Optimal Power Flow model with energy storage, an extension towards large integration of renewable energy sources.

Transcription

1 Preprints of te 9t World Congress Te International Federation of Automatic Control Optimal Power Flow model wit energy storage, an extension towards large integration of renewable energy sources. Alessio Maffei Daniela Meola Giancarlo Marafioti Giovanni Palmieri Luigi Iannelli Geir Matisen Eilert Bjerkan Luigi Glielmo Dipartimento di Ingegneria, Università del Sannio, Piazza Roma, 8 Benevento, Italy, {amaffei, palmieri, daniela.meola, luigi.iannelli, luigi.glielmo}@unisannio.it SINTEF ICT - Applied Cybernetics, O.S. Bragstads plass D, 734 Trondeim, Norway, {giancarlo.marafioti, geir.matisen}@sintef.no NTE Nord-Trøndelag Elektrisitetsverk AS, Norway, ebj@enfo.no Abstract: Te integration of renewable energy sources (RES) into modern electrical grids contributes to satisfying te continuously increasing energy demand. Tis can be done in a sustainable way since renewable sources are bot inexaustible and non-polluting. Different renewable energy devices, suc as wind power, ydro power, and potovoltaic generators are available nowadays. Te main issue wit te integration of suc devices is teir irregular generation capacity (in particular for wind and solar energy). Terefore energy storage units are used to mitigate te fluctuations during generation and supply. In tis paper we formulate a model for te Alternate Current Optimal Power Flow (ACOPF) problem consisting of simple dynamics for energy storage systems cast as a finite-orizon optimal control problem. Te effect of energy storage is examined by solving a Norwegian demo network. Te simulation results illustrate tat te addition of energy storage, along wit demand based cost functions, significantly reduces te generation costs and flattens te generation profiles. Keywords: AC optimal power flow, power system economics, power transmission, power distribution control, renewable energy sources. INTRODUCTION Te electric power industry as lived a significant expansion and growt over te course of te past two decades. Te penetration of renewable sources, suc as wind, ydro and solar, is increased by te requirements of te governments in order to acieve goals related to emission reduction and energy independence. However, teir intermittent nature may ave negative effects on te entire grid. One of te most viable solutions is te integration of Energy Storage Systems (ESS), wic mitigates against fluctuations in generation and supply. However, tey add anoter degree of complexity to te sceduling of power flows. Tus our interest in improving algoritms for power flow optimization. To acieve bot operational reliability and financial profitability, a more efficient utilization and control of te existing transmission and distribution system infrastructures is required. All tese factors contribute to te increasing need of fast and reliable optimization metods tat can ad- Te researc leading to tese results as received funding from te European Union s Sevent Framework Programme (FP7/-5) for te ICT-based Intelligent management of Integrated RES for te smart grid optimal operation under grant agreement n dress bot security and economical issues simultaneously, supporting power system operation and control. In tis scenario, te microgrid concept is a promising approac. Usually described as a confined cluster of loads, storage devices, and small generators, tese autonomous networks can operate in island mode or in parallel wit te main grid to supply power to te loads, Lasseter and Paigi (4), Hatziargyriou et al. (7). In addition a microgrid can purcase and sell power from te public distribution grid troug te Point of Common Coupling PCC. Te optimization of te microgrid operations is extremely important in order to manage its energy resources in a cost-efficient way, Hatziargyriou et al. (7), (8). Te set of optimization problems in electric power systems engineering is known collectively as Optimal Power Flow (OPF). It is one of te most important problem regarding andling large-scale power systems in an effective and efficient manner and, it falls into te well-researced sub-fields of constrained nonlinear optimization. Te OPF concept was first introduced by Carpentier (96). He included te transmitted power problem in a simple optimal Economic Dispatc (ED). His work as been widely applied in power Copyrigt 4 IFAC 9456

2 systems analysis, Momo (989); in addition see te surveys in Cowdury and Raman (99); Huneault and Galiana (99); Momo et al. (999a); K.S.Pandya (8); Momo (), for a broader view and details. In general, OPF is a nonlinear optimization problem, wic seeks to optimize te operation of an electric power system (power generation and transmission) wile satisfying operational and pysical constraints imposed by Kircooff s laws and functional limits on te decision variables, Momo et al. (999a). Te solution tecnique for OPF problems was first proposed by Dommel and Tinney (968) based on Newton-Rapson metod. Since tat time, several matematical metods ave been employed to solve OPF, suc as linear, nonlinear, quadratic, mixed integer programming, interior-point metods and Newton-based metods, Huneault and Galiana (99); Momo (989); Momo et al. (999b). Te aim of OPF is to set te power network variables in order to meet te energy demand in te most economically manner, wile simultaneously keeping all constraints witin specific bounds, imposed on te pysical systems. Tis paper aims to formulate a model for te Alternate Current Optimal Power Flow (ACOPF) problem consisting of simple dynamics for energy storage systems. Te ESS is needed to mitigate irregular generation from renewable energy sources wile guaranteeing te network power balance. Tere are several commercial software tools available to simulate and solve power flow problem, suc as MAT- POWER Zimmerman et al. (), PSAT Milano (3), GridLabD (3) and GAMS (3). In our work, te solution of te ACOPF is modelled by using te software GAMS and InterPSS in a JAVA framework. Te InterPSS coice as been motivated by te free and open source distribution of te simulation platform. In addition, InterPSS core can be integrated into custom made software/simulators in combination wit GAMS to define and efficiently solve large scale optimization problems. Te paper is organized as follows. In Section 3 te nonlinear non-convex optimization problem is given, were microgrid components models are presented in details. Simulation results are sown in Section 4 and te effect of energy storage is examined by solving a Norwegian demo network. Finally, te conclusion and future work are presented.. NOMENCLATURE Variables C g i : Cost function of generator at bus i [$] Ci b : Cost function of storage at bus i [$] S g i : Complex power generated at bus i [MVA] P g i : Active power generated at bus i [] Q g i : Reactive power generated at bus i [MVAr] S ij : Complex power flow from bus i to bus j [MVA] P ij : Active power flow from bus i to bus j [] Q ij : Reactive power flow from bus i to bus j [MVAr] V i : Voltage magnitude at bus i [pu] Ṽ i : Voltage pasor at bus i Ĩ i : Current pasor at bus i θ i : Voltage angle at bus i [ ] b i : State of Carge (SOC) of te storage unit at bus i [] r i : Power excanged wit te storage unit at bus i [] Sets G: Set of buses wit generator D: Set of buses wit load R: Set of buses wit renewable generator B: Set of buses wit storage units N : Set of all buses wit cardinality n Parameters c i : Cost coefficients, bus i [$/] c p, c s : Purcasing/selling prices [$/] c b i : Storage cost coefficients, bus i [$/] Ỹ ij : Complex series admittance, line ij [pu] Ỹij S : Complex sunt admittance, line ij [pu] B ij : Series susceptance, line ij [pu] G ij : Series conductance, line ij [pu] Bij S: Sunt susceptance, line ij [pu] G S ij : Sunt conductance, line ij [pu] S i d: Complex power load, bus i [MVA] Pi d: Active power load, bus i [] Q d i : Reactive power load, bus i [MVAr] S i res : Complex renewable power, bus i [MVA] Pi res : Renewable active power, bus i [] Q res i : Renewable reactive power, bus i [MVAr] P g imin, P g imax : Generator active power bounds, bus i [] Q g imin, Qg imax : Generator reactive power bounds, bus i[mvar] S ij max : Rating of line ij [MVA] V i min, V i max : Min and max voltage magnitudes, bus i [pu] θ imin, θ imax : Minimum and maximum pases, bus i [ ] θ ijmax : Maximum angle difference between bus i-j [ ] B i : Maximum storage unit capacity, bus i [] b loss i : Storage energy loss, bus i [] r rated : Maximum power supplied by te storage [] T : Sampling time [] 3. SYSTEM DESCRIPTION, MODELLING, CONSTRAINTS AND PROBLEM FORMULATION Te OPF formulation models te entire network (i.e. generators, loads, storage units and transmission lines). Furtermore, te interaction wit te utility grid troug te PCC or slack bus needs to be modelled as well (ere indicated wit te index i = ). Next sections are inspired by te idea developed in Gayme and Topcu (). 3. Storage Dynamics We consider te following discrete time model of an energy storage unit (tat can represent a battery, for example). b(t) = b(t ) r(t) b loss t r rated r(t) r rated t () b(t) B t wit given initial energy level b(). We denote by b(t) te level of te energy stored at time t (divided by T ) and by r(t) te power excanged wit te storing device at time t. Te b loss term denotes a constant stored energy degradation in te sampling interval. Note tat te power excanged at time t, r(t), can eiter be negative (te storage unit is carging) or positive (te storage unit is discarging). A bus tat does not include a storage device as B i = i / B. () Te cost function for te storage units depends only on te actual capacity of storage: 9457

3 C b (t) = c b (B b(t)) t (3) were c b imposes a penalty proportional to te deviation of te stored energy level from te unit capacity, Atwa and El-Saadany (); Gayme and Topcu (). 3. Buses We consider tat eac bus may ave eiter generators or loads, or neiter, or one of tem. Furter, except for te PCC, te following pysical constraints need to be satisfied: V min V (t) V max t (4a) θ min θ(t) θ max t. (4b) Indeed te slack bus complex voltage is given as a reference and it is modelled as an uncontrollable source V min = V max = V (5a) θ min = θ max = (5b) were V is te PCC voltage magnitude given as a network parameter. 3.3 Generators A generator is modelled as a controllable complex power injection at a specific bus S g (t) = P g (t) + jq g (t) t (6) subject to te following lower and upper bounds: P g min P g (t) Pmax g t (7a) Q g min Qg (t) Q g max t. (7b) We assume te following generation linear cost function: C g (t) = cp g (t) t. (8) Furtermore, not all buses ave connected generators P g i,min = P g i,max = i / G (9a) Q g i,min = Qg i,max = i / G. (9b) 3.4 Interaction wit te utility grid Wen grid-connected, te microgrid can purcase and sell energy from/to te utility grid. We consider tis feature by modelling te PCC discontinuous linear cost function C (t) = c p max(, P g (t)) + cs min(, P g (t)) t. () Furtermore we define te following negative lower bound on its injected active power P g min () to take into account bidirectional power flows. In summary wen te microgrid sells power to te utility grid P g is less tan zero and C gives a negative contribution to te OPF objective function; vice-versa, wen te microgrid purcases power from te utility grid, te PCC beaves as a generator (P g min and C gives a positive contribution). 3.5 Loads and Renewable Energy Sources Loads and renewable energy resources are uncontrollable quantity because tey do not depend on te optimization variables. Terefore, tey are represented at eac time by fixed real and reactive power values (consumed/delivered) i Ṽ i Ĩ i Ĩ S Ỹ ij S Ỹ ij Fig.. π-model of a power transmission line. Ĩ j Ṽ j j Ỹ ij S at te bus, and tey give a negative/positive contribute to te power balance equations S d (t) = P d (t) + jq d (t) t (a) S res (t) = P res (t) + jq res (t) t. (b) Note tat demand and renewable unit profiles are timevarying and te network needs a forecast on te future loads and renewable power supply. Tere exist several matematical models tat allow us to forecast te renewable power distribution in time, e.g. solar irradiance, wind speed and biomass modelling, Atwa and El-Saadany () and Atwa et al. (). 3.6 Transmission lines We use te well known π equivalent circuit in Fig. to model te transmission lines Toro (99). Te circuit current-voltage relation is given by te Kircoff s equation, ] ] ] [Ĩi [Ỹff Ỹ = ft [Ṽi (3) Ĩ j Ỹ tf were Ĩ and Ṽ are te current and voltage pasors and Ỹ tt Ṽ j Ỹ ff = Ỹtt = Ỹij + Ỹ S ij Ỹ ft = Ỹtf = Ỹij. (4a) (4b) Te complex power, injected at node i troug te transmission line ij, is: S ij = ṼiĨ i = Ṽi(Ỹff Ṽi + ỸftṼj). (5) Terefore active and reactive power flow, on a line ij are written as non-linear functions of te i s and j s complex bus voltage P ij (t) = V i (t) (G ij + GS ij ) V i (t)v j (t)g ij cos(θ ij (t)) V i (t)v j (t)b ij sin(θ ij (t)) i, j, t (6a) Q ij (t) = V i (t) (B ij + BS ij ) V i (t)v j (t)g ij sin(θ ij (t)) +V i (t)v j (t)b ij cos(θ ij (t)) i, j, t (6b) were θ ij = θ i θ j. Furtermore, for eac transmission line we ave a maximum apparent power S ij max, wic give te following upper bound S ij (t) = P ij (t) + Q ij (t) S ij max. (7) At eac time te nodal bus injections ave to matc te injections from loads and generators to ave te 9458

4 power system balance. In te traditional ACOPF, tis is expressed by active and reactive power balance functions of bus voltages and generator injections. Tat is P g i Q g i res (t) + Pi (t) Pi d (t) j (t) + Qres i (t) Q d i (t) j P ij (t) + r i (t) = i, t Q ij (t) = i, t. (8a) (8b) Te model takes into account te power losses on te AC transmission lines, and if we consider te sum of (8a) on te entire network we obtain i= P g i + n i= P res i + r i (t) = i= Pi d + P loss (9) i= Te same appens for te reactive power (8b). 3.7 Te Optimization Problem Te ACOPF optimal operational scedule consists in taking decisions on ow muc generators and storage units must produce to cover te entire network load wile satisfying pysical bounds, minimizing te cost function on generators, storage level, and te power excanged wit te utility grid. Combining te expressions above, te OPF formulation wit storage dynamics is: min X T t= i= C g i (t) + i B C b i (t) + C (t) () +V i (t)v j (t)b ij cos(θ ij (t)) i, j, t () P ij (t) + Q ij (t) S ij max (3) P g i res (t) + Pi (t) Pi d (t) j P ij (t) + r i (t) = i, t (4) Q g i (t) + Qres i (t) Q d i (t) Q ij (t) = i, t j (5) P g min P g i (t) P max g i, t (6) Q g min Qg i (t) Qg max i, t (7) P g i,min = P g i,max = i / G (8) Q g i,min = Qg i,max = i / G (9) b i (t) = b i (t ) r i (t) b loss i i, t (3) r rated r i (t) r rated i, t (3) b i (t) B i i, t (3) B i = b loss i = i / B (33) V i min V i (t) V i max i, t (34) θ min θ ij (t) θ max i, j, t (35) V min = V max = V (36) θ min = θ max = (37) P g min (38) were X = {P g i, Qg i, V i, θ i, r i, b i } is te set of optimization variables. Next section illustrates simulation based results obtained by te solution of tis problem for a Case Study. 4. SIMULATION RESULTS Te ACOPF problem ()-(38) is a non-convex optimization problem wose objective function as discontinuous first order derivatives and we solve it troug Interior Point OPtimizer solver (IPOPT) Ye () in GAMS environment. 4. Case Study Te OPF formulation wit storage dynamics () is implemented for a case study witin te I3RES project. Te network, sown in Fig., represents te feeder supplying some residential loads in Steinkjer, Norway. Currently te network consists of: a ydro power plants wit generators, 3 loads, 49 link buses (i.e. witout generation nor load) and 84 transmission lines. In te future energy storage units and RESs may be included into te network. Te parameters and te boundary conditions associated wit te generators and te PCC bus are given in Table. Te cost of power purcased/sold from/to te main grid troug te PCC can be found in Norway (3), were te linear cost coefficient c b for te storage unit is equal to. $/. Te voltage magnitude limits for all buses are set to.95 V i.5 pu and te pase sift between te connected buses is set to. Eac transmission line as a maximum apparent power S ij max = 7. MVA. Te storage is limited by a capacity B of 4 wit s. t. P ij (t) = V i (t) (G ij + te maximal carge/discarge power rate r rated equal GS ij ) V i (t)v j (t)g ij cos(θ ij (t)) to and a storage energy loss b loss of.. Te power profile generated by a wind farm depends on V i (t)v j (t)b ij sin(θ ij (t)) i, j, t () Q ij (t) = V i (t) (B ij + many factors, e.g. speed of wind, weater, number of wind BS ij ) V i (t)v j (t)g ij sin(θ ij (t)) turbines, and it as been considered as given in Cen et al. (3). A sampling time of one our as been cosen and te simulations ave been performed over one day. Te simulation results illustrate te advantages of including an energy storage under a stressed load demand. During low demand, te energy is stored in te storage unit and ten released wen te load/demand is ig, smooting te total power injected into te grid. Te progress of te reactive power is neglected since it does not contribute to te function cost. 4. Network beaviour under stressed load Te simulation as been performed using data from te stressed network situation on te t January 3. Note tat wen te storage unit is not considered into te network (see Fig. 3), te active power injected by te two Table. Case Study: generation unit parameters. Unit S min S max c g c s c p [MVA] [MVA] [$/] [$/] [$/] Hydro.6 Hydro. PCC

5 6 Power profiles wit storage unit 5 Generator Wind farm Load Fig. 4. Case Study: active power ourly evolution wit storage. 6 Storage capacity Storage capacity profile 5 PCC Load Generator Storage + wind farm Linking bus Fig.. Case Study: Demo Steinkjer network topology Generator Wind farm Load Power profiles witout storage unit Fig. 3. Case Study: active power ourly evolution witout storage. ydro power units is equal to te active power loads minus te power injected by te RESs. Te cost function value is 39$. Fig. 4 and Fig. 5 sows te advantages of using te reservoir storage: te storage unit avoids peaks (from pm to pm) by reducing te cost value to 76$, tat is a reduction of 4.6% Fig. 5. Case Study: storage active power capacity trend. If te power demand (including losses) is low enoug, te ydro units can sell te surplus to te utility grid (see Fig. 6, from am to am); oterwise te network needs to purcase power from it (from pm to pm). Comparing Fig. 6a and Fig. 6b, it is wort noting tat te optimal control leads a power purcase reduction from te utility grid in te ours wen te generator can not provide te power required by te network. Wen a no stressed load profile is considered te advantages of te energy storage unit presence may be diminised. Te amount of time it takes to run a simulation depends on many factors, including te network s complexity and te computer s clock speed. We tested te OPF formulations and algoritm on a PC wit Intel 3. GHz i7 processor, GB memory. Te simulation time of case study is 53 s witout storage unit and 578 s including storage equations in te model. 5. CONCLUSION Tis paper presents interesting results igligting te benefits obtained by extending te traditional optimal power flow problem wit an energy storage device. Tis is 946

6 Power sold/purcase to/from utility grid (a) Witout storage unit. Power sold/purcase to/from utility grid (b) Wit storage unit. Fig. 6. Case Study: active power excanged wit te utility grid troug te PCC bus. quite relevant, particularly wen renewable energy sources are integrated into te existing distribution/transmission grid, because teir intermittent production affects negatively te energy balance in te grid. Energy storage units may be installed to mitigate tis generation irregularity. However, te traditional tools/algoritms implemented to optimally balance te power flow in te grid do not consider explicitly storage units. Te results ave been obtained by simulating a real network in a real situation of load and tey confirm te benefits of aving a storageaware OPF algoritm, wic in practice yields economical benefits. Furter work will be addressed to investigate different scenarios and optimization algoritms. REFERENCES Y.M. Atwa and E.F. El-Saadany. Optimal allocation of ESS in distribution systems wit a ig penetration of wind energy. Power Systems, IEEE Transactions on, 5 (4):85 8,. ISSN doi:.9/tp- WRS Y.M. Atwa, E. F. El-Saadany, M. M A Salama, and R. Seetapaty. Optimal renewable resources mix for distribution system energy loss minimization. Power Systems, IEEE Transactions on, 5():36 37,. ISSN doi:.9/tpwrs J. Carpentier. Contribution to te economic dispatc problem. Bull Soc. France Elect, 8:43 447, 96. N. Cen, Z. Qian, I.T. Nabney, and X. Meng. Wind power forecasts using gaussian processes and numerical weater prediction, 3. ISSN B.H. Cowdury and S. Raman. A review of recent advances in economic dispatc. Power Systems, IEEE Transactions on, 5(4):48 59, 99. ISSN doi:.9/ H.W. Dommel and W.F. Tinney. Optimal power flow solutions. Power Apparatus and Systems, IEEE Transactions on, PAS-87(): , 968. ISSN doi:.9/tpas GAMS. GAMS, 3. URL ttp:// D. Gayme and Ufuk Topcu. Optimal power flow wit distributed energy storage dynamics. In American Control Conference (ACC),, pages ,. GridLabD. Gridlab-d, 3. URL ttp:// N. Hatziargyriou, H. Asano, R. Iravani, and C. Marnay. Microgrids. Power and Energy Magazine, IEEE, 5(4):78 94, 7. ISSN doi:.9/mpae M. Huneault and F.D. Galiana. A survey of te optimal power flow literature. Power Systems, IEEE Transactions on, 6():76 77, 99. ISSN doi:.9/ S.K.Josi K.S.Pandya. A survey of optimal power flow metods. Journal of Teoretical and Applied Information Tecnology, 4(5):45 458, 8. R.H. Lasseter and P. Paigi. Microgrid: a conceptual solution. In Power Electronics Specialists Conference, 4. PESC 4. 4 IEEE 35t Annual, volume 6, pages Vol.6, 4. doi:.9/pesc Federico Milano, 3. URL www3.uclm.es/profesorado/federico.milano/ dome.tm. J.A. Momo. Optimal power flow wit multiple objective functions. In Power Symposium, 989., Proceedings of te Twenty-First Annual Nort-American, pages 5 8, 989. doi:.9/naps J.A. Momo. Electric power system applications of optimization. CRC Press,. J.A. Momo, R. Adapa, and M.E. El-Hawary. A review of selected optimal power flow literature to 993. part : Nonlinear and quadratic programming approaces. Power Systems, IEEE Transactions on, 4():96 4, 999a. ISSN doi:.9/ J.A. Momo, M.E. El-Hawary, and R. Adapa. A review of selected optimal power flow literature to 993. part : Newton, linear programming and interior point metods. Power Systems, IEEE Transactions on, 4():5, 999b. ISSN doi:.9/ Statistic Norway. Norway Energy Pricing, 3. URL ttp:// statistikker/elkraftpris/. V. Del Toro. Electric Power Systems. Englewood Cliffs, volume. Prentice-Hall, Strategic deployment document for europe electricity networks, 8. URL ttp:// Yinyu Ye. Interior Point Algoritms: Teory and Analysis. Wiley-Interscience,. R.D. Zimmerman, C.E. Murillo-Sancez, and R.J. Tomas. Matpower: Steady-state operations, planning, and analysis tools for power systems researc and education. Power Systems, IEEE Transactions on, 6 (): 9,. ISSN doi:.9/tp- WRS