Optimal Power Flow Formulations in Modern Distribution Grids

Transcription

1 P L Power Systems Laboratory Optimal Power Flow Formulations in Modern Distribution Grids Student Etta Shyti Supervisors Stavros Karagiannopoulos Dmitry Shchetinin Examiner Prof. Dr. Gabriela Hug Project Number 1820 Report Date July 25, 2018 Power Systems Laboratory ETH Zurich

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3 Abstract In this semester thesis two different optimal power flow formulations for active distribution networks are compared in terms of optimality and computational speed. More specifically, on the one hand we examine the full AC OPF formulation, which is implemented in MATLAB and solved using IPOPT, and on the other the existing Backward Forward Sweep approach for the OPF [1], which was already implemented in MATLAB using YALMIP by one of the supervisors and solved using GUROBI. Some control measures available to the distribution system operator from the distributed energy resources present in the network are also implemented, such as active power curtailment, reactive power control, controllable loads and battery energy storage systems. Two case studies are considered in order to first compare the results of the different formulations, and in the second case the effects of the various control measures. The results of the simulations for these two case studies are presented and evaluated. Finally, conclusions are drawn and future work and further steps are proposed. iii

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5 Acknowledgements I would like to sincerely thank my supervisors, Stavros and Dmitry, for their guidance and support, and for helping me gain this valuable experience regarding optimal power flow formulations at the power systems lab. v

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7 Contents List of Acronyms ix 1 Introduction Motivation Contribution Optimal Power Flow Standard AC OPF BFS approach Implementation Case studies Input data Results and Evaluation 11 5 Conclusion 21 Bibliography 23 vii

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9 List of Acronyms DSO DER LV DN OPF PF BFS BIBC BCBV DG APC RPC CL BESS Distribution System Operator Distributed Energy Resources Low voltage Distribution Network Optimal Power Flow Power Flow Backward Forward Sweep Bus-Injection to Branch-Current Branch-Current to Bus-Voltage Distributed Generators Active Power Curtailment Reactive Power Control Controllable Load Battery Energy Storage System ix

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11 Chapter 1 Introduction 1.1 Motivation Modern power systems have been developed from a fundamental change in structure of the traditional power systems. The main driver is the increasing number of distributed energy resources (DERs), which are being installed in the medium voltage (MV) and low voltage (LV) networks. Distribution System Operators (DSOs) are facing new challenges due to the uncertainty of renewable resources, which tend to bring fluctuations of power flows, overvoltages and undervoltages in the grid. Figure 1.1: Modern power systems [2] 1

12 2 CHAPTER 1. INTRODUCTION Therefore, DSOs need to use all of the available measures at hand to ensure a safe and reliable operation of the distribution networks (DNs), while supplying the total load demand, minimizing costs and fulfilling all operating constraints. When DERs are controllable and dispatchable, as is assumed in this semester thesis, they can provide flexibility for the DSOs and new opportunities to allow them to use active control measures in real time operation. Hence, DSOs need to make significant changes both in the planning stage and in the operational stage and exploit the operational flexibility of the DERs. The main focus during this project is specifically the optimal power flow (OPF) calculation, which has many applications, such as state estimation, economic dispatch, demand response, contingency analysis, etc. In DNs it is very important for finding the optimal operating points for the available controllable DERs in the system. There has been continuous research in the field of computational algorithms for OPF and there have been recent advancements and higher attention due to the increasing number of electric vehicles, energy storage systems, inverter based PVs, flexible loads etc. 1.2 Contribution The goal of the thesis is to compare known OPF formulations and solution techniques for DNs in terms of robustness, computational speed and applicability. The versions of the AC OPF for DNs that are investigated are the standard AC OPF and the AC OPF based on the Backward Forward Sweep (BFS) approach. First, the full AC OPF problem is formulated and implemented. Next, the already implemented BFS OPF [1] is studied and fully understood. Finally, the different OPF calculations are simulated using a benchmark distribution test system and are compared based on optimality, operating point and speed. The remainder of this thesis is structured as follows: Chapter 2 introduces the OPF in general terms and elaborates more on the standard AC OPF formulation and the BFS approach. Chapter 3 provides some details regarding the implementation and describes the considered case studies. Chapter 4 presents and evaluates the results of the case studies and Chapter 5 finally draws the main conclusions from the results of this thesis, followed by an outlook for possible future steps and further developments.

13 Chapter 2 Optimal Power Flow Optimal power flow (OPF) is a crucial computation to allow DSOs to find the optimal dispatch of the control measures available from the DERs in active DNs. The total load demand needs to be supplied while minimizing costs and power losses in the network. The DSOs must also ensure that all the network operational constraints are not violated, such as the limits on branch currents, bus voltage magnitudes etc. In general the goal of an optimization problem is to minimize an objective function f(x), which, for power systems, can be the cost of generation, power losses etc., such that it satisfies a set of equality constraints g(x) and inequality constraints h(x), e.g. power balance equations, voltage, current and generation limits etc. The optimization problem variables x are separated into state variables, which consist in voltage magnitudes, angles etc., and control variables, which can include amounts of active power curtailment and reactive power generation from the inverter based distributed generators, amount of shifting for controllable loads etc. [3] minimize x f(x) subject to g(x) = 0 h(x) 0 Since the power balance equations are nonlinear and non-convex, this leads to complex OPF calculations and raises tractability concerns. Due to the nature and characteristics of DNs certain approximations and linearizations applied to the power balance equations for transmission networks are not valid, which would simplify the problem. Additionally, solving such an optimization problem for active DNs can become quite computationally intense, considering the inter-temporal constraints which need to be fulfilled for the different active control measures. [1] OPF FORMULATIONS There has been a lot of research in order to find new ways to formulate and solve the multi-period AC OPF. Some of the techniques that have been proposed consist of linear approximations of the AC Power Flow (PF), heuristics, convex relaxations etc., but most of them are mainly developed for transmission grids and are not suitable for distribution networks. Another approach focuses on the possibility to solve the power flow problem using iterative methods based on the BFS PF. These exploit the radial DN topology and have been used in multi-period three-phase systems with unbalanced loading, meshed configurations and AC feasible solutions considering uncertainties. 3

14 4 CHAPTER 2. OPTIMAL POWER FLOW 2.1 Standard AC OPF In this section a formulation of the full AC OPF for active distribution grids with DERs is presented, which is the one considered in this project.[1] The objective function, shown in Eq. (2.1), aims at minimizing the cost of active power curtailment P curt,j,t and reactive power control Q ctrl,j,t for bus j and time t over all buses N b, and the losses in the grid P sub,t for time t, considering the entire time horizon N hor. Active power curtailment (APC) is defined as P curt,j,t = Pg,j,t max P f g,j,t, which is the difference between the maximum active power available from the distributed generators (DGs) and the actual power produced. Reactive power control (RPC) is defined as Q ctrl,j,t = Q f g,j,t, which is the reactive power injected into the node. The total losses in the system are calculated by summing the power demanded from or fed into the substation P sub,t = in node 2 at time t over the entire time horizon N hor. The corresponding P f g,2,t price factors are c P = 3 CHF kw h, c Q = CHF kw h and c L = 0.3 CHF N hor min u t=1 N b j=1 kw h. ( c T P P curt,j,t + c T Q Q 2 ) ctrl,j,t + c T L P sub,t t (2.1) In order to keep a power balance in the grid the exact AC power flow equations, shown in Eqs. (2.2) and (2.3), bring non-linearity to the optimization problem. The active power injections, P f inj,j,t, defined in Eq. (2.4), are expressed in terms of the active power generated by the DGs, P f g,j,t, the amount of active power demand from the load after control and load shifting, P f lflex,j,t, and the charging and discharging power of the energy storage system, PB,j,t ch and P dis B,j,t. Regarding the reactive power injections, Qf inj,j,t, shown in Eq. (2.5), they are defined in terms of the reactive power injected from the DGs, Q f g,j,t, and the reactive power demand after load shifting, P f lflex,j,t tan (θ load). P f inj,j,t = V bus,k,t Q f inj,j,t = V bus,k,t N b m=1 N b m=1 V bus,m,t (G km cos θ km,t + B km sin θ km,t ) (2.2) V bus,m,t (G km sin θ km,t + B km cos θ km,t ) (2.3) P f inj,j,t = P f g,j,t P f ch lflex,j,t (PB,j,t PB,j,t) dis (2.4) Q f inj,j,t = Qf g,j,t P f lflex,j,t tan (θ load) (2.5) There are limits for the magnitudes of the bus voltages V bus,j,t, shown in Eq. (2.6), the fixed reference slack bus voltage magnitude V slack and angle θ 1, shown in Eqs. (2.7) and (2.8), and thermal loading limits I br,i,t, shown in Eq. (2.9), which define the maximum branch currents I i,max allowed on the lines. V min V bus,j,t V max (2.6) V slack = 1 (2.7) θ 1 = 0 (2.8) I br,i,t I i,max (2.9)

15 2.1. STANDARD AC OPF 5 Next, there are the inverter based DGs limits for active power and reactive power generation, shown in Eqs. (2.10) and (2.11). It can be noticed that the considered control method sets limits on the reactive power generation of the PVs depending on the active power generation and maximum allowed power factor cos (φ max ). P min g,j,t P f g,j,t P max g,j,t (2.10) tan (φ max ) P f g,j,t Qf g,j,t tan (φ max) P f g,j,t (2.11) The considered controllable load (CL) allows for a specific amount of load to be shifted as shown in Eq. (2.12), under the condition that the total daily load demand is satisfied as expressed in Eq. (2.13). P lflex,j,t = P l,j,t + P l,j,t (2.12) N hor t=1 P l,j,t = 0 (2.13) The battery energy storage system (BESS) has limits on the available energy capacity, Ej,t bat, defined in terms of the installed capacity, Ebat inst,j, and the minimum and maximum allowed states of charge, SoCmin bat and SoCbat max. The initial energy level available, Ej,1 bat, is set to half the installed capacity of the battery as shown in Eq. (2.16). Regarding the charging power and discharging power, they are both positive as defined in Eqs. (2.17) and (2.18) and the constrains in Eqs. (2.19) and (2.20) make sure the battery does not charge and discharge at the same time, using the small value of ˆη = SoCmin bat Einst,j bat Ej,t bat SoCmax bat Einst,j bat (2.14) E bat j,t Ej,1 bat = 0.5 E bat ( = Ej,t 1 bat + η bat PB,j,t ch P dis inst,j (2.15) B,j,t η bat ) t (2.16) P ch B,j,t 0 (2.17) P dis B,j,t 0 (2.18) B,j,t (P l,j,t Pg,j,t max ) ˆη (2.19) B,j,t (P l,j,t Pg,j,t max ) ˆη (2.20) P ch P dis The power flow equations are nonlinear, which leads to higher complexity for solving the OPF problem. In the following section, another approach based on the BFS PF is considered, which avoids the use of the non-convex AC PF.

16 6 CHAPTER 2. OPTIMAL POWER FLOW 2.2 BFS approach As already mentioned, the BFS PF exploits the radial structure of the DNs. In order to provide a short overview of this iterative approach the grid shown in Fig. 2.1 is considered. Two matrices are required for performing the PF: the Bus-Injection to Branch-Current (BIBC), shown in Fig. 2.2a, which defines the topology of the network, and the Branch- Current to Bus-Voltage (BCBV), shown in Fig. 2.2b, a matrix composed of the impedances of the lines. After forming the two matrices there are two sweeping steps. During the first step, namely the backward sweep step, the branch currents are calculated from the injected currents, and in the second one, namely the forward sweep, the branch currents are used to update the voltages. The sweeping steps are performed repeatedly in a loop until the differences between the voltage magnitudes of the current iteration with the previous one are below a certain predetermined threshold. This algorithm is shown in Fig. 2.3 and yields a full PF calculation. [1] [3] Figure 2.1: Example of radial distribution grid [3] (a) Bus-Injection to Branch- Current (BIBC) [3] (b) Branch-Current to Bus-Voltage (BCBV) [3] Figure 2.2: Matrices required for the BFS PF

17 2.2. BFS APPROACH 7 Figure 2.3: BFS algorithm [1] BFS-OPF IMPLEMENTATION First, the bus voltage magnitudes and angles and the iteration counter are initialized. Next the BFS based OPF approach incorporates the BFS PF by performing one single sweeping iteration described above instead of the equations for the non-convex exact AC power flow. Solving the BFS OPF problem provides the optimal set-points and the operating point, which is updated in the next step by performing the full BFS PF until convergence. This allows for the operating point to be projected into the feasible domain of the exact AC power flow equations. The iterative approach uses the updated operating point as a starting point for the next iteration of the multi-period BFS OPF and repeats until the mismatches between the voltage magnitudes resulting from the OPF and those resulting from the full PF are below a specified threshold. Since this algorithm updates the resulting voltages and currents from the solution of the OPF problem for each iteration by running the BFS PF, it expedites and facilitates in general the convergence of the multi-period BFS OPF. [1]

18 8 CHAPTER 2. OPTIMAL POWER FLOW Figure 2.4: Multi-period BFS OPF [1]

19 Chapter 3 Implementation The goal and contribution from this semester thesis lies in the realization of a standard AC OPF centralized scheme for operating a LV DN model in order to compare its results with the already implemented BFS OPF approach. The OPF was modelled as an optimization problem as presented in Section 2.1 and all the parameters were set the same as in case of the existing BFS OPF. The work was carried out in MATLAB R2016b and the time horizon considered was 24 hours. The active control measures from DERs that were implemented are APC, RPC, which are both provided by the inverter based DGs, CL and BESS. AC OPF The AC OPF was modeled in MATLAB and then it was passed directly to the solvers. The chosen solvers were IPOPT [4] and the TOMLAB nonlinear solver SNOPT [5]. A wrapper function was provided by one of the supervisors, Dmitry, to make it easier to switch between them, which prepared the data according to the specified solver and performed preprocessing steps. A MATLAB structure called model was defined to describe the optimization problem and contained all the necessary parameters for the solvers. The variables considered: V, theta, δp P V, Q P V, δp L, P ch, P dis, E b, P sub. First, the lower and upper bounds for the variables were set, then the objective function was formulated, followed by the linear constraints and the nonlinear constraints. Some MATPOWER functions were used in order to help with the nonlinear constraints, specifically for the right hand side of the active and reactive power balance equations shown in Eqs. (2.2) and (2.3), and for calculating the current magnitudes in Eq. (2.9) to check if they were within the set limits. Additionally, the gradient and hessian of the objective function and the Jacobian of the nonlinear constraints were provided manually. They were required for IPOPT, while for the case of SNOPT they contributed significantly to making the solution process faster. Finally, the Hessian of the Lagrangian was also provided manually by Dmitry, speeding up the process further. BFS OPF The BFS OPF implementation as shown in Fig. 2.4 was provided by Stavros and was modeled using the YALMIP MATLAB toolbox for optimization modeling. The solver that was chosen in this case was GUROBI [6]. 9

20 10 CHAPTER 3. IMPLEMENTATION 3.1 Case studies After implementing the AC OPF with all the available control measures, two case studies were considered: Case study A: Using only APC as a control measure, the results obtained from running the AC OPF and the BFS OPF were compared in terms of objective function, speed and operating point. Case study B: To identify their individual contribution, the AC OPF was simulated using each of the different control measures provided by the DERs - APC, RPC CL and BESS. The comparison was based again on the objective function value, speed and operating point. 3.2 Input data The network used in the case studies presented above is based on the benchmark radial LV grid from CIGRE and a single phase was considered in a balanced system. The data for the PV injection profiles is real data obtained from PV stations in Zurich and the load data consists of typical profiles provided by CIGRE. There are four PVs installed in nodes 12, 16, 18 and 19. Additionally, in node 16 there is a BESS unit present, which has an energy capacity of 26 kwh, as well as a CL with 5 kw available to the DSO for shifting. A summer day simulation is considered in order to include violations of network constraints. The acceptable limits for voltage magnitudes are 0.9 p.u. as lower limit and 1.04 as upper limit, while for branch currents a maximum value of 1 p.u. is allowed. Figure 3.1: CIGRE LV benchmark grid [7]

21 Chapter 4 Results and Evaluation Case study A After running the simulations for both the AC OPF and BFS OPF, the results obtained show that the BFS yields a higher objective function value of around 50 CHF as shown in the graph in Fig. 4.1, while both solvers IPOPT and SNOPT achieve the same result for the ACOPF. Solving the BFS OPF seems to be only slightly slower than solving the AC OPF using IPOPT, while SNOPT achieves a solution very fast as can be noticed in Fig However, the solving times vary with every simulation. OBJECTIVE FUNCTION (KCHF) AC OPF (IPOPT) AC OPF (SNOPT) BFS OPF SOLVER TIME (s) 0.28 AC OPF (IPOPT) AC OPF (SNOPT) BFS OPF Figure 4.1: Resulting objective function and needed time from the solver 11

22 12 CHAPTER 4. RESULTS AND EVALUATION In terms of operating point the AC OPF and the BFS OPF achieve results that are quite close, but with small differences. As it can noticed from the graphs in Fig. 4.2 the voltage magnitudes look almost the same with small deviations amounting to a root mean square error (RMSE) of e-04. Figure 4.2: Results for voltage magnitudes

23 13 The results for APC also look quite similar as seen in Fig. 4.3, but the values are slightly higher for the BFS. These small differences are mainly responsible for the higher value of the objective function for hte BFS, due to the high price coefficient (c P = 3). More specifically the overall cost of PV curtailment for the AC OPF amounts to kchf, while for the BFS OPF is kchf. Figure 4.3: Results for PV curtailment

24 14 CHAPTER 4. RESULTS AND EVALUATION Likewise, as it can be observed in Fig. 4.4 the line loadings are very similar, with the AC OPF not exceeding 81.2% and BFS OPF 80.3%. However, the overall costs of losses in the grid are again slightly higher for the BFS with a value of kchf, while for the AC OPF they amount to kchf. Figure 4.4: Results for line loadings In summary, the BFS OPF achieves a slightly higher objective function value, due to the small differences in the operating point, such as the amount of APC. One possible reason could be that the BFS OPF might be stuck at a local optimum and results in a different operating point. When solving the ACOPT with IPOPT, the BFS is almost as fast, however it is quite slow when compared to using SNOPT. IPOPT also requires the gradient and hessian of the objective function and the Jacobian of the nonlinear constraints, which need to be provided manually, which is not the case for SNOPT.

25 15 Case study B In order to compare the influence of the different active control measures available from the DERs, the AC OPF was simulated adding one measure at a time. The results show that the objective function decreases continuously as shown in Fig. 4.5, achieving the lowest value when all four measures are used. The cost coefficient for RPC is three orders of magnitude lower than the one for APC, and CL and BESS have no operational cost and do not contribute to the objective function. Thus, being able to use the cheaper options and lower the amount of APC explains the decrease in objective function value. On the other hand, the solver time increases each time a new control measure is introduced, since having an additional element brings more complexity to the problem and requires more time to be solved. OBJECTIVE FUNCTION 2.46 (kchf) APC + RPC + Flex + Bat SOLVER TIME (s) APC + RPC + Flex + Bat Figure 4.5: Resulting objective function and needed time from the solver

26 16 CHAPTER 4. RESULTS AND EVALUATION In terms of operating point, there are lower voltage magnitudes when using all measures APC, RPC, CL and BESS, compared to using only APC, as it can be noticed in Fig Figure 4.6: Results for voltage magnitudes

27 17 There is considerably lower PV curtailment as well, as shown in Fig When using all four measures APC, RPC, CL and BESS there is curtailment only in node 16 and up to 36%, compared to having only APC available, where node 16 is curtailed up to 45% and there is curtailment in node 19 as well. Figure 4.7: Results for PV curtailment

28 18 CHAPTER 4. RESULTS AND EVALUATION Regarding RPC, it is almost always used at the maximum when needed since it is an effective and cheap option with a low cost coefficient c Q = The amount of RPC remains almost constant while adding other measures, as can be noticed in Fig Figure 4.8: Results for Q P V generation

29 19 From Fig. 4.9 it can be observed that using all of the available control measures leads to higher loading of lines in general. Especially lines 2-3 and 3-4 are loaded at full capacity for a few hours, while on the other hand when only APC is used, lines are loaded at a maximum of 81.2%. Figure 4.9: Results for line loadings Summarizing, using APC, RPC, CL and BESS leads to the lowest cost for the DSO, but requires the longest computation time. When all four control measures are used voltage magnitudes are lower, while also allowing for higher PV generation and less PV curtailment compared to using only APC. RPC is used at its maximum and kept almost constant even after adding the CL and BESS. However, line loadings are generally higher with two lines being loaded at maximum for a certain amount of time.

30 20 CHAPTER 4. RESULTS AND EVALUATION

31 Chapter 5 Conclusion Conlcusion During this semester project the full AC OPF was implemented in MATLAB and compared to the existing BFS-approach OPF in terms of optimal solution and computational speed. The formulation of the optimization problem was the same for both methods, except for the power balance equations, which were approximated with a single sweep of the BFS PF in the case of the BFS OPF. All the parameters were kept identical in order to allow for a comparison. The controllable DERs that were modelled were inverter-based distributed generators, which offer APC and RPC, a Cl and a BESS. Two case studies were considered in order to compare the results between the AC OPF and the BFS OPF and to distinguish between the role of each of the available control measures from the DERs. From the results of the simulations it was concluded that for: Case study A: The BFS approach is quite accurate compared with the standard AC OPF It leads to a slightly higher objective function value and more PV curtailment However it achieves almost the same operating point BFS could be considered as a good approximation for the case under study. Case study B: Using APC, RPC, flexible load and batteries leads to the lowest cost, but highest computation time However, some of the lines in the system are loaded at full capacity for a few hours APC is used less when the other control measures are available RPC is used whenever needed at maximum since it is cheap and effective If DERs are controllable and dispatchable they are very beneficial to the DSOs Further steps After having set up the AC OPF optimization problem, some further steps and alternatives for the implementation could be: 21

32 22 CHAPTER 5. CONCLUSION Considering a larger network as input data Having a larger share of DERs (more PVs, Cls and BESS) Adding On-Load-Tap-Changers (OLTC) Applying convex relaxations to AC OPF In this work, we only consider balanced, single-phase system operation, but the framework can be extended to three-phase unbalanced networks.

33 Bibliography [1] S. Karagiannopoulos, L. Roald, P. Aristidou, and G. Hug, Operational planning of active distribution grids under uncertainty, IREP, Espinho, Portugal, [2] svg. [3] P. G. Hug, Power System Analysis: Lecture Notes. Power Systems Laboratory (EEH), ETH Zurich, [4] COIN-OR, Introduction to ipopt: A tutorial for downloading, installing, and using ipopt, Ipopt/doc/documentation.pdf?format=raw. [5] K. Holmström, A. O. Göran, and M. M. Edvall, User s guide for tomlab/snopt, [6] G. optimization, Gurobi optimizer reference manual, com/documentation/8.0/refman/index.html. [7] K. Strunz, E. Abbasi, C. Abbey, C. Andrieu, F. Gao, T. Gaunt, A. Gole, N. Hatziargyriou, and R. Iravani, Benchmark Systems for Network Integration of Renewable and Distributed Energy Resources. CIGRE, Task Force C6.04,