Optimal Wind Power Portfolio Allocation to Decrease Wind Variability System Impacts

Transcription

1 Optimal Wind Power Portfolio Allocation to Decrease Wind Variability System Impacts Final Report Josh Novacheck 4/2/4 Abstract Wind power can be a challenging power source to integrate into the grid because of sudden variability in its output over short periods time. The fast ramping up or down of its power output can have negative consequences for grid operations, including increased costs. One method to deal with ramp rate variability of wind farms is to increase additional wind power that is subject to different wind patterns. In essence, diversifying the wind power portfolio. However, diversification of wind power can lead to tradeoffs with average power output from the cumulative wind power output. Therefore, multi-objective optimization is used determine optimal portfolios that are diverse and maximize power output. A pareto frontier is then used to better analyze the tradeoffs between maximum power output and decreased variability.

2 Introduction One of the largest challenges to integrating large quantities of wind power into the grid is the variability in its power output. The fast ramping up or down of its power output can have negative consequences for grid operations, including increased costs, inefficient operation of conventional generators, and the need for additional ancillary services [,2,3]. One method to deal with the variability of power output from individual wind farms is to develop and interconnect other wind farms that are subject to different wind patterns, essentially diversifying the wind power portfolio. If proper wind power portfolios are chosen, the cumulative power output of the portfolio will smoother relative to the output from individual wind farms. However, reduced variability through wind portfolio diversification may require the development of lower quality wind sites, resulting in a tradeoff between output variability and the average power output of the entire portfolio. The goal of this project is to enhance techniques used to optimize diversified wind power systems and to understand the parts of the system of have the biggest impact on reducing wind power variability. To optimize diversified wind, a modified version of the multi-objective optimization method called Mean-Variance Portfolio optimization (MVP) is implemented. The details of the modified MVP will be explained in a later section, but the two objectives are to minimize the ramp rate variability of the power output and to maximize the average power output of the cumulative wind power output. Ramp rate variability is defined as the change of the cumulative power output from time step to time step. Ten-minute interval wind power data from NREL s Eastern Wind Dataset [4] is used in the optimization to develop the set of optimal wind power portfolios. The ramp rate for each portfolio is calculated by taking the difference in the power output from one time step to the next. Therefore, the ramp rate in this project has units of change in power over ten minutes (MW/min). MVP has been used before to analyze wind power diversity. Hansen (25) [5], Degeilh and Singh (2) [6], Roques et al (29) [7], and Rombauts et al (2) [8] all use MVP to some extent to demonstrate impacts of wind power diversification. However, this project is original because it uses both finer temporal and geographical resolution wind power output. Additionally, this study minimizes ramp rate variability of the overall portfolio rather than the deviation from the average power output as most past studies using MVP have done. This difference is important because reducing the magnitude and frequency of large ramps (up or

3 down) in the wind power output generally minimize the negative system level impacts of wind power variability; this is not necessarily true when the deviation from the portfolio s average output is reduced. Minimizing the deviation from the average also penalizes solutions with periods of high power output. Typical average power output from wind farms is about 35-4% [4]. Therefore, the portfolio operating at close to % of its potential increases the power output variance, penalizing the portfolio. By minimizing the ramp rate variance instead, the portfolio is not penalized for operating well above its average power output, as long as the change in output was gradual to reach the high output. Finally, deviation from average power output is generally not completely normally distributed, a trait required when using MVP. The ramp rate variability is closer to being normally distributed, reducing the bias from using MVP. This project is also unique because it will treat the overall objective differently. This project compares the optimal placing of wind power sites at different wind penetration levels within a system. Under the different penetration levels the wind power portfolios will be required to deliver a fixed amount of electricity. This allows for a more direct comparison of costs associated with different wind power deployment strategies. Therefore, instead of the objective of maximizing average power output typical of other MVP work, this project minimizes the installed capacity of the wind power portfolio. Some studies have attempted to model transmission in MVP optimization. But to accurately model the impact of transmission constraints on a wind farm s ability to deliver power to load, all of the other generators that use the same transmission lines must also be modeled. This dramatically increases the model complexity. For that reason, transmission constraints are be ignored in the modeling and optimization for this project. Constraints to the system include, meeting a required amount of delivered wind energy and a maximum size constraint for each individual wind farm. The maximum size of each wind farm will be the same as the size assumed in the NREL Eastern Wind Dataset [4]. Decision variables are the installed wind power capacity at each wind farm site. The allocation of the total installed capacity will determine the overall portfolio s ramp rate variability based on the covariance in the change in power output of each site to the other sites. 2

4 Design Problem Problem Statement The design problem this project will attempt to address is the need to reduce wind power variability, while still delivering as much wind energy as possible. Diversifying the wind power portfolio has been shown to decrease variability of the cumulative power output [2,9]. Diversification, however, may lead to significant tradeoffs in the average power output if optimization is not employed. This optimization will produce a pareto frontier, ensuring that every decrease in the output variability results in as small of a decrease in the average power output as possible. As mentioned earlier, others have used MVP to investigate this design problem. Hansen (25) [5] initially proposed the idea while investigating the feasibility of methods to provide capacity credit for interconnected wind farms. Using three wind farms, Hansen (25) [5] demonstrates how using MVP can guide strategies to allow for more economical wind power development. The optimization used in this paper only minimizes deviation from average output, rather than decreased ramping variation. Degeilh and Singh (2) [6] also laid out an approach to optimize the siting of wind farms using Mean Variance Portfolio Theory. The optimization technique maximizes average power output, while minimizing the portfolio s output variability. The method used to reduce variability only minimizes deviation from the mean output, rather than minimizing ramping amplitude from one time step to the next. Using power output data from the NREL wind dataset from their interconnection study, the optimal portfolios are applied to a loss of load probability (LLOP) to attempt to find the value of a wind portfolio that is less variable. Roques et al (29) [7] applies Mean Variance Portfolio Theory to find optimal wind power portfolios in Europe using wind power data from Spain, France, Germany, Denmark, and Austria. They used two methods, one that minimized wind ramping variability and the other that minimized variance from average output during peak hours. In all cases, Roques et al (29) [7] found that the current wind power portfolio and projected portfolios could achieve significant reductions in variability at the same power output by using Mean Variance Portfolio theory to better plan wind projects. While the authors went through multiple iterations of the optimization to add additional realism to the results (including adding cross border transmission constraints and inter-country wind resource potential), the assumption that all wind power within a country 3

5 was perfectly correlated does not put any value on diversified wind resources within an individual country. In addition, their treatment of transmission constraints does not account for power from other sources also using the lines to deliver power from one country to the next. Rombauts et al (2) [8] explores in greater depth the issue of cross-border transmission-capacity constraints when using MVP to determine the efficient wind farm build out. While they tackle the problem in greater depth than Roques et al 29 [7], they acknowledge that their generalized solution does not accurately model the situation where there is no transmission constraints, calling into question their methodology for including the crossborder transmission-capacity constraints. This project approaches the design problem and optimization strategy slightly differently than the past MVP studies. The motivation, as described earlier, to take a different approach is to better decrease the ramp rate variability and to introduce more realism to the model. This project also adds an additional constraint that the cumulative wind power portfolio delivers a minimum amount of wind energy. This constraint allows for better comparison of the portfolios along the pareto frontier because each portfolio must deliver the same amount of energy. Earlier studies allowed the total delivered energy to change along the pareto frontier. This project also compares the sensitivity of the results to the minimum delivered energy constraint to determine if diversified wind provides more benefits under high wind energy penetrations. Mathematical Model Objective Functions The two objectives of this design problem are:. Minimize the installed capacity of wind power, and 2. Minimize the ramp rate variability from time step to time step. The first objective is simply the sum of the installed capacity at each site, as shown below, Min n i= x i () where x i is the installed capacity in MW at site i and n is the number of wind sites in the system being considered for wind power development. This objective essentially maximizes the average power output, because to minimize installed capacity while meeting the minimum delivered energy constraint (described later) requires the highest average power output sites be developed first. The second objective is more complex and involves minimizing the entire portfolio s ramp rate variance. The variance is a statistical measure (square of the standard deviation) gives 4

6 a sense of the spread in the ramp rates. By minimize the ramp rate variance of the interconnected wind farms, the frequency and magnitude of large ramps up or down in cumulative wind power output will decrease. The ramp rate variance of a portfolio P is given by, σ 2 n P = 2 i= x i σ 2 n i + 2 i<j x i x j σ ij (2) where σ i 2 is the variance in the ramp rate of site i, and σ ij is the ramp rate covariance between sites i and j. A derivation of the ramp rate variance can be found in Appendix A. The ramp rate variance is a quadratic equation that can be can be simplified into matrix form. Min x T Πx (3) where x is vector of all x i, and Π is the covariance matrix. The covariance matrix is a symmetric matrix where each entry is the covariance in the ramp rate from one site to another site. The matrix is shown below, σ 2 σ n Π = [ ]. (4) σ n 2 σ n Where σ n is the covariance between the ramp rate of site n and site. The variances in the ramp rates at each individual site are along diagonal of the covariance matrix. Each covariance is calculated by comparing the ramp rates of wind sites within in the study region. The Hessian of the ramp rate variance is simply 2Π. The covariance matrix is by definition positive semidefinite, and therefore the ramp rate variance is convex. Minimizing this function results in power output with smoother output and a low frequency of large ramp rate events. Constraints The first constraint is the requirement that a minimum amount wind energy be delivered from the portfolio s wind farms over one year, (8766 hrs year ) x i CF i E min [MWh/year] (5) where CF i is the capacity factor (average power output as a percentage of total installed capacity) of site i and E min is the minimum energy the portfolio must produce in MWh/year. The right hand side is multiplied by 8766 hours per year to convert between power and energy. In this project, multiple minimum energy values will be chosen to examine the importance of wind power diversification under larger penetrations of wind energy. They will take the form of a percentage of load (demand for electricity) within the geographic area to be considered ( E min = 5

7 % of load, 2% of load, 3% of load, and 4% of load). Defining the minimum energy constraint as a percentage of load makes this constraint analogous to state level Renewable Portfolio Standards policies in place in many states in the U.S. which require electric utilities met particular percentages of their retail sales (load) from renewable energy, such as wind energy. Another constraint deals with the individual size constraints of each wind farm in the portfolio, x i x i,max (6) where x i,max is the maximum installed capacity allowed at site i. The maximum installed capacity is defined by the data source of the wind farm power output, the NREL Eastern Wind Dataset [4]. Decision variables The decision variables in this system are the installed capacity at each wind farm site, x i. The wind sites will come from the NREL Eastern Wind Dataset [4], which defines wind power outputs for each site in ten-minute time intervals for three representative years. The geography chosen to investigate wind power diversity is the Upper Midwest portion of the Midcontinent Independent System Operator (MISO). MISO is the system operator for this region and makes system level decisions on how to deal with wind power variability. Therefore it makes sense to consider this entire interconnected geography for wind power diversification. The geography includes portions or all of North Dakota, South Dakota, Minnesota, Iowa, Wisconsin, Illinois, Indiana, and Michigan. Within this region there are 572 wind farm sites in the NREL database, meaning there will be 572 decision variables in the optimization problem. A map of all 572 sites is shown in Appendix B. A feasible solution exists as long as at least one wind farm has non-zero capacity and it delivers enough wind energy to meet the minimum requirement over one year. In the NREL database there is more than enough wind potential to meet even a minimum energy constraint requiring % of load be met by wind energy. Model Analysis From monotonicity analysis, it is known that for both objectives, the minimum energy constraint will be active. The objective to minimize the total installed capacity is monotonically increasing with respect to every decision variable, while the minimum energy constraint is monotonically decreasing for every decision variable. Therefore, by the first monotonicity 6

8 principle, the minimum energy constraint must be active. The problem is also well bounded for the minimum installed capacity objective function. For the objective of minimizing the portfolio s ramp rate variance, the minimum energy constraint is also active. As stated earlier, the ramp rate variance function is convex and quadratic. Therefore the stationary point will be the global minimum. By definition, the minimum possible variance is zero. In a real system this would mean the real value never deviates from the average or expected value. In the wind power portfolio system this is only possible when the ramp rates are perfectly anti-correlated with each other (increase in output at one wind farm is meet with an equal magnitude decrease in output at other wind farms), or when no wind farms are built (x = ). In real systems the wind farm portfolio will never be perfectly anti-correlated. Therefore, the unconstrained minimum of the ramp rate variance will occur when x =. Because equation 6 is also a constraint (and does not all form negative xi), the minimum energy constraint will be active as it does not allow all x i =. Design Optimization The results of the multi-objective optimization were obtained by first finding the optimal solution for each objective. LINPROG in Matlab was used to solve the minimum installed capacity objective, while QUADPROG was used to solve the minimum for the ramp rate variance. To obtain the pareto frontier between the optimal solutions, the minimum installed capacity objective is converted to a linear constraint. The allowed installed capacity is then changed in equal increments and the quadratic optimization of the ramp rate variance is solved to define the curve of the pareto frontier. Figure shows the pareto frontier results for the four different minimum wind energy requirements. Each point on the line represents a different sample point. A subset of maps of the geographic location of wind farms chosen in the optimization are shown in Appendix C. On an absolute scale, the pareto frontiers cover a larger range of ramp rate variances as the wind energy requirement increases. While not as obvious, the range covered by the installed capacity also increases. 7

9 Ramp Rate Variance (MW 2 /min 2 ) 25, 2, 5,, % 2% 3% 4% 5,, 2, 3, 4, 5, 6, 7, Installed Capacity (MW) Figure. Pareto frontiers for four distinct minimum wind energy requirements. The minimum wind energy requirement is expressed as a percentage of total regional demand. Figure 2 shows the same results, but normalized to the objective values of the wind power portfolio with the smallest installed capacity, for each minimum wind energy pareto frontier. The normalized view reveals the difference between the different minimum energy requirements. At increased wind energy, achieving the optimal ramp rate variance requires a smaller relative increase in installed capacity than at lower wind energy requirements. At 4% wind, only a 9.9% increase in the installed capacity is required to achieve optimal ramp rate variance, while an increase of 3.6% capacity is required in the % wind case. This is caused by an increase in natural, or unintentional, wind diversity as the minimum wind energy requirement increases. The natural diversity is caused by more wind farm sites needing to be constructed to fulfill the minimum energy requirement and the size constraint binding further capacity from being added at the highest quality wind power sites. 8

10 Normalized Ramp Rate Variance % 2% 3% 4% Normalized Installed Capacity Figure 2 Pareto frontiers normalized to the wind power portfolio with the smallest installed capacity The normalized pareto frontier also reveals that there is relatively less reduction in ramp rate variance that can be achieved under higher wind energy requirements. In both views of the praetor frontier, the slope at different points can be interpreted as the tradeoff between the two objectives. The largest reductions in ramp rate variance come with only a minimal increase in the installed capacity. This can be seen on the far left end of the pareto frontier. The pareto frontiers quickly flatten out, meaning to achieve the last feasible reductions in the ramp rate variance requires significant increases in the installed capacity. The constraint activity agrees with the earlier monotonicity analysis. In all cases the minimum energy constraint is active. The Lagrange multipliers for the minimum energy constraint change along the pareto frontier. They are the highest at the point of minimum installed capacity and decrease as ramp rate variance is reduced. This means the system is more sensitive to the minimum energy constraint at reduced installed capacity. The Lagrange multiplier is significantly larger at the minimum installed capacity point than any other point on the pareto frontier. This is due to the fact that any further constraining of the system by the minimum energy constraint at this point would result in an infeasibility. Therefore the numerical computation of the Lagrange multiplier becomes difficult at that point due to a discontinuity. A full table of Lagrange multipliers can be found in Appendix D. The decision variable values change significantly across the pareto frontier. Figure 3 shows how the change for each of the minimum energy requirements. Each horizontal line 9

11 represents one of the wind farm site decision variables. The decision variables are sorted with the highest capacity factor sites (highest wind power potential) on top and the lowest capacity factor sites on the bottom. Moving from left to right on the x-axis represents moving along the pareto frontier from minimum installed capacity to minimum ramp rate variance. The darker the line, the closer the site is to its maximum capacity. These figures show there is a gradual spreading out of the installed capacity across decision variables as ramp rate variance is reduced. The darkest region of the plot in the upper left-hand corner demonstrates the importance of the building at the high capacity factor sites when minimizing the total installed capacity is the primary objective. Wind Farm Site % Wind Pareto Frontier Fraction.5 of Xi,max Wind Farm Site % Wind Pareto Frontier Fraction.5 of Xi,max % Wind 4% Wind Wind Farm Site Fraction.5 of Xi,max.4.3 Wind Farm Site Fraction.5 of Xi,max Pareto Frontier Pareto Frontier.2. Figure 3 Change in the decision variable values as a fraction of their upper bound along the pareto frontier. The figures also show there are sites that are consistently near capacity or at capacity across the pareto frontier. There are two types or sites that are favored along the pareto frontier. The first are the sites with high capacity factors. These sites contribute to minimizing the installed capacity. The other type of sites are sites with lower capacity factors but decrease the

12 ramp rate variance because their ramp rate covariance relative to the rest of the portfolio is low. These figures can help identify sites that are of particular importance to reducing the ramp rate variance of an entire portfolio. In a real system, the sites that are at full capacity for a large portion of the pareto frontier should be investigated to determine whether the upper bound capacity at those sites could be relaxed. % Wind.5 2% Wind Wind Farm Site % of Total Portfolio Installed Capacity.5 Wind Farm Site % of Total Portfolio Installed Capacity Pareto Frontier Pareto Frontier 3% Wind.5 4% Wind Wind Farm Site % of Total Portfolio Installed Capacity.5 Wind Farm Site % of Total Portfolio Installed Capacity Pareto Frontier Pareto Frontier Figure 4 Change in the decision variable values along the pareto frontier as a percentage of each variables contribution to the total. Figure 4 shows a similar result, but with the color of the line based on the percentage of the entire portfolio s installed capacity that is located at each site. This plot helps identify sites that also help decrease the ramp rate variance of the entire portfolio, but have relatively high upper bounds on their site capacity. Sites that are at maximum capacity along the pareto frontier and sites that make up large portions of the overall portfolio together can be assumed to be the most important decision variables.

13 Discussion Impact of Reduced Variance As stated earlier, the primary motivation to minimize the ramp rate variance of a cumulative wind power portfolio is to reduce the frequency and magnitude of large ramping events, in either direction. These events are difficult for the system to respond to and have negative economic and environmental impacts. Unfortunately, the ramp rate variance results obtained from the optimization are difficult to interpret in terms of how they impacted the large ramping events. Figure 5 shows one way to assess how well diversified wind can decrease the frequency and magnitude of large ramping events. For each minimum wind energy constraint, Figure 5 shows the cumulative frequency of the absolute ramp rates (ramp up and down combined) of four wind power portfolios along the pareto frontiers. The four portfolios shown on the graph each represent interesting points along the pareto frontier. Both extremes of the pareto frontier are shown, along with two interior portfolios. One interior portfolio is the portfolio one step from the from the minimum installed capacity portfolio, this represents the largest decrease in ramp rate variance between points on the pareto frontier. The other interior portfolio represents the portfolio whose variance is % higher than the variance of the minimum ramp rate variance portfolio. As the ramp rate variance of a portfolio decreases, the cumulative frequency in Figure 5 shifts closer to the y-axis. Note the scale on the x-axis changes for each plot. 2

14 % Wind 2% Wind.9.9 Cumulative Frequency Min Installed Capacity Min Variance Portfolio 2 Portfolio 2 Cumulative Frequency Min Installed Capacity Portfolio 2 Portfolio Min Installed Capacity Wind Power Absolute Ramp Rate (MW/min) Wind Power Absolute Ramp Rate (MW/min) 3% Wind 4% Wind.9.9 Cumulative Frequency Min Installed Capacity Portfolio 2 Portfolio Min Variance Cumulative Frequency Min Installed Capacity Portfolio 2 Portfolio 9 Min Variance Wind Power Absolute Ramp Rate (MW/min) Wind Power Absolute Ramp Rate (MW/min) Figure 5 Cumulative frequency distribution of the absolute ramp rate for different points along the pareto frontier. Figure 5 shows that minimizing the ramp rate variance decreases the frequency and magnitude of large ramping events relative to the minimum installed capacity portfolio. Figure 5 also shows reducing the variance to within % of the minimum ramp rate variance has nearly an identical impact on the reduction of large ramping events. Table shows the ramping magnitudes of the 9 th percentile ramping event for each of the four portfolios in Figure 5 and minimum wind energy requirements. The 9 th percentile point is equivalent to stating the wind power portfolio will ramp up or down at a magnitude equal to or greater than the value listed in the table % of the time. Similar to the cumulative frequency plots, the table demonstrates a decrease in the magnitude and frequency of large ramping events, and the largest reductions occur in the step along the pareto frontier closest to the minimum installed capacity extreme. 3

15 9th Percentile Absolute Ramp Rate (MW/Min) % Wind 2% Wind 3% Wind 4% Wind Min Installed Wind Capacity Portfolio 2 (One step from Min Installed) % Higher than Min Variance Min Ramp Rate Variance Table 9th percentile absolute ramp rates for different points along the pareto frontier. Practical Considerations When designing a wind power portfolio for a region, there are some practical constraints and penalties that were not taken into account in the mathematical model. The two main practical considerations are the desire to minimize the number of wind farms needed, and to avoid building too small of a wind farm. While most of the cost of building wind farms is dependent on the installed capacity, one of the objectives of the model, there also exist fixed costs. These costs can include the cost of obtaining construction permits, meeting with the local community to discuss the project, increasing transmission lines due to the geographic spread of the wind farms, and other upfront costs. The impact of these fixed costs are greatly minimized by reducing the number of wind farms needed to meet the objective and to avoid building sites with only a small amount of wind power capacity. Both of the practical constraints could be modeled and used in the optimization. However, they would either unrealistically constrain the system or add discreteness to the design. This project intentionally ignored these practical considerations to simplify the model and analysis. The practical considerations can be taken into account after the initial optimization. Using the results from Figures 3 and 4, the most important sites for reducing ramp rate variance can be identified. After identifying those sites, the optimization can be run again on the smaller decision variable set. The results of this optimization would result in fewer total wind farms being built and fewer wind farms with small installed capacities, with only small tradeoffs in the ramp rate variances. 4

16 Choosing a Wind Power Portfolio Ideally, this analysis will result in a recommendation on which wind power portfolio should be chosen for each of the different minimum wind energy requirements. This project has narrowed down the choices to portfolios along the pareto frontier, but without further weighting or valuation of the costs and benefits of diverse wind, any recommendation would be subjective. The costs of wind power are for the most part quantifiable. There are capital costs required to build wind farms that are primarily a function of the installed wind power capacity (although, there are other practical considerations that should be accounted for as discussed earlier). However, the benefits of reduced ramp rate variance that are achieved through diversifying the wind power resource are more difficult to quantify because their benefits are seen on a systems level. The next step of this project will be to input different wind power portfolios into a power systems model. Using the results on how the system changes under varying degrees of wind power variability, the economic and environmental benefits of decreased variability can be quantified. A cost/benefit analysis can then be done to choose wind power portfolios that achieve the optimal tradeoff between installed wind power capacity and the ramp rate variance of the wind power portfolio. Acknowledgements This project will support my master s thesis research into the value of wind diversification to the power system. This research is being completed with the help and guidance of my research advisor Dr. Jeremiah Johnson, a Research Scientist in the Center for Sustainable Systems in the School of Natural Resources and Environment. 5

17 References. Katzenstein, W. and J. Apt, The cost of wind power variability. Energy Policy, 22. 5: p Milligan, M. and B. Kirby, An Analysis of Sub-Hourly Ramping Impacts of Wind Energy Balancing Area Size. WindPower 28 Proceedings, 28: p Kirby, B. and M. Milligan, Cost-causation-based tariffs for wind ancillary service impacts. WINDPOWER 26 JUN 4, National Renewable Energy Labortory, Transmission Grid Integration: Eastern Wind Dataset Hansen, L., Can Wind be "Firm" Resource? A North Carolina Case Study. Duke Law & Policy Forum, 25: p Degeilh, Y. and C. Singh, A quantitative approach to wind farm diversification and reliability. International Journal of Electrical Power & Energy Systems, 2. 33(2): p Roques, F., C. Hiroux, and M. Saguan, Optimal wind power deployment in Europe-A portfolio approach. Energy Policy, 2. 38(7): p Rombauts, Y., E. Delarue, and W. D'Haeseleers, Optimal portfolio-theory-based allocation of wind power: Taking into account cross-border transmission-capacity constraints. Renewable Energy, 2. 36(9): p Katzenstein, W., E. Fertig, and J. Apt, The variability of interconnected wind plants. Energy Policy, 2. 38(8): p

18 Appendix A σ 2 P = Var(P) = Var( i= x i r i ) σ ij = Cov(r i, r j ) n Where, x i is the installed capacity at site i and r i is the set of the ramp rates at site i. By definition, taking the variance of two weighted sets summed together gives, Var(x i r i + x j r j ) = x i 2 Var(r i ) + x j 2 Var(r j ) + x i x j Cov(r i, r j ) Combining the above equations and summing over the entire set of i =,,n gives σ 2 n P = 2 i= x i σ 2 n i + 2 i<j x i x j σ ij 7

19 Appendix B NREL Wind Sites within the Study Area. Number in circles indicates there are that many wind farms in close proximity on the map. 8

20 Appendix C 4% Wind/Min Installed Capacity 4% Wind/Min Ramp Rate Variance 9

21 Appendix D Lagrange Multiplier Table % Wind 2% Wind 3% Wind 4% Wind Portfolio Energy Energy Energy Energy Min Installed Capacity Min Ramp Rate Variance

22 Appendix E Matlab code to model and solve optimization system %***read_file.m*** %***Created Jan 2 24 by Josh Novacheck %***Completed April 2 24 %Reads in NREL Eastern Wind Dataset csv files %and calculates the ramp rate data array for each file %then determines the covariance of the ramp rates clear; % Clear the workspace close all; % Close all windows filename = dir('nrel Wind Data\*.csv'); nfiles = size(filename,); n = size(csvread(['nrel Wind Data\',filename().name],3,3),)-; ramp = zeros(n,nfiles); power = zeros(n+,nfiles); site_info = csvread('site capacity\site capacity.csv',,5); site_cap = site_info(:,); site_cf = site_info(:,2); for j = :nfiles power(:,j) = (csvread(['nrel Wind Data\',filename(j).name],3,3)/site_cap(j)); ramp(:,j) = power(2:n+,j)-power(:n,j); end covariance = cov(ramp); correlation = corrcoef(ramp); %solution = min_var(site_cf,site_cap,covariance); %***min_var.m*** %***Created Feb 2 24 by Josh Novacheck %Solves the pareto fronteir for the required energy using bounding method function solution = min_var(site_cf,site_cap,covariance,nfiles) % initial point nsites = nfiles; x = ones(nfiles,); Emin = ; %determined by % of load [MWh] %intial constraint set up (only minimum energy requirement) A = [-8766*site_cf']; %[MWh] b = [-Emin]; %[MWh] %upper and lower bounds on decision variables lb = zeros(nfiles,); ub = site_cap; % NREL site capacity is set as upper bound %Hessian and gradient calc for min variabiliy objective H = 2*covariance; f_var = zeros(nsites,); 2

23 options = optimoptions('quadprog','algorithm','interior-pointconvex','display','iter'); %portofolio of minimum variability (quadratic programing) [x_var,fval_var,exitflag,output,lambda] = quadprog(h,f_var,a,b,[],[],lb,ub,x,options); %installed capacity of minimum variability fcap_max = sum(x_var); %Gradient calc for min installed capacity objective f_cap = ones(nsites,); %portfolio of minimum installed capcaity (linear programing) opt2 = optimoptions('linprog','algorithm','interiorpoint','display','iter'); [x_cap,fval_cap,exitflag,output,lambda] = linprog(f_cap,a,b,[],[],lb,ub,[],opt2); %variability of minimum installed capacity fvar_max = variability(x_cap,covariance); %pareto frontier calc n = 2; %number of steps along pareto fronteir pcap = linspace(fval_cap,fcap_max,n); %capacity points along pareto %pareto_var = zeros(,n); %create array to hold variance of each portfolio coeff = ones(,nsites); %create array of ones to use in capacity constraint %x_star = zeros(nsites,n); %create array to hold x values of each portfolio solution = struct('x',[],'pvar',[],'pcap',[],'lambda',[]); %create structure to save results for i=:n %set constraints to account for restricted capacity A = [-8766*site_cf';coeff]; %[MWh; MW/number of turbines] b = [-Emin;pcap(i)]; %solve for minimum variance with capacity constraint [x_star,pvar,exitflag,output,lambda] = quadprog(h,f_var,a,b,[],[],lb,ub,x,options); % save pareto frontier point information solution.x = [solution.x, x_star]; solution.pvar = [solution.pvar, pvar]; solution.pcap = [solution.pcap, pcap(i)]; solution.lambda = [solution.lambda, lambda]; end 22