ( ξ +) , s ( ξ − ) = s ( ξ +) , · · · w ( ξ − ) = n · w ( ξ +) , w ( ξ − ) = n · w ( ξ +) , · · · . (43)The above derivation for the perturbation terms is almost the same as in (21) and is henceomitted in this paper. Appendix D Detailed Settings of Numerical Demonstrations
The appendix is described the detailed settings of the numerical demonstrations in Section 4based on Mizuta et al. [19].In both the feeder models, we assume that the secondary voltage of the transformer isset as 6.6 kV, which is the normal condition in Japan’s high-voltage distribution networks.The condactance and susceptance of each feeder are common and constant in x as in (6).The feeder’s resistance (or reactance) are set at 0.227 Ω/km (or 0.401 Ω/km). These values17re from [19] and based on the standard setting in Japan. In the following, we use per-unitsystem [10] for numerical simulations of the nonlinear ODE model. The conductance G (orsusceptance B ) of the simple model per unit-length is calculated as 3.881 (or 6.856) in per-unit system ( G/B is about 5.661 × − ; it is typical in practice). Similarly, G (or B ) of thepractical model is also calculated as 2.329 (or 4.113).For the practical feeder model in Figure 6, it has multiple bifurcation points and 9 chargingstations denoted by circled numbers . The sum of the lengths of all the feeders is 2.52 km. Thesecondary voltage at the bank is regulated at 6.6 kV, and the loading capacity of the bank isset at 20 MVA. The model has 103 pole transformers distributed along the feeders. All thepole transformers are connected to a total of 1001 residential households. The residential loadsused here are based on practical measurement at 19 o’clock in summer in Japan. The detaileddata on the feeders and residential loads could not be published following an agreement withthe utility company. References [1] Committee on Climate Change. 2019 Net Zero: The UK’s contribution to stopping globalwarning. Tech Rep.[2] IEA. 2019. Global EV Outlook 2019, IEA, Paris: 140–141.[3] Ipakchi A, Albuyeh F. 2009. Grid of the future.
IEEE Power and Energy Mag. : 52–62.[4] Arias NB, Hashemi S, Andersen PB, Træholt C, Romero R. 2019. Distribution systemservices provided by electric vehicles: Recent status, challenges, and future prospects. IEEE Trans. on Intell. Transp. Syst. : 4277–4296.[5] IEA. 2018. Nordic EV Outlook 2018: Insights from leaders in electric mobility, IEA,Paris: 61–62.[6] Kempton W, Tomic J. 2005. Vehicle-to-grid power fundamentals: Calculating capacityand net revenue. J. Power Sources : 268–279.[7] Tomic J, Kempton W. 2007. Using fleets of electric-drive vehicles for grid support.
J.Power Sources : 459–468.[8] Ota Y, Taniguchi H, Nakajima T, Liyanage KM, Baba J, Yokoyama A. 2012. Au-tonomous distributed V2G (Vehicle-to-Grid) satisfying scheduled charging.
IEEE Trans.on Smart Grid : 559–564.[9] Toh CK, Sanguesa JA, Cano JC, Martinez FJ. 2020. Advance in smart roads for futuresmart cities. Proc. R. Soc. A : 20190439.[10] Kersting WH. 2012
Distribution System Modeling and Analysis , Third Edition. Florida,FL: CRC Press.[11] Clement-Nyns K, Haesen E, Drisen J. 2010. The impact of charging plug-in hybridelectric vehicles on residential distribution grid.
IEEE Trans. on Power Syst. : 371–380. 1812] Clement-Nyns K, Haesen E, Drisen J. 2011. The impact of vehicle-to-grid on the distri-bution grid. ELectr. Power Syst. Res. : 185–192.[13] Yilmaz M, Krein PT. 2013. Review of the impact of vehicle-to-grid technologies ondistribution system and utility functions. IEEE Trans. on Power Electr. : 5673–5689.[14] Dixon J, Bell K. 2020. Electric vehicles: Battery capacity, charger power, access tocharging and the impacts on distribution networks. eTransportation , article no. 100059.[15] Chertkov M, Backhaus S, Turtisyn K, Chernyak V, Lebedev V. 2011. Voltage collapseand ODE approach to power flows: Analysis of a feeder line with static disorder inconsumption/production. arXiv:1106.5003.[16] Baek S, Susuki Y, Ota Y, Hikihara T. 2016. Analysis of distribution voltage profileby ODE model incorporated with power demand/supply data. in Proc. 60th AnnualConference of the Institute of Systems, Control, and Information Engineers : 4 pages.[In Japanese][17] Susuki Y, Baek S, Ota Y, Hikihara T. 2016. Computer simulation of distribution voltageprofile using a nonlinear ODE.
IEICE Tech. Rep. : 15–20. [In Japanese][18] Susuki Y, Mizuta N, Kawashima A, Ota Y, Ishigame A, Inagaki S, Suzuki T. 2017. Acontinuum approach to assessing the impact of spatio-temporal EV charging to distri-bution grids. in
Proc. IEEE 20th International Conference on Intelligent TransportationSystems , 2372–2377.[19] Mizuta N, Susuki Y, Ota Y, Ishigame A. 2018. Synthesis of spatial charging/dischargingpatterns of in-vehicle batteries for provision of ancillary service and mitigation of voltageimpact.
IEEE Syst. J. , 3443–3453.[20] Keller HB. 1968 Numerical Methods for Two-Point Boundary Value Problems . New York,NY: Dover Books on Mathematics.[21] Domokos G, Holmes P. 2003. On nonlinear boundary-value problems: ghosts, parasitesand discretizations.
Proc. R. Soc. A : 1535–1561.[22] Chowdhury A, Tanveer A, Wang X. 2020. Nonlinear two-point boundary value problems:applications to a cholera epidemic model.
Proc. R. Soc. A : 20190673.[23] Guckenheimer J, Holmes P. 1983
Nonlinear Oscillations, Dynamical Systems, and Bi-furcations of Vector Fields . New York, NY: Springer.[24] Tadano H, Susuki Y, Ishigame A. 2019. An asymptotic expansion of distribution voltageprofile using nonlinear ODE model: An application of perturbation method. in
Proc.Power and Energy Society Meeting, IEE of Japan , 111–112. [In Japanese][25] Yamamoto T, Oishi S. 2006. A mathematical theory for numerics treatment of nonlineartwo-point boundary value problems.
Japan J. Indust. Appl. Math. , 31–62.[26] Hale JK. 2009 Ordinary Differential Equations . New York, NY: Dover Books on Math-ematics. 19 P o w e r d e n s it y [ pu ] -0.08-0.06-0.04-0.020 V o lt a g e g r a d i e n t [ pu ] Asymptotic (up to 1st)Asymptotic (up to 2nd)Asymptotic (up to 3rd)Asymptotic (up to 4th)Nonlinear ODE V o lt a g e a m p lit ud e [ pu ] Location [km] -0.3-0.25-0.2-0.15-0.1-0.05 V o lt a g e ph a s e [r a d ] Figure 4: Validation of proposed asymptotic expansions (7) for simple distribution feedermodel in Figure 3. The power density function p ( x ) and associated numerical solutions ofthe nonlinear ODE model (3) are also shown.20 Location [km] -0.12-0.1-0.08-0.06-0.04-0.020 V o lt a g e a m p lit ud e [ pu ] Asymptotic (up to 4th)Nonlinear ODE
Figure 5: Impact assessment of the EV charging using (7). The orange line shows theassessment result of (7) under the asymptotic expansion up to 4th-order term and ε ev = 0 . ε .The blue line shows the associated numerical result of the nonlinear ODE.Figure 6: Model of multiple feeders based on a practical distribution grid in residential areain Japan. The 9 EV charging stations virtually installed and denoted by circled numbers .21 L o ca ti on [ k m ] -0.1-0.0500.050.1 Power density [pu] -0.400.4 L o ca ti on [ k m ] Voltage gradient [pu] -4 -0.400.4 L o ca ti on [ k m ] Voltage amplitude [pu] -3 Location [km] -0.400.4 L o ca ti on [ k m ] Voltage phase [rad] -4 Figure 7: Visualization of the differences between asymptotic expansion (7) up to 4th and thenonlinear ODE (3). The power density function p ( x ) used for simulations is also visualizedon the top. 22igure 8: Configuration of three feeders A, B, and C connected at one bifurcation point T.The arrows represent the reference directions of current flows. We consider the neighborhoodat the short distance δδ