Micro Water-Energy Nexus: Optimal Demand-Side Management and Quasi-Convex Hull Relaxation
Qifeng Li, Suhyoun Yu, Ameena S. Al-Sumaiti, Konstantin Turitsyn
MMAY 2018 1
Micro Water-Energy Nexus: Optimal Demand-SideManagement and Quasi-Convex Hull Relaxation
Qifeng Li,
Member, IEEE,
Suhyoun Yu, Ameena S. Al-Sumaiti, and Konstantin Turitsyn,
Member, IEEE
Abstract —This paper investigates the water network’s poten-tial ability to provide demand response services to the powergrid under the framework of a distribution-level water-energynexus (micro-WEN). In particular, the hidden controllability ofwater loads, such as irrigation systems, was closely studied toimprove the flexibility of electrical grids. A optimization modelis developed for the demand-side management (DSM) of micro-WEN, and the simulation results assert that grid flexibility indeedbenefits from controllable water loads. Although the proposedoptimal DSM model is an intractable mixed-integer nonlinearprogramming (MINLP) problem, quasi-convex hull techniqueswere developed to relax the MINLP into a mixed-integer convexprogramming (MICP) problem. The numerical study shows thatthe quasi-convex hull relaxation is tight and that the resultingMICP problem is computationally efficient.
Index Terms —Convex hull, convex relaxation, demand re-sponse, microgrid, water-energy nexus
I. I
NTRODUCTION M ODERN day water and power systems are closelyintertwined as a coupled system, commonly referred toas the water-energy nexus (WEN) [1]–[5]. On one hand, mostof the water facilities consume electrical energy. For instance,ground water pumping and seawater desalination accountfor roughly 12 % of the total electric energy consumptionin the Arabian Gulf regions [2]. On the other hand, waterusage is inevitable in refining fuels and generating electricpower. Despite the two networks’ inevitable interdependency,water and energy networks have traditionally been operatedindependently from one another, and the idea of co-operatingthe two in parallel has long been glossed over.In recent years, researchers have started to direct theirattention to water system’s potential ability to provide demandresponse (DR) services [6]. It is believed that higher cost-efficiency can be achieved by co-operating the water andpower systems . In 2016, PG & E built up an efficiency part-nership with the water utility in the city of San Luis Obispowhich is the first-of-its-kind . The power sector suffers from an This work is supported by the MI-MIT Cooperative Program under grantMM2017-000002.Q. Li, S. Yu, and K. Turitsyn are with the Department of MechanicalEngineering, Massachusetts Institute of Technology, Cambridge, MA, 02139USA, e-mail: { qifengli, syu2, turitsyn } @mit.edu.A. Al-Sumaiti is with the Department of Electrical Engineering & ComputerScience, Masdar Institute, Khalifa University of Science and Technology, AbuDhabi, UAE, e-mail: [email protected] California water utilities stake out new role in energy programs to financetheir future, available at http://artemiswaterstrategy.com/slopgepartnership/. First-of-its-kind efficiency partnership with PG & E expected to saveSLO millions in energy bills, available at http://kcbx.org/post/first-its-kind-efficiency-partnership-pge-expected-save-slo-millions-energy-bills stream/0. unprecedented amount of network uncertainty with the ever-increasing penetration of intermittent renewable energy andelectro-mobility [7], and the idea of exploiting the flexibilityof water network has come under the spotlight as a possiblesolution. The co-operation of two systems allows the watersystem to fast and accurately response to the energy imbalancecaused by the uncertain renewable energy generation or evencontingencies on the electricity side. With this solution, theoverall security and reliability of water and power systemswill benefit from the nexus operation mode.This paper focuses on developing a mathematical tool toassess the flexibility and responsiveness of a given micro-WEN, wherein both the water and energy networks are atdistribution levels, to the energy imbalance. The AC powerflow integrated with battery energy storage systems (BESSs)and high penetration of renewable energy sources [8] is usedin the developed mathematical model. The water pipe net-work model is characterized by the directed Darcy-Weishachequation [9] and allow for water flow directions to reverse,and the on/off status of pumps are represented by integervariables. The resulting mathematical model for the micro-WEN is unfortunately a nonlinear mixed-integer model. Tomitigate its intractability, a quasi-convex hull relaxation [8],[10] that is sufficiently tight and efficient is developed for themicro-WEN model.Much of the micro-WEN’s flexibility will rise from thewater sector, as electricity-driven water services—pumping,cooling, desalination, water treatment—are time-flexible. Fur-ther flexibility can be introduced by including controllablewater loads such as irrigation services, which this paper usesto demonstrate hidden DR capabilities of the water network.The flexibility of pumps is constrained by water tank capac-ities that are physically limiting by nature, but incorporatingcontrollable discharge scheme for water tanks should resolvethe issue. Moreover, pumps, tanks and irrigation systems areto be jointly used to create virtual energy storages to powergrids to alleviate the stress on the WEN in case of limitedenergy supply. II. P
ROBLEM F ORMULATIONS
A. Physical Model of Micro-WEN
The schematic of the micro-WEN is given in Figure 1.The electricity side is a distribution network, or a micro-grid, integrated with renewable energy and BESSs. The waterside consists of a pipe network, pumps, utility- and customer-owned tanks and irrigation systems. EV’s and water treat-ment facilities—including desalination, water and waste water a r X i v : . [ m a t h . O C ] M a y AY 2018 2
Distribution grid/Microgrid
PCC
Tank
PumpPump
Pump
Water Distribution Network
Battery Battery
RenewablesPower
Demands
Valve
Diesel Generators EVEV
Homes &BusinessWaste watertreatment plantReservoirs/lakes/wellsWatertreatment plant
Lawn wateringWaste water
Figure 1. A physical structure of the micro water-energy nexus. treatment and recycling—are not considered for the sake ofsimplicity, but those two elements can be easily incorporatedand will be considered in future research.It has been discussed in [1] that such a micro-WEN is afundamental infrastructure of a smart building/city/village. Un-der the environment of smart buildings [11]/cities [12]/villages[13], all infrastructures will be connected through the emergingInternet of Things techniques. In order to operate and controlsuch a connected physical system, it is essential to develop themathematical model and design the optimization algorithmsfor optimal resource allocation.
B. Mathematical Model of Micro-WEN
This section introduces a multi-period mathematical modelof the micro-WEN. Unless otherwise stated, the subscript t denotes the discrete time period. The structure of an AC-microgrid or an electric distribution system is usually radial.Consequently, we use a DistFlow model [14] integrated withrenewable generation and batteries to describe the electricitynetwork. The detailed model is given as P G i,t + P RE i,t − P Pump i,t − P L i,t + P ES i,t = r ij I ji,t − P ji,t + (cid:88) k ∈D i P ik,t (1a) Q G i,t + Q RE i,t − Q Pump i,t − Q Li,t + Q ES i,t = x ij I ji,t − Q ji,t + (cid:88) k ∈D i Q ik,t (1b) V i,t − V k,t + ( r ik + x ik ) I ik,t = 2( r ik P ik,t + x ik Q ik,t ) (1c) P ik,t + Q ik,t = V i,t I ik,t , ( ik ∈ E E ) (1d) P ik,t + Q ik,t ≤ S ik , ( ik ∈ E E ) (1e) ≤ I ik,t ≤ I ik , (1f) V i ≤ V i,t ≤ V i (1g) P G i , Q G i ≤ P G i,t , Q G i,t ≤ P G i , Q G i , (1h)where i ∈ N E and ik ∈ E E . For the sake of simplicity, thispaper only consider the balanced cases. It has been proved inliterature that the convex relaxations of the DistFlow for bal-anced networks can be easily extended to the unbalanced cases under some mild approximations. Therefore, the proposedapproach in this paper can be easily leveraged to the cases ofthree-phase unbalanced distribution networks or microgrids.The following nonlinear model of a battery energy storageunit is incorporated into the overall mathematical model ofthe micro-WEN. Please refer to [8] for more details about thisBESS model. For ∀ i ∈ N SE , we have ( r Batt i + r Cvt i )( P ES i,t ) + r Cvt i ( Q ES i,t ) = L ES i,t V i,t (2a) ( P ES i,t ) + ( Q ES i,t ) (cid:54) ( S ES i ) (2b) E ES i (cid:54) E ES i, − t (cid:88) t =0 ( P ES i,t + L ES i,t ) (cid:54) E ES i . (2c)We make the following assumptions for the water distri-bution networks: (1) The pipe network is a directed graph G W = ( N W , E W ) with incidence matrix A such that A ik ∈{− , , } for all i, k ; (2) A pump is considered as a typeof pipe that imposes a head gain when the pump is on andclosed otherwise; (3) The pump converts the electric powerinto a mechanical power at a constant efficiency of η ; (4) Thepower factors of pumps are fixed, namely P Pump k,t /Q Pump k,t isconstant. The resulting model can be expressed as: (cid:88) k ∈E W A ik f k,t = f G i,t − f UT i,t − f CT i,t , ( i ∈ N W ) (3a) y i,t − y j,t + h i − h j = R P k sgn( f k,t ) f k,t , ( k ∈ E W \ E PW ) (3b) y i,t − y j,t + h i − h j + y G k,t = R P k f k,t if α k,t = 1 f k,t = 0 , if α k,t = 0 , ( k ∈ E PW ) (3c) y G k,t = B k f k,t + C k ( k ∈ E PW ) (3d) S w i (cid:54) S w i, + t (cid:88) t =0 w UT i,τ (cid:54) ¯ S w i , ( i ∈ N Sw ) (3e) f ≤ f t ≤ f , (3f) y ≤ y t ≤ y, (3g) w G i ≤ w Gi,t ≤ w G i , ( i ∈ N G W ) (3h) w S i ≤ w S i,t ≤ w S i , ( i ∈ N SW ) (3i) AY 2018 3 where A is a |N W | × |E W | incidence matrix and pipe k con-nects nodes i and j . Equation (3a) represents the mass balanceof the water network; constraints (3b) and (3c) formulate thehydraulic characteristics of a normal pipe and the pipe with apump installed respectively; constraint (3e) denotes the state ofcharging of the water tanks; (3f)-(3i) are system constraints; sgn( f ) = − if f ≤ or, otherwise sgn( f ) = 1 . When α k,t = 1 , the quantity f k,t in constraint (3c) is nonnegative.In model (3), the quantity w CT i,t represents the uncontrollablewater load. Similar to the uncontrollable electric load, it is agiven value at each period.The pumps are considered as constant-speed motors in thispaper. The hydraulic characteristics of a constant-speed pumpis generally approximated by a quadratic function of the waterflow across the pump, i.e. y G = a f + a f + a [17] and[18]. Contribution of the nonlinear a f is usually very smallcompared to the linear ones a f + a . Thus, equation (3d)captures the head gain of a pump simply making a = 0 . Thefollowing constraints act as the mathematical link between thedistribution network (1)–(2) and the WDS (3): ηP Pump i,t = f k,t y G k,t = a ,k f k,t + a ,k f k,t , (4)where i ∈ N PE and k ∈ E PW . C. A Co-optimization Framework of Water and Electricity
Based on the mathematical model of the micro-WEN in-troduced above, this subsection introduces a co-optimizationframework for water and energy networks. The objective ofthis co-optimization problem is to minimize the total energycost for meeting the demands of both electricity and water.We formulate the energy cost as C ( P G i,t ) = (cid:88) t ( c t P G1 ,t + (cid:88) i ∈N GE / PCC ( c ,i P Gi,t + c ,i ( P Gi,t ) )) , (5)where P G ,t denotes the power from the grid via PCC (i.e.the serial number of PCC is 1); c t can be considered as thenodal prices at PCC which are obtained by solving the securityconstrained economic dispatch (SCED) by ISOs/RTOs. As aresult, the co-optimization model is min P Gi,t (5) s.t. (1) − (4) , (CO-OPT)which is a mixed-integer nonlinear programming (MINLP).III. Q UASI -C ONVEX H ULL R ELAXATIONS
The MINLP problem is computationally intractable, espe-cially for large-scale systems. To reduce the computationalburden, this section relaxes the MINLP into a mixed-integerconvex programming problem [19] with high-tightness.
A. Convex Hull Relaxations of Constraints (1d) and (2a)
Within the circular bounds (1e) and (2b), the feasible setsof equations (1d) and (2a) can be captured by the followinggeneral formulation: Ω = x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ax + bx = x x x + x ≤ cx , x ≤ x , x ≤ x , x xy x x Figure 2. The quasi-convex hull of the hydraulic characteristic of a normalpipe. where x = [ x x x x ] T , a ≥ b , and x x ≤ c ≤ x x .By generalizing the theorem presented in [8], we have thefollowing Lemma. Lemma . The convex hull of nonconvex set Ω is Ω = CH (Ω ) = x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ax + bx ≤ x x ( a − b ) x + x x ≤ ac ( a − b ) x + x x ≤ acD T x − d ≤ x + x ≤ cx , x ≤ x , x ≤ x , x where D = [0 0 k k ] T is a coefficient vector, d is a scalar,and their values are given by k = ac, k = x x , d = ac ( x + x ) if x x ≤ ac ≤ x x k = x x , k = ac, d = ac ( x + x ) if x x ≤ ac ≤ x x k = x , k = x , d = ac + x x if x x , x x ≤ ack = x , k = x , d = ac + x x if ac ≤ x x , x x Proof : See appendix. (cid:3)
For the case of (1d), a = b = 1 , c = S ik , and ( x , x , x , x ) = ( V i , , V i , I ik ) , assume that S ik ≤ V i I ik .Within the system bounds (1e) - (1g), the convex hull of (1d)is given as (cid:40) P ik,t + Q ik,t ≤ V i,t I ik,t S ik V i + V i V i I ik ≤ S ik ( V i + V i ) . (6)For the case of (2a), a = r Batt i + r Cvti , b = r Cvti , c =( S ESi ) , and ( x , x , x , x ) = ( V i , , V i , + ∞ ) . It is obviousthat V i L ESi,t ≤ ( S ESi ) ( r Batt i + r Cvti ) ≤ V i L ESi,t . Hence, withinthe system bounds (1g) and (2b), the convex hull of (2a) is ( r Batt i + r Cvti )( P ESi,t ) + r Cvti ( Q ESi,t ) ≤ L ESi,t V i,t r Batt i ( Q ESi,t ) + V i L ESi,t ≤ ( S ESi ) ( r Batt i + r Cvti )( S ESi ) V i + V i V i L ESi,t ≤ ( S ESi ) ( V i + V i ) . (7) B. Quasi-Convex Hull Relaxation of (3b)
With the left-hand-side replaced by an auxiliary vari-able y and the right-hand-side replaced by a general term R P sgn( f ) f , function (3b) yields the blue curve, as shown inFigure 2, in the ( f, y )-plane. It is relaxed into the red polygon AY 2018 4 as shown in the figure. The red polygon is not exactly theconvex hull of (3b). However, it is very close to the convexhull from the perspective of tightness and, therefore, is calleda quasi-convex hull. Its mathematical formulation is given as y i,t − y j,t + h i − h j (cid:54) (2 √ − R Pk f k f k,t + (3 − √ R Pk f k (cid:62) (2 √ − R Pk f k f k,t + (3 − √ R Pk f k (cid:62) R Pk f k f k,t − R Pk f k (cid:54) R Pk f k f k,t − R Pk f k . (8) C. Convex Hull Relaxation of Constraint (3c)
The nonconvex constraint (3c) contains a logic proposition.To eliminate the if expression, we use the big- M techniqueto rewrite constraint (3c) as R Pk f k,t − Y ≤ (9a) Y − R Pk f k,t ≤ (9b) ≤ f k,t ≤ M ∗ α (9c)where Y = y i,t − y j,t + h i − h j + y Gk,t + M ∗ (1 − α ) and Y = y i,t − y j,t + h i − h j + y Gk,t + M ∗ ( α − . Note that theexpression (9a) is convex, while (9b) is a concave constraint.The convex hull of (9b) can be obtained through a geometricapproach as shown in Figure 3. Its mathematical expressionis given as Y − R Pk f k,t f k,t ≤ . (10) y x y ax bx y ax b x x Figure 3. Convex hull of a parabola.
It can be observed, by comparing Figures 2 and 3, thatone can construct a tighter relaxation for the hydraulic char-acteristic of a pipe if the direction of water flow is given. Aplanning problem of gas networks is discussed in [20] wherethe authors introduced additional binary variables and bilinearequations to relax the Weymouth equation. The Weymouthequation is similar to constraint (3b). However, the convexrelaxation of the Weymouth equation developed in [20] isnot necessarily tighter than the proposed relaxation (8) dueto the introduced bilinear equations. Moreover, the auxiliarybinary variables and constraints in [20] are not desirable for anoperation problem which is sensitive to computational time.
D. Convex Hull Relaxation of Constraint (4)
Constraint (4) is a quadratic equation which can be consid-ered as the intersection of a convex inequality and a concaveinequality. The geometric approach introduced in Figure 3 canbe used to construct the convex hull of the concave inequality.As a result, the convex hull constraint (4) is given by ηP Pump i,t ≥ a k f k,t + b k f k,t (11a) ηP Pump i,t ≤ ( a k f k,t + b k ) f k,t (11b)where f k,t is nonnegative since the direction of pump flowsis determined. E. Quasi-Convex Hull Relaxation of (CO-OPT)
To sum up, the quasi-convex hull relaxation of the overallco-optimization problem (CO-OPT) is min (5) s.t. (1a-c, e-i), (2b-c), (3a, d-i), (6)(7) , (8) , (9a, c), (10) and (11) , (C-CO-OPT)which is a mixed-integer convex quadratically-constrainedquadratic programming (MICQCQP) problem.The basic idea of the quasi-convex hull relaxation of anoptimization problem is replacing the nonconvex constraintswith their convex hulls or quasi-convex hulls. As discussedin [8], [10], the concept of convex hull is attractive since theextreme points of a convex hull generally belong to its originalnon-convex set. If the objective function is a convex functionand monotonic over the convex hull, the optimal solution isusually located at one of the extreme points, implying that theoptimal solution obtained by solving the convex relaxation ismost likely the exact globally optimal solution of the originalproblem. Unfortunately, for many of the nonlinear nonconvexsets, it is extremely hard to formulate their convex hulls. Forsuch cases, an interesting alternative is to construct a convexinner approximation of a nonconvex set [21]. Compared withthe convex relaxation, the foremost advantage of the convexinner approximation is guaranteeing the feasibility of theobtained solutions to the original noncovex set.A characteristic of the MINLP problem (CO-OPT) is thatthe integer variables only exist in linear terms of constraints.For the purpose of improving the computational efficiency, itis wise to relax such a mixed-integer problem into a mixed-integer convex problem. The convex relaxations which aretight for the continuous cases are equivalently tight for thediscrete cases since the nonconvex terms that need to berelaxed do not contain integer variables.IV. A F LEXIBLE I RRIGATION S CHEME
A. Flexibility of Irrigation Systems
It is straightforward to improve the grid flexibility byallowing for controllability of electric loads. From a differentangle, this subsection explores opportunities for improving theDR capacity of water systems by investigating the flexibilityof customer-owned water tanks and irrigation systems whichare water loads rather than electric loads. An intuitive inter-pretation is that customers can use the superfluous energy to
AY 2018 5
Figure 4. Topology of the test system. pump and store water. However, there are some certain volumelimits on tanks. Thus, it is necessary to develop a coordinationstrategy for charging and discharging tanks based on the multi-period energy imbalance and the flexible water consumption.Assuming that the crops growth is not sensitive to thewatering time, the irrigation process is considered flexible. Todevelop a mathematical model of such a flexible irrigationsystem, we have the following assumptions: i) the irrigationflow is fixed and the irrigation volume is a function of thewatering time; ii) for a given season, the total amount ofwater is fixed, which can be represented as turning on theirrigation system for totally N hours per day. Consequently,the mathematical model is given by S wi (cid:54) S wi, + t (cid:88) t =0 ( f CTi,t − f Di,t − kα i,t ) (cid:54) ¯ S wi (12a) (cid:88) t α i,t = N, (12b)which are mixed-integer linear. B. A Mixed-integer Convex DSM Scheme of Water Systems
By incorporating the model (12) of flexible irrigation sys-tems into (C-CO-OPT), we have the following DSM scheme min (5) s.t. (1a-c, e-i), (2b-c), (3a, d-i), (6)(7) , (8) , (9a, c), (10), (11) and (12) , (C-DSM)which is also an MICQCQP problem.V. C ASE S TUDY
A. Introduction to the Test System
The micro-WEN for the case study is composed of the IEEE13-bus system and an 8-node WDS from the EPANET manual[15]. The topology of the test micro-WEN is given in Figure4. We assume that the 13-bus microgrid is integrated with highpenetration of PV resources. Figure 4 shows the shape of atypical average load in summer [16]. The 24-hour load profileof the 13-bus system is generated by applying this load shapewith the load provided by IEEE as the load at 9 am. Further
Figure 5. The shape of a typical average load in summer.Table IPV
SYSTEM , BESS,
AND P UMP P ARAMETERS
PV location (bus
633 (0.5 MW), 680 (0.2 MW), 684 (0.5 MW) 34.68%
BESS location (bus
684 (1.15 MVA, 2.5 MWh), 692 (1.41 MVA, 3.2 MWh)
Pump location (bus
633 ( b =0.3 p.u., c =0.4 p.u.), 652 ( b =0.3 p.u., c =0.4 p.u.) detailed information about the PV systems, BESSs, and pumpsare given in Table I.Pumps deliver 30.48 meters and 15.24 meters of headrespectively at a flow of 0.038 m /s. The tank is 18.3 metersin diameter and 5.1 meters in depth. For the 24-hour demandprofiles and the lengths of pipes of the water system, pleaserefer to the EPANET manual. The parameter R Pij is calculatedby (the subscript ij is eliminated for the sake of simplicity) R P = 8 f Lπ gD where f is the coefficient of surface resistance, D and L arethe diameter and length of pipe respectively, and g is thegravitational acceleration. B. Tightness of the QCH Relaxation
The tightness of the proposed convex relaxation is firstevaluated by comparing solutions obtained by solving (CO-OPT) and (C-CO-OPT), respectively. Using the JuMP packageof Julia [22], the optimization problems were solved in a MACcomputer with a 64-bit Intel i7 dual core CPU at 2.40 GHzand 8 GB of RAM. The MINLP problem (CO-OPT) andits quasi-convex hull relaxation (C-CO-OPT) are solved bycalling BONMIN [23] and GUROBI [24] solvers respectively.The simulation results are tabulated in Table II. The firstand foremost improvement brought by convexification is in thecomputational efficiency. The required CPU time has been sig-nificantly reduced by solving the quasi-convex hull relaxation.Note that BONMIN is an open source solver with a limitedcomputational capacity. However, MINLP problems are NP-hard to solve. Even using some well-designed commercialsolvers, like KNITRO [25], the CPU time of solving a MINLPproblem is still not comparable to that of solving a MICQCQPproblem of a similar size.The proposed quasi-convex hull relaxation is exact for thetest case in this paper. It means that the optimal solution
AY 2018 6
Figure 6. The 24-hour nodal price at PCC. obtained by solving (C-CO-OPT) is the exact global optimalsolution of its original nonconvex problem (CO-OPT) withzero optimality gap. The numerical results in this subsectiondemonstrate the potential of the proposed quasi-convex hullrelaxation for convexifying similar MINLP problems withhigh-accuracy.
C. Improvement in System Security
In current practice, the electrical and water systems areoperated separately by the electrical and water utilities re-spectively. First, the water utility tries to minimize the energyconsumption by doing a day-ahead optimal pump schedulingbased on the day-ahead water demand forecast (see the for-mulation (OPS) in [1]). Then, with the schedule of energyconsumption reported by the water utility, the power operatorsolves a multi-period optimization problem (see the formula-tion (UC) in [1]) to minimize the total energy cost for meetingall electricity demands based on the day-ahead forecast ofelectricity demands, where the diesel generators and BESSunits are control devices. It can be observed from Table IIIthat the co-optimization produces a lower-cost solution underthe same penetration level.In this section, we also evaluate the improvement in systemsecurity introduced by co-optimizing the electrical and waternetworks. There is a certain limit on the penetration of PV thatthe distribution system can accommodate. A PV generationthat exceed this limit will cause security problem to the system.The simulation results tabulated in Table III demonstrate that,by considering the pumps as controllable loads, the PV pene-tration which can be accommodated by the power distributionsystem is increased by 33 % . In reality, the proposed co-operation scheme of micro-WEN allows the electricity-drivenwater facilities response to energy uncertainties in the powergrid and, consequently, improve the system’s security. D. Efficiency of the DSM Scheme of Tanks
This subsection evaluates the efficiency for the demandresponse of the flexible irrigation system (12) by comparingthe results of (C-CO-OPT) and (C-DSM). Both problemsare MICQCQP and solved by GUROBI on the computermentioned in the previous subsection. The results are tabulatedin Table IV. Problem (C-DSM) has more integer decision vari-ables than (C-CO-OPT). However, the CPU time for solving (C-DSM) is not significantly larger than that for solving (C-CO-OPT) due to the property that the nonlinear terms areconvex.In the studied case, 30 % of the total water load is forirrigation and, namely, flexible. It can be observed from TableIV that the total operational costs of the mirco-WEN can bereduced by considering the flexibility of the irrigation systems.A considerable cost saving can be expected if: 1) the proposedapproach is applied to a larger system, 2) the flexibility ofother water facilities, such as water/waste water treatment,desalination, recycling, and cooling, is also included.VI. C ONCLUSION AND F UTURE W ORK
This paper introduces a mixed-integer nonlinear mathemat-ical model for the distribution-level water-energy nexus, orthe micro-WEN. A co-optimization framework for water andenergy distribution networks is built upon this mathematicalmodel. Based on the convex hulls or quasi-convex hulls of thesystem components, a tight mixed-integer convex relaxation isdeveloped to improve the computational efficiency of solvingthe co-optimization framework. When the proposed approachis applied to solve the co-optimization problem of a micro-WEN which consists of a 13-bus distribution system and an8-node water distribution network, the CPU time reduces fromnearly 2 hours to less than 1 second.To further explore the capacity of water distribution systemsfor providing demand response service to the power grids,this paper developed an optimal demand response frameworkconsidering a flexible irrigation system. Simulation results onthe test micro-WEN demonstrated the DR potential of watersystems. In the future work, we will consider the flexibilityof other water facilities, such as water/waste water treatment,desalination, recycling, and cooling, in the DSM model.The water-energy nexus is a fundamental infrastructure inthe building/city/village as both water and electricity are life-lines of humans. The findings of this research should be a goodfit to the research framework of the smart building/city/village.A
PPENDIX
The proof for only the case, where k = ac, k = x x ,and d = ac ( x + x ) is provided due to the page limit. Theset Ω represents a convex solid, in the x -space, that consistsof 5 (linear) facets and 4 (nonlinear) surfaces. The relation Ω = CH (Ω ) means CH (Ω ) ⊆ Ω and Ω ⊆ CH (Ω ) . (i) CH (Ω ) ⊆ Ω Let Ω = x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ax + bx ≤ x x x + x ≤ cx , x ≤ x , x ≤ x , x Ω = x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( a − b ) x + x x ≤ acx + x ≤ cx , x ≤ x , x ≤ x , x Ω = x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( a − b ) x + x x ≤ acx + x ≤ cx , x ≤ x , x ≤ x , x AY 2018 7
Table IIR
ESULTS OF T IGHTNESS
Problem MathematicalClassification Solver OptimalSolution ($) OptimalityGap CPU Time (CO-OPT) MINLP BONMIN 1463 - ≈ < APABILITY OF
PV P
ENETRATION
Operation Scheme Penetration Cap Optimal Solution ($)
Independent Optimization 83.23% 1579Co-optimization 110.98% 1463Table IVR
ESULTS OF E CONOMIC E FFICIENCY
Problem MathematicalClassification Solver OptimalSolution ($) CPU Time (C-CO-OPT) MICQCQP GUROBI 1463 < < Ω = x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D T x − d ≤ x + x ≤ cx , x ≤ x , x ≤ x , x , which are convex sets. It is straight forward to know that Ω ⊆ Ω . Under the condition x + x ≤ c , equation ax + bx = x x implies ( a − b ) x + x x = a ( x + x ) ≤ ac . Any point ( x , x ) that satisfies ( a − b ) x + x x ≤ ac will also satisfy ( a − b ) x + x x ≤ ac . Therefore, Ω ⊆ Ω . Similarly, we canprove that Ω ⊆ Ω . Equation ax + bx = x x also implies x x ≤ ac since ax + bx ≤ ac . From Figure 7, it sufficesto show that Ω ⊆ Ω . The convex hull of Ω is defined asthe intersection of all convex relaxations of Ω [26]. Hence, CH (Ω ) ⊆ (Ω ∩ Ω ∩ Ω ∩ Ω ) = Ω . x x x x x x x x ac acx x x x ac x x Figure 7. Convex hull relaxation of x x ≤ ac . The shaded area is theoriginal nonconvex set while the trapezoid region with red boundaries is itsconvex hull. (ii) Ω ⊆ CH (Ω ) If a linear cut is valid for the convex set CH (Ω ) , it willalso be valid for any subset of CH (Ω ) . Note that “a linearinequality is valid for a set” means the inequality is satisfiedby all its feasible solutions [27]. On the other hand, Ω is a subset of CH (Ω ) if any valid cut of CH (Ω ) is also valid for Ω according to the properties of supporting hyperplanes [26].Let α T x ≤ β denote any given valid linear cut for CH (Ω ) ,it should also be valid for Ω . The cut α T x ≤ β is valid for Ω if it is valid for all the surfaces of Ω .The mathematical formulation of a surface can be obtainedby changing one inequality constraint in Ω into an equality.For instance, Surf and Surf are two surfaces of solid Ω . Itis straight forward to show that α T x ≤ β is valid for Surf since, in reality, Surf = Ω . In this appendix, we prove that α T x ≤ β is valid for Surf as an example. Surf = x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ax + bx = x x ( a − b ) x + x x ≤ ac ( a − b ) x + x x ≤ acD T x − d ≤ x + x ≤ cx , x ≤ x , x ≤ x , x Surf = x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ax + bx ≤ x x ( a − b ) x + x x = ac ( a − b ) x + x x ≤ acD T x − d ≤ x + x ≤ cx , x ≤ x , x ≤ x , x Let ˜ x = (˜ x , ˜ x , ˜ x , ˜ x ) denote any given point on Surf ,where ( a − b )˜ x + x ˜ x = ac . Let’s consider the two chosenpoints ˆ x = (ˆ x , , ˜ x , ˜ x , ˆ x , ) and ˆ x = (ˆ x , , ˜ x , ˜ x , ˆ x , ) which are located in the original feasible set Ω . That means a ˆ x i, + b ˜ x = ˜ x ˆ x i, for i = 1 , . By carefully choosingthe values of ˆ x , and ˆ x , inside the required bounds, we canmake conditions ˆ x , ≤ ˜ x ≤ ˆ x , and ˆ x , ≤ ˜ x ≤ ˆ x , hold.As a result, it suffices to show that the following conditionholds: ˜ x = γ ˆ x + γ ˆ x where γ and γ are nonnegative, and γ + γ = 1 . Therealways exist two points ˆ x and ˆ x that satisfy the aboveconditions for any given point ˜ x on Surf as long as theintersection of Ω and Surf is nonempty. In reality, Surf is redundant if its intersection with Ω is an empty set. Wedon’t need to consider the case that Surf is redundant.Given that ˆ x and ˆ x belong to the original set Ω , thelinear cut α T x ≤ β is valid for both of them. Therefore, wehave α T ˜ x = γ α T ˆ x + γ α T ˆ x ≤ γ β + γ β = β, which means α T x ≤ β is also valid for ˜ x and, consequently, Surf since ˜ x represents any point on Surf . We do notprovide the proof showing that the linear cut is also valid for AY 2018 8 the rest of surfaces due to the page limit. 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